aa r X i v : . [ m a t h . K T ] M a r ON THE BEILINSON FIBER SQUARE
BENJAMIN ANTIEAU, AKHIL MATHEW, MATTHEW MORROW, AND THOMAS NIKOLAUS
Abstract.
Using topological cyclic homology, we give a refinement of Beilinson’s p -adic Goodwillieisomorphism between relative continuous K-theory and cyclic homology. As a result, we generalizeresults of Bloch–Esnault–Kerz and Beilinson on the p -adic deformations of K-theory classes. Fur-thermore, we prove structural results for the Bhatt–Morrow–Scholze filtration on TC and identifythe graded pieces with the syntomic cohomology of Fontaine–Messing. Contents
1. Introduction 21.1. Fiber squares 21.2. p -adic deformations of K-theory classes 41.3. The motivic filtration on TC 5Notation 7Acknowledgments 72. The Beilinson fiber square 72.1. Background 72.2. Pullback squares 92.3. Fiber sequences up to quasi-isogeny 123. Quasi-isogenies of cyclotomic spectra 143.1. Preliminaries 153.2. Quasi-isogenies on THH 163.3. Quasi-isogenies and the Beilinson fiber sequence 174. Application to p -adic deformations 194.1. The Bloch–Esnault–Kerz theorem 204.2. Generalization of Beilinson’s obstruction; proof of Theorem F 235. The motivic filtration on TC 245.1. Review of [BMS19] 255.2. Relative THH and its filtration 285.3. Preliminary connectivity bounds 305.4. Frobenius nilpotence on b ∆ R /p , and proof of Theorem 5.1 (2) 315.5. Proofs of the connectivity bounds (Theorem 5.1(1)) for the Z p ( i ) 345.6. Rigidity, and proof of Theorem 5.2 356. The comparison with syntomic cohomology 396.1. Syntomic cohomology 396.2. The Z p ( i ) in equal characteristic p Date : March 30, 2020.2010
Mathematics Subject Classification.
Key words and phrases. p -adic K -theory, cyclic homology, deformation of algebraic cycles, motivic cohomology. Z p ( i ) FM and Z p ( i ) 457. Examples 507.1. K-theory of p -adic fields 507.2. Perfectoid rings 537.3. Application to p -adic nearby cycles 54Appendix A. Twisted Tate diagonals 56Appendix B. Categorical lemmas 62References 641. Introduction
Fiber squares.
For any ring R , one has its connective algebraic K-theory K( R ) and its negativecyclic homology HC − ( R ); they are related via the Goodwillie–Jones trace map tr GJ : K( R ) → HC − ( R ),often interpreted and referred to as a Chern character, [Lod98, Ch. 8]. Moreover, when R is a Q -algebra, the map tr GJ induces an isomorphism on relative theories for nilimmersions, via the followingtheorem of Goodwillie. Theorem 1.1 (Goodwillie [Goo86]) . If I ⊆ R is a nilpotent ideal in an associative Q -algebra R , thenthe commutative square K( R ) / / trGJ (cid:15) (cid:15) K( R/I ) trGJ (cid:15) (cid:15) HC − ( R ) / / HC − ( R/I ) is cartesian, i.e., the Goodwillie–Jones trace map induces an equivalence tr GJ : K( R, I ) ≃ HC − ( R, I ) on relative theories. Here for a pair (
R, I ) with I ⊆ R an ideal, we write K( R, I ) for the fiber of K( R ) → K( R/I ), andsimilarly for other functors such as HC − and so on.In order to extend Goodwillie’s theorem to more general rings, one uses topological cyclic homologyTC( R ), introduced in [BHM93] in the p -complete case and in [DGM13] integrally, and the cyclotomictrace tr : K( R ) → TC( R ), which refines the Goodwillie–Jones trace map. Theorem 1.2 (Dundas–Goodwillie–McCarthy [DGM13]) . If I ⊆ R is a nilpotent ideal in an asso-ciative ring R , then the commutative square K( R ) / / tr (cid:15) (cid:15) K( R/I ) tr (cid:15) (cid:15) TC( R ) / / TC(
R/I ) is cartesian, i.e., the cyclotomic trace induces an equivalence tr : K( R, I ) ≃ TC(
R, I ) on relativetheories. Topological cyclic homology is thus the primary tool in calculations of relative K-theory (see forexample [Mad94, HM97b, HM03]), but it is a significantly more complicated invariant than cyclichomology. However, recently Beilinson [Bei14] gave a version of Goodwillie’s original result in a p -adic setting, when the ideal in question is ( p ) and R is assumed to be complete along ( p ). The first goalof this paper is to construct a variant of the Chern character and prove a strengthening of Beilinson’stheorem. N THE BEILINSON FIBER SQUARE 3
Throughout this paper, we fix a prime number p . We use the convention that the modifier “ Z p ”refers to p -adic completion of an object, and “ Q p ” to the rationalization of the p -completion; forexample K( R ; Z p ) denotes the p -complete K-theory of R , and K( R ; Q p ) denotes the rationalization ofK( R ; Z p ). Similarly, the modifier “ Q ” refers to rationalization. We denote by HP (resp. HC) periodiccyclic (resp. cyclic) homology. Theorem A.
For an associative ring R , there is a natural p -adic Chern character map tr crys : K( R/p ; Q p ) → HP( R ; Q p ) (1) which fits into a natural commutative square K( R ; Q p ) / / trGJ (cid:15) (cid:15) K( R/p ; Q p ) trcrys (cid:15) (cid:15) HC − ( R ; Q p ) / / HP( R ; Q p ) . (2) If R is commutative and henselian along ( p ) then this square is cartesian, thereby giving an equivalence K( R, ( p ); Q p ) ≃ ΣHC( R ; Q p ) . In [Bei14], Beilinson constructs a natural equivalence K cts ( R, ( p ); Q p ) ≃ ΣHC( R ; Q p ) under theassumption that R is p -complete with bounded p -power torsion, R/p has finite stable range, and therelative K-theory term K( R, ( p )) is replaced by the “continuous” relative K-theory K cts ( R, ( p )) =lim ←− K( R/p n , ( p )); this replacement does not affect the conclusion if R is commutative thanks to[CMM18, Theorem 5.23]. Beilinson’s arguments rely on some p -adic Lie theory.In this paper, we will construct the map (1) using the description of topological cyclic homologyfrom Nikolaus–Scholze [NS18], as a consequence of B¨okstedt’s calculation of THH( F p ). Together withthe rigidity results of Clausen–Mathew–Morrow [CMM18], we explain a short, homotopy-theoreticproof of Theorem A. In fact, Theorem A and all the corollaries listed below hold for any (possiblynon-commutative) ring R if we replace K-theory by TC (see Theorem 2.12); the henselian conditionis only needed to translate between K-theory and TC.Next, we observe some consequences of and complements to Theorem A. In [Bei14], slightly morethan an equivalence of rational spectra K( R, ( p ); Q p ) ≃ ΣHC( R ; Q p ) is proved: there is a natural zig-zag of “quasi-isogenies” of spectra before inverting p . By definition, a quasi-isogeny is a map which isan equivalence up to uniformly bounded denominators in any finite range of degrees. We also obtainthe same conclusion in our setting and can keep track of the denominators at least in some range. Corollary B.
Let R be a commutative ring which is henselian along ( p ) . Then there is a naturalzig-zag of quasi-isogenies between K( R, ( p ); Z p ) and ΣHC( R, ( p ); Z p ) . If R is moreover p -torsion free,then there are isomorphisms π i K( R, ( p ); Z p ) ≃ π i ΣHC( R, ( p ); Z p ) for i ≤ p − . A similar result for an arbitrary nilpotent ideal, albeit in a smaller range of degrees (depending onthe exponent of nilpotence of the ideal), is proved in [Bru01]. The argument we use here seems to bespecial to the ideal ( p ).As explained above, one could formulate Theorem A entirely in the language of topological cyclichomology, completely avoiding the mention of K-theory. At some point in the proof, however, wetranslate back into K-theory and use a homology argument. Therefore, we also offer an alternativepurely cyclotomic proof of this step. This relies on a study of quasi-isogenies in the homotopy theory Since the methods are different, we do not know if our identification on fiber terms is the same as Beilinson’s. The stable range of a ring R was defined in [Bas64] (see also [Bas68, V.3]) and is sometimes, as in [Bei14], calledthe stable rank. BENJAMIN ANTIEAU, AKHIL MATHEW, MATTHEW MORROW, AND THOMAS NIKOLAUS of cyclotomic spectra, based on the t -structure introduced by Antieau–Nikolaus [AN]. The key stepis an extension of a theorem of Geisser–Hesselholt [GH11] and Land–Tamme [LT19]. Theorem C.
Let f : A → A ′ be a map of connective associative ring spectra. Suppose that (1) f is a quasi-isogeny of spectra and (2) the map π ( f ) : π ( A ) → π ( A ′ ) is surjective with nilpotent kernel.Then THH( A ; Z p ) → THH( A ′ ; Z p ) is a quasi-isogeny in cyclotomic spectra and in particular theinduced map TC( A ; Z p ) → TC( A ′ ; Z p ) is a quasi-isogeny. p -adic deformations of K -theory classes. In our first main application of Theorem A, we gen-eralize work of Bloch–Esnault–Kerz [BEK14b] and Beilinson [Bei14] on the formal p -adic deformationof rational K-theory classes. Let us first recall the motivation for their work.Fix a complete discretely valued field K of mixed characteristic (0 , p ) with ring of integers O K and perfect residue field k as well as a proper smooth scheme X → Spec( O K ) with special fiber X k and generic fiber X K . Given X , we can consider the algebraic de Rham cohomology H ∗ dR ( X K /K )of the generic fiber, together with its Hodge filtration Fil ≥ ⋆ H ∗ dR ( X K /K ); these are finite-dimensional K -vector spaces, and arise as the cohomology groups of objects Fil ≥ ⋆ R Γ dR ( X K /K ) in the derivedcategory of K .As usual, we have a Chern characterch : K ( X ; Q ) ։ K ( X K ; Q ) → H evendR ( X K /K ) . (3)A foundational motivating question is to determine the image of this map: in other words, to determinewhich cohomology classes come from algebraic cycles on X K (or equivalently on X ). A conjectureof Fontaine–Messing, a p -adic analog of the variational Hodge conjecture of Grothendieck [Gro66,Footnote 13], predicts that this question can essentially be reduced from mixed to equal characteristic.To formulate the conjecture, we consider also the (absolute) crystalline cohomology H ∗ crys ( X k ) of thespecial fiber, a family of finitely generated W ( k )-modules. By the de Rham-to-crystalline comparison[BO83], we have an isomorphism H ∗ crys ( X k ) ⊗ W ( k ) K ∼ = H ∗ dR ( X K /K ) . Finally, we have the crystallineChern character map [Gro85], ch crys : K ( X k ) → M i ≥ H i crys ( X k ; Q p ) , (4)leading to a commutative diagramK ( X ; Q ) ch (cid:15) (cid:15) / / K ( X k ; Q ) ch crys (cid:15) (cid:15) H evendR ( X K /K ) ≃ / / / / H evencrys ( X k ) ⊗ W ( k ) K. (5) Conjecture 1.3 ( p -adic variational Hodge conjecture) . Let α ∈ H evendR ( X K /K ). Then α belongs tothe image of the Chern character from K ( X ; Q ) if and only if(1) the image of α under the de Rham-to-crystalline isomorphism in H evencrys ( X k ) ⊗ W ( k ) K belongsto the image of the crystalline Chern character from K ( X k ; Q ) and(2) the class α belongs to L i Fil ≥ i H i dR ( X K /K ) ⊆ H evendR ( X K /K ).For further details and arithmetic applications of the p -adic variational Hodge conjecture, we referto [Eme97].Motivated by Conjecture 1.3, Bloch–Esnault–Kerz [BEK14b] considered the following p -adic defor-mation question, which starts with a K -class on the special fiber (rather than a cohomology class)and asks when it lifts infinitesimally. N THE BEILINSON FIBER SQUARE 5
Question 1.4 (The p -adic deformation problem) . Given the data as above, define the “continuous”K-theory K cts ( X ) def = lim ←− K( X/π n ) , where π is a uniformizer of O K . Given a class x ∈ K ( X k ; Q ), when does it belong to the image ofthe reduction map from the continuous K-theory K cts0 ( X ; Q ) = π (K cts ( X )) Q ?Since the map K( X ) → K cts ( X ) is generally not an equivalence, the p -adic deformation problemdoes not imply Conjecture 1.3. However, the p -adic deformation problem is a (pro)-infinitesimal one,so it can be studied using methods of topological cyclic homology. Using the Beilinson fiber square,we answer the p -adic deformation problem as follows; in [BEK14b], this result is proved for a smoothprojective scheme of dimension d < p + 6 when K is unramified. Theorem D.
Let X be a proper smooth scheme over O K . A class x ∈ K ( X k ; Q ) lifts to K cts0 ( X ; Q ) if and only if ch crys ( x ) ∈ L i ≥ H i crys ( X k ; Q p ) is carried by the de Rham-to-crystalline comparisonisomorphism to a class in L i ≥ Fil ≥ i H i dR ( X K /K ) ⊆ L i ≥ H i dR ( X K /K ) . Our main observation is that Theorem A together with Hochschild–Kostant–Rosenberg comparisonsbetween cyclic and de Rham cohomology yield a fiber squareK cts ( X ; Q ) (cid:15) (cid:15) / / K( X k ; Q ) (cid:15) (cid:15) Q i ∈ Z Fil ≥ i R Γ dR ( X K /K )[2 i ] / / Q i ∈ Z R Γ dR ( X K /K )[2 i ] . (6)Moreover, on K , one checks that the vertical map on the right-hand side induces the crystallineChern character (4), at least up to scalars, implying the result. For this argument, it is crucial thatone has the fiber square (6), rather than a fiber sequence alone.One can also generalize the above questions to higher K-theory. In [Bei14], the Beilinson fibersequence is used to prove that if x ∈ K j ( X k ; Q ), then there exists a natural obstruction class in L i ≥ H i − j dR ( X K ) / Fil ≥ i H i − j dR ( X K ) which vanishes if and only if x lifts to the continuous K-theoryK cts i ( X ; Q ); however, [Bei14] does not identify the class with the crystalline Chern character for i = 0.Here we also extend this result to an arbitrary quasi-compact and quasi-separated (qcqs) scheme withbounded p -power torsion, using p -adic derived de Rham cohomology [Bha12] and results of [Ant19]. Theorem E.
Let X be a qcqs scheme with bounded p -power torsion. For each n we write X n for X × Spec Z Z /p n , and put K cts ( X ) def = lim ←− K( X n ) . Given a class x ∈ K j ( X ; Q ) , there is a natural class c ( x ) ∈ M i ≥ H i − j ( L Ω X /L Ω ≥ iX ) Q p , where L Ω X is the p -adic derived de Rham cohomology of X with the derived Hodge filtration L Ω ≥ ⋆X .The class x lifts to K cts i ( X ; Q ) if and only if c ( x ) = 0 . The motivic filtration on TC . In [BMS19], Bhatt–Morrow–Scholze discovered a fundamentaladditional structure on the p -adic topological cyclic homology TC( − ; Z p ) of p -adic commutative rings:a “motivic filtration” Fil ≥ ⋆ TC( − ; Z p ) on TC( − ; Z p ) with associated graded terms denoted Z p ( i )[2 i ].The objects Z p ( i ) thus obtained are related to integral p -adic Hodge theory and can be defined (inde-pendently of topological cyclic homology) as a type of filtered Frobenius invariants on the prismaticcohomology [BS19]. They are known explicitly in some cases: in characteristic p > BENJAMIN ANTIEAU, AKHIL MATHEW, MATTHEW MORROW, AND THOMAS NIKOLAUS over O C (for C a complete algebraically closed nonarchimedean field), they can be identified withtruncated p -adic nearby cycles.Recall also that for p -adic (commutative) rings, TC( − ; Z p ) is p -adic ´etale K-theory in nonnegativedegrees [GH99, CMM18, CM19]. Therefore, it is expected (but not known in mixed characteristic)that the filtration Fil ≥ ⋆ TC( − ; Z p ) is the ´etale sheafified motivic filtration on algebraic K-theory, andthat the Z p ( i ) are p -adic ´etale motivic cohomology, at least where all of these objects are defined.One also has constructions of Schneider and Sato [Sat07] of “ p -adic ´etale Tate twists,” which satisfya type of arithmetic duality. In general, one expects that the Z p ( i ) should be related to importantfoundational questions in arithmetic geometry and K-theory. An advantage of the construction ofFil ≥ ⋆ TC( − ; Z p ) and the Z p ( i ) as in [BMS19] is that it works in a much more general setting (forthe quasisyntomic rings; see Section 5.1 below for a review) than existing approaches to motiviccohomology. Moreover, its definition is extremely direct: it is simply a sheafified Postnikov tower(albeit for a “large” topology).Using Theorem A, we will give a description of the Z p ( i ) for i ≤ p − Q p ( i ) for all i in terms of syntomic cohomology as considered by Fontaine–Messing [FM87] and Kato [Kat87]. Inparticular, this construction gives a description of the Z p ( i ) (with the above restrictions) that reliesonly on derived de Rham theory, rather than prismatic theory. Our result is an analog of a result ofGeisser [Gei04] for ´etale motivic cohomology for smooth schemes over Dedekind rings.To formulate the result, we write L Ω R for the p -adic derived de Rham cohomology for a commu-tative ring R equipped with its derived Hodge filtration L Ω ≥ ⋆R [Bha12]. The object L Ω R carries acrystalline Frobenius ϕ : L Ω R → L Ω R . For i < p , one has a “divided” Frobenius ϕ/p i : L Ω ≥ iR → L Ω R .Using the techniques of [BMS19] (in particular, quasisyntomic sheafification) applied to the Beilinsonfiber square, we deduce our next theorem. Theorem F.
Let R be a quasisyntomic ring. (1) For each i ≥ , there is an identification Q p ( i )( R ) ≃ fib( ϕ − p i : L Ω ≥ iR → L Ω R ) Q p . (2) For i ≤ p − , there is an identification Z p ( i )( R ) ≃ fib( ϕ/p i − id : L Ω ≥ iR → L Ω R ) . We explicitly analyze Theorem F in three cases in which one has alternate descriptions of the Z p ( i ): rings of integers in p -adic fields, perfectoid rings, and formally smooth O C -algebras where C is an algebraically closed, complete nonarchimedean field of mixed characteristic. The first caserecovers classical calculations of the rational p -adic K-theory of p -adic fields; the second case recoversthe fundamental exact sequence in p -adic Hodge theory; the last case recovers results of Colmez–Nizio l [CN17] on p -adic vanishing cycles, albeit only in the formally smooth case.Finally, Theorem F provides a complete computation of low-degree or rationalized TC in terms ofsyntomic cohomology. This computation relies on the following connectivity estimate about the Z p ( i )and about the filtration on TC( − ; Z p ). The estimate for algebras over a perfectoid ring is stated in[BMS19, Constr. 7.4]; the argument for all quasisyntomic rings relies on the use of relative topologicalHochschild homology and the spectral sequence of Krause–Nikolaus [KN19]. Theorem G. If R is a quasisyntomic ring, then Z p ( i )( R ) ∈ D ≤ i +1 ( Z p ) . If R is w -strictly local (e.g.,strictly henselian local), then Z p ( i )( R ) ∈ D ≤ i ( Z p ) . Consequently, Fil ≥ i TC( R ; Z p ) is concentrated inhomological degrees ≥ i − (and pro-´etale locally ≥ i ). Corollary H. If R is any commutative ring, then there is a natural equivalence TC( R ; Q p ) ≃ M i ≥ fib( ϕ − p i : L Ω ≥ iR → L Ω R ) Q p . N THE BEILINSON FIBER SQUARE 7
Notation.
Throughout, we write Sp for the ∞ -category of spectra. Given a ring R , we write D ( R )for the derived ∞ -category of R .We will use homological indexing conventions indicated with a subscript when referring to spectraand cohomological indexing conventions indicated with a superscript when referring to objects of thederived category. For instance, given n , we write Sp ≥ n (resp. Sp ≤ n ) for spectra with homotopy groupsconcentrated in degrees ≥ n (resp. ≤ n ); we write D ( R ) ≤ n (resp. D ( R ) ≥ n ) for objects of D ( R ) withcohomology groups concentrated in degrees ≤ n (resp. ≥ n ).We write HH( R ) for the Hochschild homology of R , always relative to Z and always computed in aderived sense (also known as Shukla homology), and we let THH( R ) denote the topological Hochschildhomology of R . We write HC − ( R ) = HH( R ) hS for negative cyclic homology and HP( R ) = HH( R ) tS for periodic cyclic homology. For a scheme X , we let L Ω X denote its p -completed derived de Rhamcohomology (relative to Z ) and L Ω ≥ iX for the i th stage of the (derived) Hodge filtration. We denotethe Hodge-completed variants by c L Ω X and c L Ω ≥ iX , respectively. Acknowledgments.
We are very grateful to Peter Scholze for suggesting this question to us and forsharing many insights. We would also like to thank Johannes Ansch¨utz, Alexander Beilinson, BhargavBhatt, H´el`ene Esnault, Lars Hesselholt, Moritz Kerz, Arthur-C´esar Le Bras, Wies lawa Nizio l, and SamRaskin for helpful conversations. The third author would like to thank Uwe Jannsen, Moritz Kerz,and Guido Kings for organising and inviting him to the 2014 Kleinwalsertal workshop on Beilinson’spaper [Bei14].The first two authors would like to thank the University of Bonn and the University of M¨unsterfor their hospitality. This material is based upon work supported by the National Science Foundationunder Grant No. DMS-1440140 while the first three authors were in residence at the MathematicalSciences Research Institute in Berkeley, California, during the Spring 2019 semester. The first authorwas supported by NSF Grant DMS-1552766. This work was done while the second author wasa Clay Research Fellow. The fourth author was funded by the Deutsche Forschungsgemeinschaft(DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044 390685587,Mathematics M¨unster: Dynamics–Geometry–Structure.2.
The Beilinson fiber square
Background.
We review some background on the theory of cyclotomic spectra and topologicalcyclic homology as in [NS18], of which we will use the p -typical variant. This theory uses the ∞ -category Fun( BS , Sp) of spectra equipped with S -actions. Given a spectrum X equipped with an S -action, we can form the homotopy S -orbits X hS , the homotopy S -fixed points X hS , and the S -Tate construction X tS . These are related by a natural fiber sequence Σ X hS → X hS → X tS ,which we will use constantly and without further comment. See for example [NS18, Cor. I.4.3]. Definition 2.1 (Nikolaus–Scholze [NS18]) . We let CycSp denote the symmetric monoidal, stable ∞ -category of cyclotomic spectra . An object of CycSp consists of a spectrum X equipped with an S -action and an S -equivariant cyclotomic Frobenius map ϕ p : X → X tC p .Given X ∈ CycSp, we writeTC − ( X ) = X hS and TP( X ) = X tS . Our conventions are slightly different from those of [NS18], which requires a Frobenius map for each prime number q and not only the fixed prime p . What we call a cyclotomic spectrum is called a p -typical cyclotomic spectrum by[AN], and what we write as CycSp is written as CycSp p in [AN]. For p -complete objects (in which we will primarily beinterested), there is no difference between a cyclotomic spectrum in the sense of [NS18] and the definition used here. BENJAMIN ANTIEAU, AKHIL MATHEW, MATTHEW MORROW, AND THOMAS NIKOLAUS
We will define only p -complete TC for X ∈ CycSp and will assume that X is bounded below. Wehave two maps can : TC − ( X ; Z p ) → TP( X ; Z p ) and ϕ : TC − ( X ; Z p ) → TP( X ; Z p ) . By definition, canis the canonical map from S -invariants to the Tate construction, and ϕ is induced from the Frobenius ϕ p . The p -complete topological cyclic homology TC( X ; Z p ), for X ∈ CycSp bounded below, can becomputed as the fiber of the difference of the two maps, i.e.,TC( X ; Z p ) = fib (cid:0) can − ϕ : TC − ( X ; Z p ) → TP( X ; Z p ) (cid:1) . (7) Remark 2.2.
We will use throughout the basic fact that if X ∈ CycSp has underlying n -connectivespectrum, then TC( X ; Z p ) is ( n − Example 2.3. (1) Given a ring R , we can form the topological Hochschild homology THH( R )as a cyclotomic spectrum.(2) Given a spectrum Y , we let Y triv be the cyclotomic spectrum where we view Y as a spectrumwith trivial S -action and with cyclotomic Frobenius given by the natural map Y → Y hC p → Y tC p .(3) For a spectrum X with S -action we get a cyclotomic spectrum by letting ϕ p : X → X tC p bezero (as an S -equivariant map). Remark 2.4.
Let Y be a bounded below spectrum of finite type, meaning that each π i Y is a finitelygenerated abelian group. If X ∈ CycSp is p -complete and bounded below, then there is a naturalequivalence TC( X ⊗ S Y triv ; Z p ) ≃ TC( X ; Z p ) ⊗ S Y. (8)This is a consequence of the fact that the functor TC( − ; Z p ) commutes with geometric realizations ofconnective cyclotomic spectra, since it is exact and carries n -connective objects into ( n − Y is only assumed to be bounded below, as long as the right side of (8) is p -adically completed, by [CMM18, Th. 2.7].As a simple exercise with the above definitions, we prove the following result for use below. Thishas been used in other references as well, e.g., [HN19, Sec. 1.4]. Proposition 2.5. If X ∈ CycSp is bounded below and
TP( X ; Z p ) = 0 , then we have natural equiva-lences TC( X ; Z p ) ≃ (Σ X hS ) ∧ p ≃ TC − ( X ; Z p ) .Proof. In fact, the formula (7) shows that TC( X ; Z p ) = TC − ( X ; Z p ). Now TP( X ; Z p ) is the cofiberof the norm map (Σ X hS ) ∧ p → TC − ( X ; Z p ). Since we have assumed TP( X ; Z p ) = 0, we have(Σ X hS ) ∧ p ≃ TC − ( X ; Z p ). Combining the two identifications, we conclude. (cid:3) Next, we apply this to a specific crucial example.
Construction 2.6 (The cyclotomic spectrum Z hC p ) . Recall first that the cyclotomic trace (or adirect construction) gives a map Z triv → THH( F p ) (9)in CycSp, and that as objects of Fun( BS , Sp) we have THH( F p ) ≃ τ ≥ ( Z tC p ) by [NS18, Sec. IV-4];here we obtain the S -action on Z tC p via the sequence C p → S z z p −−−−→ S . This is a refinement of(and deduced from) B¨okstedt’s calculation of THH( F p ). Consequently, there is a cofiber sequence inCycSp, Z hC p → Z triv → THH( F p ) , (10)where Z hC p is a cyclotomic spectrum with underlying spectrum with S -action Z hC p ∈ Fun( BS , Sp).Note that ( Z hC p ) tC p ≃ Z triv → THH( F p ) induces an equivalence on TP( − ; Z p ), i.e., Z tS p ≃ THH( F p ) tS = TP( F p ). We N THE BEILINSON FIBER SQUARE 9 obtain by Proposition 2.5 that TC( Z hC p ; Z p ) ≃ Σ( Z p ) hS . Our next observation is that this remainstrue after tensoring with any bounded below cyclotomic spectrum. Lemma 2.7. If X ∈ Fun( BS , Sp) , then ( X ⊗ S Z hC p ) tS is p -adically zero (here we use the diagonal S -action).Proof. The spectrum ( X ⊗ S Z hC p ) tS is a module over ( Z hC p ) tS . Since ( Z hC p ) tS vanishes p -adicallyby the Tate fixed point lemma [NS18, Lem. I.2.2 and Lem. II.4.2], the lemma follows. Alternatively,one can easily verify that ( F p ) hC p ∈ Fun( BS , Sp), is induced from the trivial subgroup, which forcesthe p -adic Tate vanishing. (cid:3) Combining Proposition 2.5 and Lemma 2.7, we conclude that if X is any bounded below cyclotomicspectrum, then there are equivalencesTC( X ⊗ S Z hC p ; Z p ) ≃ Σ(( X ⊗ S Z hC p ) hS ) ∧ p ≃ TC − ( X ⊗ S Z hC p ; Z p ) . (11)2.2. Pullback squares.
Next, we establish some pullback squares involving cyclotomic spectra andgive a proof of Theorem A.
Proposition 2.8.
Let X ∈ CycSp be a bounded below cyclotomic spectrum. Then the commutativesquare
TC( X ⊗ S Z triv ; Z p ) (cid:15) (cid:15) / / TC( X ⊗ S THH( F p ); Z p ) (cid:15) (cid:15) TC − ( X ⊗ S Z triv ; Z p ) / / TC − ( X ⊗ S THH( F p ); Z p ) (12) is cartesian, where the horizontal maps arise from the map Z triv → THH( F p ) in CycSp of Construc-tion 2.6 and the vertical maps are the canonical maps
TC( − ; Z p ) → TC − ( − ; Z p ) arising from thedefinition of TC( − ; Z p ) .Moreover, there is a natural fiber sequence (Σ( X ⊗ S Z hC p ) hS ) ∧ p → TC( X ⊗ S Z triv ; Z p ) → TC( X ⊗ S THH( F p ); Z p ) . (13) Proof.
Since TC( Z ; Z p ) for a bounded below cyclotomic spectrum Z is an equalizer of two mapsTC − ( Z ; Z p ) ⇒ TP( Z ; Z p ), the statement that (12) is cartesian follows from the fact that X ⊗ S Z triv → X ⊗ S THH( F p ) induces an equivalence on TP( − ; Z p ), via Lemma 2.7. Moreover, the fiber sequence(13) then follows from (12) via taking fibers, and using Lemma 2.7 again to replace homotopy fixedpoints by homotopy orbits. Alternatively, to prove that (12) is cartesian, one observes that the fibersof the horizontal arrows are TC( X ⊗ S Z hC p ; Z p ) and TC − ( X ⊗ S Z hC p ; Z p ) and these are naturallyequivalent as in (11). (cid:3) Corollary 2.9.
For every connective ring spectrum R we have a natural fiber sequence of p -completespectra Σ (cid:0) THH( R ; Z p ) ⊗ S Z hC p (cid:1) hS → TC( R ; Z p ) ⊗ S Z → TC( R ⊗ S F p ; Z p ) Proof.
We apply Proposition 2.8 to X = THH( R ; Z p ). We have that THH( R ) ⊗ S THH( F p ) ≃ THH( R ⊗ S F p ) which gives the identification of the third term. For the identification of the termin the middle we observe that TC( X ⊗ S Z triv ; Z p ) ≃ TC( X ; Z p ) ⊗ S Z by (8). Note finally that forbounded below spectra, tensoring with Z preserves p -completeness as Z is finite type. (cid:3) Next, we study what happens in (12) after rationalization.
Corollary 2.10.
Let X ∈ CycSp be a bounded below, p -complete cyclotomic spectrum. Then thereexists a natural map TC( X ⊗ S THH( F p ); Z p ) → ( X ⊗ S Z triv ) tS which fits into a natural commutativesquare TC( X ⊗ S Z triv ; Z p ) (cid:15) (cid:15) / / TC( X ⊗ S THH( F p ); Z p ) (cid:15) (cid:15) ( X ⊗ S Z triv ) hS / / ( X ⊗ S Z triv ) tS . (14) Moreover, this square becomes cartesian after inverting p .Proof. We can vertically extend the cartesian square (12) via the canonical maps ( − ) hS → ( − ) tS . Inthis case, as we saw earlier, the map ( X ⊗ S Z triv ) tS → ( X ⊗ S THH( F p )) tS is an equivalence. Using thisidentification, we obtain the commutative square (14). The fact that (14) is cartesian after inverting p follows from the facts that (12) is cartesian and that ( X ⊗ S THH( F p )) hS → ( X ⊗ S THH( F p )) tS ≃ ( X ⊗ S Z triv ) tS becomes an equivalence after inverting p . (cid:3) Remark 2.11 (Effective bounds for the denominators in Corollary 2.10) . For future reference, itwill be helpful to give a more effective version of Corollary 2.10. Consider the total cofiber (cofiberof horizontal cofibers) of the square (14). This is given by Σ ( X ⊗ S THH( F p )) hS because (12) ishomotopy cartesian. If X is connective, then it follows that the τ ≤ i of the total cofiber is annihilatedby p i , since τ ≤ i − THH( F p ) is S -equivariantly annihilated by p i .Consequently, we can deduce the following fiber square, which is the basic TC-theoretic result fromwhich the Beilinson fiber square is a consequence. Theorem 2.12.
Let R be a ring (or more generally, a connective associative H Z -algebra spectrum).Then there is a natural commutative square of spectra TC( R ; Z p ) (cid:15) (cid:15) / / TC( R ⊗ S F p ) (cid:15) (cid:15) HC − ( R ; Z p ) / / HP( R ; Z p ) , (15) which becomes cartesian after inverting p . Aside from the right vertical arrow, all the maps are thecanonical ones.Proof. Via (14) for X = THH( R ; Z p ), we obtain a natural commutative diagramTC( R ; Z p ) (cid:15) (cid:15) TC(THH( R ) ⊗ S Z triv ; Z p ) (cid:15) (cid:15) / / TC( R ⊗ S F p ) (cid:15) (cid:15) (THH( R ; Z p ) ⊗ S Z triv ) hS / / (cid:15) (cid:15) (THH( R ; Z p ) ⊗ S Z ) tS (cid:15) (cid:15) HC − ( R ; Z p ) / / HP( R ; Z p ) , N THE BEILINSON FIBER SQUARE 11 where we use the natural cyclotomic map THH( R ; Z p ) → THH( R ; Z p ) ⊗ S Z triv and the natural S -equivariant map THH( R ; Z p ) ⊗ S Z → HH( R ; Z p ). The upper square is cartesian after inverting p byCorollary 2.10. The map TC( R ; Z p ) → TC(THH( R ; Z p ) ⊗ S Z triv ) ≃ TC( R ; Z p ) ⊗ S Z is an equivalenceafter inverting p . The induced map on the bottom horizontal fibers is Σ(THH( R ; Z p ) ⊗ S Z ) hS → ΣHH( R ; Z p ) hS , which is an equivalence after inverting p since THH( R ; Z p ) ⊗ S Z → HH( R ; Z p ) is anequivalence after inverting p and this property is preserved by taking S -homotopy orbits. Thus, thebottom square is cartesian after inverting p . Using these identifications, the theorem follows. (cid:3) Remark 2.13 (Effective bounds II) . Again, one can make effective the statement that (15) is cartesianafter inverting p , at least in the range ≤ p −
4. In this case, we find (via Remark 2.11) that for i ≤ p − τ ≤ i of the total cofiber of (15) is annihilated by p i . Indeed, the map on cofibers of the bottom rowsof (14) and (15) is given by Σ (THH( R ) ⊗ S Z triv ) hS → Σ HH( R ; Z p ) hS . This map is an equivalencein degrees ≤ p − Definition 2.14 (The p -adic Chern character) . Let R be a ring. Consider the map TC( R ⊗ S F p ) → HP( R ; Z p ) from above. After inverting p , in view of Theorem 3.4 below, we have an equivalenceTC( R ⊗ S F p ; Q p ) ≃ TC(
R/p ; Q p ). We therefore obtain a map β : TC( R/p ; Q p ) → HP( R ; Q p ), andprecomposing with the trace we obtaintr crys = tr ◦ β : K( R/p ; Q p ) → HP( R ; Q p ) . We call tr crys the p -adic Chern character and record that it fits into a natural commutative diagramK( R ; Q p ) / / tr (cid:15) (cid:15) K( R/p ; Q p ) tr (cid:15) (cid:15) tr crys x x TC( R ; Q p ) / / (cid:15) (cid:15) TC(
R/p ; Q p ) β (cid:15) (cid:15) HC − ( R ; Q p ) / / HP( R ; Q p ) (16)in which the bottom square is a pullback. Remark 2.15.
In recent work, Petrov–Vologodsky [PV19] have shown that if p > R is p -torsion free, then there is a natural equivalence HP( R ; Z p ) ≃ TP(
R/p ; Z p ). Thus, one could attemptto compare the p -adic Chern character tr crys with the usual trace K( R/p ; Z p ) → TP(
R/p ; Z p ). Wehave not considered this question.We can now give a quick proof of Theorem A by combining the above results with the followingtheorem. Theorem 2.16 (Clausen–Mathew–Morrow [CMM18]) . If R is commutative and henselian along ( p ) ,then the trace induces an equivalence K( R, ( p ); Z p ) ≃ TC( R, ( p ); Z p ) . Remark 2.17. If R is only associative, but p -complete and has bounded p -power torsion, then there isan equivalence lim ←− K( R/p n , ( p )) ≃ TC( R, ( p ); Z p ). This follows by the Dundas–Goodwillie–McCarthytheorem [DGM13] and the p -adic continuity of TC, cf. [CMM18, Theorem 5.19]. Proof of Theorem A.
As we have already noted, the square (15) from Theorem 2.12 is a pullback afterinverting p ; that is, the bottom square in (16) is a pullback. But the top square in (16) is a pullbackby Theorem 2.16; assembling these cartesian squares completes the proof of the theorem. (cid:3) In fact, we expect that if R is non-commutative and p -complete (or some non-commutative version of henselian)we also have an equivalence K( R, ( p ); Z p ) ≃ TC( R, ( p ); Z p ). But this is out of the scope of this paper. Fiber sequences up to quasi-isogeny.
Next, we review some definitions and terminology asin [Bei14], identify more carefully the fiber terms in the above squares, and prove Corollary B fromthe introduction.
Definition 2.18 (Isogenies and quasi-isogenies) . Given an additive category (or ∞ -category) C , wesay that a map f : X → Y is an isogeny if there exists g : Y → X and an integer N > g ◦ f = N id X and f ◦ g = N id Y . Let C be a stable ∞ -category equipped with a t -structure whichis left-complete. We say that a map f : X → Y of bounded below objects is a quasi-isogeny if thefollowing equivalent conditions are satisfied:(1) for each n , the map τ ≤ n f : τ ≤ n X → τ ≤ n Y is an isogeny in C ;(2) for each n , the map π n X → π n Y in the heart C ♥ is an isogeny.We will need some elementary observations about quasi-isogenies. Note that if one restricts to C ≥ (i.e., connective objects), then quasi-isogenies are preserved under finite colimits and geometricrealizations (but generally not under filtered colimits). Next, let C , D be stable ∞ -categories with left-complete t -structures. Given a right t -exact functor C → D (or just a right bounded exact functor),it is easy to see that F preserves quasi-isogenies. Given an ∞ -category I , we will say that a natural transformation f → g of functors f, g : I → C isa quasi-isogeny if it is a quasi-isogeny in Fun( I , C ) with the pointwise t -structure. We will say thattwo functors are naturally quasi-isogenous if they are are related by a zig-zag of quasi-isogenies offunctors. Example 2.19.
For C = Sp, the map S → Z is a quasi-isogeny but of course not an isogeny (as thereis no nontrivial map back). In fact, in Sp one has the following formality result of Beilinson [Bei14]:every bounded below spectrum X is quasi-isogenous to the spectrum L n Hπ n ( X )[ n ]. In particular,two bounded-below spectra X and Y are quasi-isogenous precisely if for each n separately the abeliangroups π n X and π n Y are isogenous. To see that every spectrum is formal in the above sense, it sufficesto observe that every k -invariant of a connective spectrum X is bounded torsion (where the torsiondegree only depends on the degree of the k -invariant and not on the specific homotopy groups). Forexplicit bounds, cf. [Mat16].This formality result of course does depend on choices and thus does not give similar results infunctor categories C = Fun( I , Sp).The fiber sequence of Corollary 2.9 is the key to obtain our version of Beilinson’s theorem [Bei14],as follows.
Theorem 2.20.
For any associative ring R the following spectra are naturally quasi-isogenous to eachother (i.e., related via a natural zig-zag of quasi-isogenies) TC( R, ( p ); Z p ) ΣHC( R, ( p ); Z p ) ΣHC( R ; Z p ) . Moreover, (a) if R is p -torsion free, then the first two are equivalent after (2 p − -truncation, and (b) if R is p -torsion free and π − (TC( R ; Z p )) = 0 , then the first two are equivalent after (2 p − -truncation. Recall that C said to be left-complete (with respect to the given t -structure) if the natural map C → lim ←− n C ≤ n isan equivalence. This is a technical condition satisfied by many stable ∞ -categories such as Sp and D ( Z ). A functor
C → D is right t -exact with respect to fixed t -structures on C and D if it restricts to a functor C ≥ → D ≥ .It is right bounded if it restricts to a functor C ≥ → D ≥ n for some n ∈ Z . This is true pro-´etale locally if R is commutative, thanks to [HM97b, Th. F]. N THE BEILINSON FIBER SQUARE 13
Proof.
For every associative ring R we have the following commutative diagram of fiber sequencesΣ (cid:0) THH( R ; Z p ) ⊗ S Z hC p (cid:1) hS / / TC( R ; Z p ) ⊗ S Z / / TC( R ⊗ S F p ; Z p ) F O O (cid:15) (cid:15) / / TC( R ; Z p ) id (cid:15) (cid:15) / / O O TC( R ⊗ S F p ; Z p ) (cid:15) (cid:15) id O O TC( R, ( p ); Z p ) / / TC( R ; Z p ) / / TC(
R/p ; Z p ) . (17)To form the above diagram, we use the map TC( R ; Z p ) → TC( R ; Z p ) ⊗ S Z induced from the map S → Z as well as the map on TC( − ; Z p ) induced by the Postnikov section R ⊗ S F p → R/p . Allhorizontal sequences in (17) are fiber sequences, either by Corollary 2.9 or by definition; that is, F isdefined as the fiber of TC( R ; Z p ) → TC( R ⊗ S F p ; Z p ).We claim now that all the vertical maps in diagram (17) are quasi-isogenies. Lemma 2.21.
The map
TC( R ; Z p ) → TC( R ; Z p ) ⊗ S Z in the diagram (17) is a natural quasi-isogenyof spectra. Moreover, its fiber is (2 p − -connective. If π − (TC( R ; Z p )) = 0 , then the fiber is (2 p − -connective.Proof. The first part follows from the observation that tensoring a quasi-isogeny (in this case S → Z )with a bounded below spectrum (here TC( R ; Z p )) is again a quasi-isogeny. Moreover, the fiber ofTC( R ; Z p ) → TC( R ; Z p ) ⊗ S Z is (2 p −
4) connective since TC( R ; Z p ) is ( − S ( p ) → Z ( p ) is (2 p −
3) connective. The last assertion follows similarly. (cid:3)
The right horizontal map TC( R ⊗ S F p ; Z p ) → TC(
R/p ; Z p ) in diagram (17) is also a quasi-isogeny.This follows from Theorem 3.4 that we will discuss and prove in Section 3 and which is purely internalto cyclotomic spectra. But we also want to give a direct proof here using K-theory and the Dundas–Goodwillie–McCarthy theorem. Proposition 2.22.
The natural map
TC( R ⊗ S F p ; Z p ) → TC(
R/p ; Z p ) is a quasi-isogeny. If R is p -torsion free then the fiber is (2 p − -connective.Proof. The map of connective ring spectra R ⊗ S F p → R/p is an isomorphism on π . Thus theDundas–Goodwillie–McCarthy theorem (for ring spectra) implies that its fiber is equivalent to thefiber of the map K( R ⊗ S F p ; Z p ) → K( R/p ; Z p )But the map R ⊗ S F p → R/p of ring spectra is a quasi-isogeny and, if R is p -torsion free, has fiberwhich is (2 p − p − loc. cit. shows that the map is truly aquasi-isogeny of functors). (cid:3) Now we know that the vertical maps in diagram (17) are quasi-isogenies, so we conclude thatΣ (cid:0)
THH( R, Z p ) ⊗ S Z hC p (cid:1) hS and TC( R, ( p ); Z p ) are quasi-isogenous to one another. Moreover, if R is p -torsion free, then the vertical maps from F have (2 p − p − π − TC( R ; Z p ) = 0). Thus, we concludethat Σ (cid:0) THH( R, Z p ) ⊗ S Z hC p (cid:1) hS and TC( R, ( p ); Z p ) are equivalent in degrees ≤ (2 p − ≤ p − π − TC( R ; Z p ) = 0. Theorem 2.20 now follows from the arguments above and thefollowing lemma. (cid:3) Lemma 2.23.
The following spectra are naturally quasi-isogenous to each other (cid:0)
THH( R ; Z p ) ⊗ S Z hC p (cid:1) hS HC( R, ( p ); Z p ) HC( R ; Z p ) and the first two are equivalent after (2 p − -truncation if R is p -torsion free.Proof. We have that THH( R ; Z p ) ⊗ S Z hC p is equivalent to the fiber ofTHH( R ; Z p ) ⊗ S Z → THH( R ⊗ S F p ; Z p )and this map sits in a commutative squareTHH( R ; Z p ) ⊗ S Z / / (cid:15) (cid:15) THH( R ⊗ S F p ; Z p ) (cid:15) (cid:15) HH( R ; Z p ) / / HH(
R/p ; Z p )of spectra with S -action. Both vertical maps are quasi-isogenies, so that we get the desired quasi-isogeny between the first two terms of the statement by taking S -orbits. The term HH( R/p ) isquasi-isogenous to 0, so that we get the last quasi-isogeny too. If R is p -torsion free, the fibers of theleft and right vertical maps are in degrees ≥ p − ≥ p −
2, respectively, so the last assertionfollows too. (cid:3)
Corollary B follows by combining Theorem 2.16 with Theorem 2.20. In particular, we have anisomorphism K ∗ ( R, ( p ); Z p ) ∼ = HC ∗− ( R, ( p ); Z p ) for ∗ ≤ p −
5, for R commutative and p -torsion free.Note also that with the same proof, we can deduce the following variant of Theorem 2.20 for arbitraryconnective Z -algebra ring spectra (also known as Z -linear dgas). Proposition 2.24. If R is a connective Z -algebra spectrum, then the fiber of TC( R ; Z p ) → TC( R ⊗ Z F p ; Z p ) is quasi-isogenous to ΣHC( R ; Z p ) and after (2 p − -truncation equivalent to the fiber of ΣHC( R ; Z p ) → ΣHC( R ⊗ Z F p ; Z p ) . Remark 2.25.
In all of the above, the denominators involved in the above quasi-isogenies are uniform:they do not depend on the choice of R . More formally, one could state all of the above quasi-isogeniesvia the ∞ -category of functors from rings R to spectra. The denominators in the next result are notindependent in the same fashion. Theorem 2.26.
Let ( R, I ) be a pair consisting of an associative ring R and a nilpotent ideal I . Thenthere is a natural zig-zag of quasi-isogenies between K( R, I ; Z p ) and ΣHC(
R, I ; Z p ) .Proof. By the Dundas–Goodwillie–McCarthy theorem, we can replace K-theory with TC. We have anatural map TC(
R, I ) → fib (TC( R, ( p ); Z p ) → TC(
R/I, ( p ); Z p )) (18)Now TC( R/p ; Z p ) → TC( R/ ( I, p ); Z p ) is a quasi-isogeny in view of Theorem 3.4 below, so that (18)is a quasi-isogeny. Combining with the quasi-isogenies of Theorem 2.20 now completes the proof. (cid:3) Quasi-isogenies of cyclotomic spectra
In this section, we systematically study quasi-isogenies in cyclotomic spectra, give another proofof Theorem A and Corollary B, and prove Theorem C, sharpening some results of Geisser–Hesselholt[GH11].
N THE BEILINSON FIBER SQUARE 15
Preliminaries.
We will apply the notion of quasi-isogeny (Definition 2.18) to the ∞ -categoryCycSp of cyclotomic spectra using the t -structure of [AN]; this t -structure is defined so that theconnective objects of CycSp are those whose underlying spectrum is connective and it is checkedin [AN, Theorem 2.1] that the t -structure is left-complete. Note that a quasi-isogeny of bounded-belowcyclotomic spectra f : X → Y is a quasi-isogeny of underlying spectra, and TC( f ; Z p ) : TC( X ; Z p ) → TC( Y ; Z p ) is also a quasi-isogeny. However, THH( F p ) ∈ CycSp has underlying spectrum quasi-isogenous to zero, but is not itself quasi-isogenous to zero because TC( F p ) ≃ TC( F p ; Z p ) is torsionfree and nonzero: π TC( F p ; Z p ) ∼ = π − TC( F p ; Z p ) ∼ = Z p .In the next result, we use the notion of TR of a cyclotomic spectrum, which plays an importantrole in the work [AN]. See [BM15] for an account of TR in the approach to cyclotomic spectra viagenuine equivariant homotopy theory. Implicitly, TR is computed with respect to our fixed prime p ,but it will not generally be p -complete unless we p -complete it forming TR( X ; Z p ) for a cyclotomicspectrum X . Proposition 3.1.
A map f : X → Y of bounded-below cyclotomic spectra is a quasi-isogeny in CycSp if and only if the map of spectra
TR( f ) : TR( X ) → TR( Y ) is a quasi-isogeny of spectra.Proof. This follows from the description of the cyclotomic t -structure of [AN]. In particular, thecyclotomic homotopy groups of X ∈ CycSp are precisely the homotopy groups of the spectrumTR( X ), together with the Frobenius and Verschiebung maps. These form p -typical Cartier modulesand the heart CycSp ♥ is equivalent to a full subcategory of the category of p -typical Cartier modules,which is essentially a module category over a certain ring. But one easily checks that a map betweenCartier modules is an isogeny precisely if the underlying map of abelian groups is an isogeny. (cid:3) Proposition 3.2.
Let X ∈ CycSp be a cyclotomic spectrum such that X is bounded-below, such thatthe Frobenius ϕ : X → X tC p is nullhomotopic in Fun( BS , Sp) , and such that X is quasi-isogenous tozero as a spectrum. Then X is quasi-isogenous to zero as a cyclotomic spectrum.Proof. The assumption that the Frobenius is nullhomotopic implies that TR( X ) is a product Q n ≥ X hC pn ,using the description of TR as an iterated pullback, cf. [AN, Remark 2.5] and [NS18, Corollary II.4.7].The assumption that X is quasi-isogenous to zero now implies that the above product is also quasi-isogenous to zero, so we conclude by Proposition 3.1. (cid:3) We observe that the theory of cyclotomic spectra admits a natural graded variant. A gradedspectrum is an object of the functor category Fun( Z ds ≥ , Sp) where Z ds ≥ denotes the discrete categoryof nonnegative integers with no non-identity morphisms; given a graded spectrum X we let X i ∈ Sp , i ≥ i th graded piece. We let GrSp denote the ∞ -category of graded spectra, whichwe consider as a symmetric monoidal ∞ -category under Day convolution using the multiplicationsymmetric monoidal structure on Z ds ≥ . A graded cyclotomic spectrum X consists of a graded spectrum X = { X i } equipped with an S -action together with a family of S -equivariant maps ϕ i : X i → X tC p pi for i ≥
0. We let GrCycSp denote the ∞ -category of graded cyclotomic spectra. Any gradedcyclotomic spectrum X = { X i } has an underlying cyclotomic spectrum L i ≥ X i , and this defines aforgetful functor GrCycSp → CycSp.More formally, the ∞ -category GrCycSp is defined as follows. We consider the ∞ -category Fun( BS , GrSp)of graded spectra equipped with an S -action. This admits a natural endofunctor F which sends { X i , i ≥ } to n X tC p pi o , where we regard X tC p pi as a spectrum with an S /C p ≃ S -action. ThenGrCycSp is defined as the ∞ -category of F -coalgebras, as in [NS18, Section II.5]. Recall that what we denote by CycSp is denoted CycSp p in [AN]. Given a graded ring spectrum R , there is a graded cyclotomic spectrum THH( R ) obtained byapplying the cyclic bar construction in the category of graded spectra. This refines the usual THH andadmits an S -action in graded spectra. See Appendix A for the details of this construction. Comparealso [Bru01] for a treatment of filtered cyclotomic spectra and filtered TC using more classical methods. Proposition 3.3.
Let X be a graded cyclotomic spectrum. If (1) the underlying spectrum of X is quasi-isogenous to zero, (2) the graded piece X is contractible, and (3) the connectivity of the pieces X i tends to infinity in i ,then X is quasi-isogenous to zero as an object of CycSp .Proof.
Given a graded cyclotomic spectrum X , for each i , we can construct a graded cyclotomicspectrum X ≤ i ∈ GrCycSp such that ( X ≤ i ) j = 0 for j > i and ( X ≤ i ) j = X i for j ≤ i and a tower ofmaps X → · · · → X ≤ n → X ≤ n − → · · · → X ≤ . This is a tower in GrCycSp, and we can consider itas a tower of underlying objects in CycSp too.We need to show that for each j , π j (TR( X )) is isogenous to zero. Our assumptions imply that π j (TR( X )) → π j TR( X ≤ n ) is an isomorphism for n ≫
0. However, the object fib( X ≤ i → X ≤ i − )defines a cyclotomic spectrum with Frobenius homotopic to zero, in view of the grading. It follows fromProposition 3.2 that TR(fib( X ≤ i → X ≤ i − )) is quasi-isogenous to zero, and by induction TR( X ≤ n )is quasi-isogenous to zero. Putting these observations together with Proposition 3.1 completes theproof. (cid:3) Quasi-isogenies on
THH . Our main result here is the following, which restates Theorem C.On TC and for discrete rings in which p is nilpotent, it is due to Geisser–Hesselholt [GH11], and themain arguments are based on theirs. Theorem 3.4.
Let f : A → A ′ be a map of connective associative ring spectra. If (i) f is a quasi-isogeny of spectra and (ii) the map π ( f ) : π ( A ) → π ( A ′ ) is surjective with nilpotent kernel,then THH( A ) → THH( A ′ ) is a quasi-isogeny in CycSp . We will first verify some special cases.
Proposition 3.5.
Let R be a connective associative graded ring spectrum. If (a) each R i , i > , is isogenous to zero as a spectrum and (b) the connectivity of the R i tends to ∞ as i → ∞ ,then the map THH( R ) → THH( R ) is a quasi-isogeny in CycSp .Proof.
Since R is a graded ring spectrum, THH( R ) admits the structure of a graded cyclotomicspectrum (refining the usual cyclotomic structure on THH( R )), and in degree zero one has THH( R ).For this, compare Appendix A, or the work of Brun [Bru01], who uses the more classical approach tocyclotomic spectra.Now we wish to apply Proposition 3.3. Consider the subcategory C ⊆
GrSp ≥ of connective gradedspectra spanned by graded spectra Z such that Z i is quasi-isogenous to zero for i > Z i grows without bound as i → ∞ . Then C is closed under tensor products andgeometric realizations. The assumptions on R imply that R ∈ C , and consequently THH( R ) ∈ C aswell. That is, THH( R ) i is quasi-isogenous to zero for i > R ) i grows without bound as i → ∞ . Thus, we can apply Proposition 3.3. (cid:3) N THE BEILINSON FIBER SQUARE 17
Proposition 3.6.
Let A be a connective associative ring spectrum. Let M be a connective ( A, A ) -bimodule which is quasi-isogenous to zero. Suppose e A is a square-zero extension of A by M , in thesense that one has a map f : A → A ⊕ M [1] in Alg /A and a pullback diagram e A (cid:15) (cid:15) / / A (cid:15) (cid:15) A f / / A ⊕ M [1] . Then the map
THH( e A ) → THH( A ) is a quasi-isogeny in CycSp .Proof.
We can form the ˇCech nerve of the map e A → A , i.e., the simplicial object . . . e A × A e A ⇒ e A .This yields a simplicial object X • of Alg which resolves A . It follows that | THH( X • ) | ≃ THH( A ) inCycSp.Now e A × A e A is a trivial square-zero extension of e A by M . It follows that THH( e A × A e A ) isquasi-isogenous to THH( e A ) by Proposition 3.5, since a trivial square-zero extension can be given agrading. Continuing in this way, it follows that all the maps in the simplicial object THH( X • ) arequasi-isogenies. Taking geometric realizations now, it follows that THH( e A ) → | THH( X • ) | ≃ THH( A )is a quasi-isogeny in CycSp. (cid:3) Proposition 3.7.
Let B be a connective associative ring spectrum and let B ′ be an object of Alg
B//B .Suppose that the augmentation map B ′ → B is a quasi-isogeny and the map π ( B ′ ) → π ( B ) hasnilpotent kernel. Then THH( B ) → THH( B ′ ) is a quasi-isogeny.Proof. Recall first that Alg
B//B is equivalent to the ∞ -category of nonunital associative algebraobjects in ( B, B )-bimodules. In particular, I = fib( B ′ → B ) has such a structure. We can workup the Postnikov tower τ ≤ ⋆ I ; since TR behaves well with respect to Postnikov towers, it suffices toprove the result for each τ ≤ n I . General results as in [Lur17, Section 7.4.1] (which go back at least to[Bas99]) now show that τ ≤ n I can be obtained in finitely many steps via square-zero extensions from B , by bimodules which are quasi-isogenous to zero. Now we conclude via Proposition 3.6. (cid:3) Proof of Theorem 3.4.
We consider the ˇCech nerve of A → A ′ . We obtain a simplicial object X • inAlg ≥ such that | X • | ≃ A ′ and such that each X i is an iterated fiber product of copies of A over A ′ . Each X i can be given the structure of an object of Alg A//A (via appropriate face and degeneracymaps), whence we conclude by Proposition 3.7 that all the maps in the simplicial object THH( X • )are quasi-isogenies in CycSp. Finally, the result now follows by taking geometric realizations. (cid:3) One important corollary of Theorem 3.4 is the following result of Geisser and Hesselholt; see [LT19]for generalizations.
Corollary 3.8 (Geisser–Hesselholt [GH11]) . If p is nilpotent in A and I ⊆ A is a two-sided nilpotentideal, then K( A, I ) ≃ TC(
A, I ) is quasi-isogenous to zero.Proof. In this case, A → A/I is a quasi-isogeny (they are both quasi-isogenous to zero) and henceTheorem 3.4 applies to prove that THH( A ) → THH(
A/I ) is a quasi-isogeny of cyclotomic spectra.This implies in particular that TC(
A, I ) is quasi-isogenous to zero. (cid:3)
Quasi-isogenies and the Beilinson fiber sequence.
Recall that Proposition 2.22, whichwas key in proving Theorem A, asserts that for every ring R the induced map TC( R ⊗ S F p ; Z p ) → TC(
R/p ; Z p ) is a quasi-isogeny. Our proof in Section 2 relied on K-theory. Altenatively we can nowalso deduce this fact directly from Theorem 3.4, which implies that THH( R ⊗ S F p ) → THH(
R/p ) is aquasi-isogeny of cyclotomic spectra and therefore TC( R ⊗ S F p ; Z p ) → TC(
R/p ; Z p ) is a quasi-isogeny of spectra. In this section we want to take this a step further and prove a cyclotomic version ofTheorem A. Theorem 3.9.
For every ring R , the following cyclotomic spectra are quasi-isogenous to each other THH ( R, ( p ); Z p ) HH( R ; Z p ) HH( R, ( p ); Z p ) where HH( R ; Z p ) and HH( R, ( p ); Z p ) are equipped with the canonical S -actions and the zero Frobenius(see Example 2.3). Moreover if R is p -torsion free then we have an equivalence of cyclotomic spectra τ cyc ≤ (2 p − THH ( R, ( p ); Z p ) ≃ τ cyc ≤ (2 p − HH( R, ( p ); Z p ) . We note that the last theorem immediately implies Theorem A by passing to TC( − ; Z p ) sinceTC( − ; Z p ) of a cyclotomic spectrum with zero Frobenius is just given by the p -completion of theshifted S -orbits. But Theorem 3.9 is strictly stronger than Theorem A since quasi-isogenies cannotbe detected on TC. The remainder of this section is devoted to proving Theorem 3.9. Lemma 3.10.
Suppose that X → Y is a quasi-isogeny of cyclotomic spectra and that M is anybounded below cyclotomic spectrum. Then, X ⊗ S M → Y ⊗ S M is a quasi-isogeny of cyclotomicspectra.Proof. We can assume M is connective, in which case the functor −⊗ S M is a right t -exact endofunctorof CycSp and hence preserves quasi-isogenies. (cid:3) We now want to apply a similar proof-strategy as in Section 2 and consider the diagramTHH( R ; Z p ) ⊗ S Z hC p / / THH( R ; Z p ) ⊗ S Z triv / / THH( R ⊗ S F p ; Z p ) (cid:15) (cid:15) THH( R, ( p ); Z p ) / / THH( R ; Z p ) O O / / THH(
R/p ; Z p ) , of cyclotomic spectra in which the horizontal rows are fiber sequences. Both vertical maps are quasi-isogenies in cyclotomic spectra: the first by Lemma 3.10 and because S triv → Z triv is a quasi-isogeny(since ( − ) triv is right t -exact) and the second by Theorem 3.4. The right vertical map has homotopyfiber in degrees ≥ p −
2, while the middle vertical map has homotopy fiber in degrees ≥ p −
3. Itfollows that THH( R, ( p ); Z p ) is quasi-isogenous in cyclotomic spectra to THH( R ; Z p ) ⊗ S Z hC p , andtheir cyclotomic (2 p − τ cyc ≤ p − are equivalent. Note that the cyclotomic Frobenius onTHH( R ; Z p ) ⊗ S Z hC p is nullhomotopic by the Tate orbit lemma.Now the next lemma finishes the proof of Theorem 3.9. Lemma 3.11.
The following cyclotomic spectra are quasi-isogenous to each other
THH( R ; Z p ) ⊗ S Z hC p HH( R ; Z p ) HH( R, ( p ); Z p ) . Moreover, the cyclotomic truncations τ cyc ≤ p − of the first and the third are naturally equivalent.Proof. We consider the squareTHH( R ; Z p ) ⊗ S Z / / (cid:15) (cid:15) THH( R ⊗ S F p ; Z p ) (cid:15) (cid:15) HH( R ; Z p ) / / HH(
R/p ; Z p )of spectra with S -action, in which the vertical maps are quasi-isogenies and the right hand terms arequasi-isogenous to zero. We consider it as a square of cyclotomic spectra by equipping all spectra withthe zero Frobenius map. It follows from Proposition 3.2 that the vertical maps are quasi-isogenies of N THE BEILINSON FIBER SQUARE 19 cyclotomic spectra. The horizontal fibers are equivalent to THH( R ; Z p ) ⊗ S Z hC p and HH( R, ( p ); Z p )which finishes the proof.Finally, the induced map of cyclotomic spectra (with zero Frobenii) THH( R ; Z p ) ⊗ S Z hC p → HH( R, ( p ); Z p ) has the property that it is an equivalence of underlying spectra in degrees ≤ p − ≤ p −
4, e.g., again using (the proof of, which describes TR) Proposition 3.2. (cid:3) Application to p -adic deformations In this section, we prove Theorems D and E. Throughout this section, let X be a quasi-compactand quasi-separated (qcqs) p -adic formal scheme with bounded p -power torsion, and write X n = X × Spec Z p Spec Z /p n . We are interested in the following invariants of X , and in particular the p -adicdeformation problem (Question 4.5 below). Definition 4.1 (Continuous invariants of formal schemes) . Let F be an invariant of schemes (suchas K , THH , HH , HC − , HP , TC). Given the formal scheme X , we define F cts ( X ) via F cts ( X ) = lim ←− n F ( X n ) . (19)If the p -adic formal scheme X arises as the p -adic completion of a scheme X , we have a naturalcomparison map F ( X ) → F cts ( X ) . (20) Proposition 4.2.
Suppose X is the p -adic completion of a qcqs scheme X with bounded p -powertorsion. Then the maps (20) for F = HH , THH , HC − , HP , TC are p -adic equivalences.Proof. Using Zariski descent on X , we may assume that X = Spec( R ) where R is a ring of bounded p -power torsion, and then X = Spf( ˆ R p ). Using the cyclic bar construction, it is not difficult to showthat THH cts ( X ; Z p ) = THH( R ; Z p ), i.e., that (20) is a p -adic equivalence for F = THH (cf. the proof of[CMM18, Theorem 5.19]). Tensoring over THH( Z ) with Z and taking S -invariants and coinvariants,we find that (20) is a p -adic equivalence for F = HC − , HP. Running the above argument with THHinstead of HH, one concludes that (20) is a p -adic equivalence for F = TC [CMM18, Theorem 5.19].See also [DM17, Cor. 4.8] for these results, when R is assumed noetherian and F -finite. (cid:3) By contrast, it is much more difficult to control (20) when F = K. We mention the two followingcases. Example 4.3 (Formal affine schemes) . Suppose X is affine, i.e., X = Spf( R ), for R a p -adicallycomplete ring with bounded p -power torsion. We can then write X as the p -adic completion (as aformal scheme) of X = Spec( R ). In this case, the comparison map (20) is a p -adic equivalence for F = K as well, cf. [CMM18, Theorem 5.23] and [GH06, Theorem C]. Example 4.4 (Proper schemes) . Suppose R is p -complete. Suppose X is the p -completion of a properscheme X → Spec( R ). The map K( X ; Z p ) → K cts ( X ; Z p ) is probably not an equivalence; compare[BEK14a, App. B] for a related counterexample in equal characteristic zero. In this case, K cts ( X ; Z p )is generally much more tractable than K( X ; Z p ) via comparisons with topological cyclic homology. Question 4.5 (The p -adic deformation problem) . Let X be a p -adic formal scheme with special fiber X as above. For i ≥
0, what is the image of the mapK cts i ( X ; Q ) → K i ( X ; Q )? (21) By the Milnor exact sequence, this is equivalent to describing the image of the map (cid:16) lim ←− n K i ( X n )) (cid:17) Q → K i ( X ; Q ). We first observe that Question 4.5 is essentially a p -adic question in TC. For each n ≥
1, letK( X n , X ) be the fiber of K( X n ) → K( X ). Since X n is a p -adic nilpotent thickening of X , the relativeK-theory K( X n , X ) has homotopy groups which are bounded p -power torsion (cf. Corollary 3.8, due to[GH11]), and the spectrum is therefore p -complete. Using the Dundas-Goodwillie-McCarthy theorem[DGM13], and taking limits, we obtain a cartesian squareK cts ( X ) / / (cid:15) (cid:15) K( X ) (cid:15) (cid:15) TC cts ( X ; Z p ) / / TC( X ; Z p ) . (22)Since the X n , n ≥ p -power torsion schemes, their TC are already p -adically complete. Using thisdiagram, we see that it suffices to determine the image of the map TC cts i ( X ; Q p ) → TC i ( X ; Q p ).In this section, we will describe an explicit obstruction class for Question 4.5 in case i = 0 (sharp-ening results of [BEK14b]) in certain geometric situations and construct general obstruction classesin all cases (after [Bei14]).4.1. The Bloch–Esnault–Kerz theorem.
In [BEK14b], Bloch–Esnault–Kerz consider Question 4.5in the case i = 0 and where X has the following form. Let K be a complete discretely valued field ofcharacteristic zero with ring of integers O K , whose residue field k is perfect of characteristic p >
0. Welet π ∈ O K be a uniformizer and denote by K the ring of fractions W ( k )[ p ]. We take X → Spf( O K )to be a smooth p -adic formal scheme, with special fiber X k → Spec( k ) and rigid analytic generic fiber X K over K . The goal is to understand the image of the map K cts0 ( X ; Q p ) → K ( X k ; Q p ).We refer to [BEK14b, Eme97] for more detailed motivation for the above question, as well as[BEK14a, Mor14, Mor19] for discussions of the analogous question in equal characteristic. Note thatwhen X arises from a smooth proper scheme X → Spf( O K ), the above question says nothing about theimage of the map K ( X ; Q ) → K ( X k ; Q ); this (at least up to homological equivalence) is the subjectof the far more difficult p -adic variational Hodge conjecture of Fontaine–Messing (Conjecture 1.3).Here we will unwind the Beilinson fiber square to answer Question 4.5 in this case in terms of thecrystalline Chern character. To begin with, we need to review the crystalline Chern character and thecrystalline to de Rham comparison. Construction 4.6 (de Rham cohomology) . Given a smooth p -adic formal scheme X → Spf( O K ), wewill consider the ( p -adic) de Rham cohomology R Γ dR ( X / O K ) ∈ D ( O K ), equipped with the descend-ing, multiplicative Hodge filtration Fil ≥ ⋆ R Γ dR ( X / O K ). When X is also assumed proper, then all ofthese are perfect complexes in D ( O K ). Furthermore, after inverting p , we write R Γ dR ( X K /K ) ∈ D ( K )and Fil ≥ ⋆ R Γ dR ( X K /K ) for the induced objects.A basic fact we will use is that when X is proper, the induced spectral sequence from the Hodgefiltration on R Γ dR ( X K /K ) degenerates after rationalization; this is the degeneration of the Hodge-to-de Rham spectral sequence for proper smooth rigid analytic varieties, proved by Scholze [Sch13]. Construction 4.7 (Comparison between crystalline and de Rham cohomology) . Given X → Spf( O K )a smooth p -adic formal scheme, we can consider the crystalline cohomology R Γ crys ( X k ) of the specialfiber as well. In the absolutely unramified case (when O K = W ( k )), the usual de Rham to crys-talline comparison theorem yields an equivalence R Γ crys ( X k ) ≃ R Γ dR ( X / O K ). In general, by [BO83,Theorem 2.4], we have a natural equivalence after rationalization R Γ dR ( X K /K ) ≃ R Γ crys ( X k ; Q p ) ⊗ K K. (23) Construction 4.8 (The crystalline Chern character) . Let Y be a regular scheme of characteristic p .Given a vector bundle V on Y , we can define Chern classes c i ( V ) ∈ H i crys ( Y ) for i ≥ N THE BEILINSON FIBER SQUARE 21 usual axioms, cf. [Gro85] (e.g., using the classical method of [Gro58]). The usual formula then yieldsa crystalline Chern character , i.e., a natural ring homomorphism into the rationalized crystallinecohomology ch crys : K ( Y ) → M i ≥ H i crys ( Y ; Q p ) , which carries the class of a line bundle L to 1 + c ( L ).Our main result is the following theorem, which extends results of Bloch–Esnault–Kerz [BEK14b].In [BEK14b], this result is proved in the case where K = K is absolutely unramified, X arises froma smooth projective scheme, and p > dim( X ) + 6. In [Bei14], it is shown that there is an obstructionin L i ≥ H i dR ( X K ) / Fil ≥ i H i dR ( X K ), but the obstruction is not identified with the Chern character; seeSection 4.2 below for more discussion. Theorem 4.9.
Let K be a complete discretely valued field of characteristic zero with ring of integers O K , whose residue field k is perfect of characteristic p > . Let X → Spf( O K ) be a proper smooth p -adic formal scheme with special fiber X k . A class x ∈ K ( X k ; Q p ) lifts to K cts0 ( X ; Q p ) if and onlyif the crystalline Chern character ch crys ( x ) ∈ L i ≥ H i crys ( X k ; Q p ) maps (via the comparison map of (23) ) to L i ≥ Fil ≥ i H i dR ( X K /K ) ⊆ L i ≥ H i dR ( X K /K ) . The proof of Theorem 4.9 will be carried out as follows. First, we give an analogous form of theBeilinson fiber square when we work relative to O K (Proposition 4.10). Next, we will show thatthe right vertical map in the p -adic Chern character can be defined entirely in terms of the specialfiber (which will use some Kan extension techniques from Appendix B), and then identify it with thecrystalline Chern character (Proposition 4.12). Theorem 4.9 will then follow directly.In the next result, we will use the continuous Hochschild (resp. negative cyclic, periodic cyclic)homology of a formal scheme over O K , defined as in Definition 4.1; note that Proposition 4.2 appliesto these relative theories too, since they can be recovered from THH. Proposition 4.10 (The fiber square relative to O K ) . Let X be a smooth formal O K -scheme. Thenthere are natural fiber squares K cts ( X ; Q p ) (cid:15) (cid:15) / / K( X k ; Q p ) (cid:15) (cid:15) TC cts ( X ; Q p ) (cid:15) (cid:15) / / TC( X k ; Q p ) (cid:15) (cid:15) HC − , cts ( X / O K ; Q p ) / / HP cts ( X / O K ; Q p ) . (24) Proof.
This will follow from the Beilinson fiber square. By Zariski descent of all terms in the formalscheme X , we can assume that X = Spf( R ) for R a formally smooth, p -complete O K -algebra. First,HH( O K ; Z p ) ≃ HH( O K /W ( k ); Z p ). Since L O K /W ( k ) is quasi-isogenous to zero, we find that themap HH( O K ; Q p ) → K given by truncation is an isomorphism. We thus conclude (via Hochschild–Kostant–Rosenberg) HH cts ( R ; Q p ) → HH cts ( R/ O K ; Q p ) is an isomorphism, whence HC( R ; Q p ) → HC( R/ O K ; Q p ) is an isomorphism too by taking S -coinvariants. Therefore, the diagramHC − ( R ; Q p ) (cid:15) (cid:15) / / HP( R ; Q p ) (cid:15) (cid:15) HC − ( R/ O K ; Q p ) / / HP( R/ O K ; Q p ) is homotopy cartesian. Combining with the Beilinson fiber square, the result now follows. (cid:3) Construction 4.11 (The p -adic Chern character map) . Since X / O K is smooth, we obtain fromHochschild–Kostant–Rosenberg type filtrations (as in [Ant19], using Adams operations as in [BMS19,Sec. 9.4] to split the filtration) natural decompositionsHP cts ( X / O K ; Q p ) ≃ Y i ∈ Z R Γ dR ( X K /K )[2 i ] , HC − , cts ( X / O K ; Q p ) ≃ Y i ∈ Z Fil ≥ i R Γ dR ( X K /K )[2 i ] . It follows that we obtain from (24) a natural mapK( X k ; Q p ) → TC( X k ; Q p ) → Y i ∈ Z R Γ dR ( X K /K )[2 i ] (25)for every smooth p -adic formal scheme X → Spf( O K ). We observe that both the source and targetactually depend only on the special fiber X k of X , thanks to Construction 4.7. Furthermore, toconstruct (25), it suffices to work with affine formal schemes over O K , by Zariski descent of thetarget, so we can assume X = Spf( R ) for R a formally smooth O K -algebra. That is, we have a naturalmap TC( R ⊗ O K k ; Q p ) → Q i ∈ Z ( R Γ crys (Spec( R ⊗ O K k ); Q p ) ⊗ K K )[2 i ].Consider the functors on smooth k -algebras A TC( A ; Q p ) and A Y i ∈ Z ( R Γ crys (Spec( A ); Q p ) ⊗ K K ) [2 i ] . (26)The left Kan extension of TC( − ; Q p ) to almost finitely presented objects of SCR k (as in Definition B.5)is TC( − ; Q p ) again, since this functor commutes with geometric realizations in SCR k ; note also thatby Theorem 3.4, TC( − ; Q p ) = TC( π ( − ); Q p ) on SCR k . By Corollary B.6 applied to the left Kanextensions of the functors (26) on smooth k -algebras (and Zariski sheafifying again), it follows that(25) actually upgrades to a natural transformation of functors in the special fiber alone. That is, forevery smooth k -scheme Z , we obtain a natural mapK( Z ; Q p ) tr −→ TC( Z ; Q p ) → Y i ∈ Z ( R Γ crys ( Z ; Q p ) ⊗ K K )[2 i ] , (27)such that (25) is obtained by taking Z = X k .Next, we identify (up to scaling factors) the map (27) on π with the crystalline Chern character. Proposition 4.12.
There exists a scalar λ ∈ K × such that for every smooth separated k -scheme Z ,the map K ( Z ; Q p ) → Q i ≥ H i crys ( Z ; Q p ) ⊗ K K of (27) is given by the crystalline Chern charactercomposed with the automorphism that multiplies the i th factor by λ i .Proof. It suffices (by the resolution property) to evaluate (27) on the class of a vector bundle on Z , andfor this we will reduce to the universal case. For this it will be convenient to extend to stacks over k as well. Given a smooth scheme or stack Z over k , let Vect n ( Z ) denote the groupoid of n -dimensionalvector bundles on Z . It follows that we obtain from (27) a natural transformation (of spaces) for allsmooth k -schemes Z , f n : Vect n ( Z ) → Ω ∞ Y i ∈ Z ( R Γ crys ( Z ; Q p ) ⊗ K K ) [2 i ] ! such that the { f n } are additive and multiplicative.Both the source and target of the f n are sheaves of spaces for the smooth or ´etale topology onsmooth k -schemes. Sheafifying for the smooth topology, we obtain such a natural transformationfor any smooth Artin stack, which still satisfies the additivity and multiplicativity properties. Bynaturality, it suffices to show that for Z = BGL n and for E the tautological n -dimensional vector N THE BEILINSON FIBER SQUARE 23 bundle, f n ( E ) is given by (up to scalars) the crystalline Chern character of E . It follows that for each n , f n ( E ) is given by a power series (with K coefficients) in the crystalline Chern classes of E , since R Γ crys ( BGL n ; Q p ) (defined via sheafification) is the polynomial ring K [ c , . . . , c n ], e.g., as in thecalculations of de Rham and Hodge cohomology of BGL n in [Tot18]. By additivity, multiplicativity,and the splitting principle to reduce to the case of line bundles, we find easily that f n must be theChern character up to normalization by powers of some constant λ . Moreover, λ = 0 by comparisonwith the left-hand-side of (24). (cid:3) Proof of Theorem 4.9.
We use the fiber square of (24). As before, we have identifications HP cts ( X / O K ; Q p ) ≃ Q i ∈ Z R Γ dR ( X K /K )[2 i ] and HC − , cts ( X / O K ; Q p ) ≃ Q i ∈ Z Fil ≥ i R Γ dR ( X K /K )[2 i ]. Using the crystalline-to-de Rham comparison (Construction 4.7) and Proposition 4.12, we see that the mapK ( X k ; Q p ) → Y i ∈ Z H i dR ( X K /K ) ≃ Y i ∈ Z H i crys ( X k ; Q p ) ⊗ K K is given up to scalar factors by the crystalline Chern character. The result now follows from Proposi-tion 4.10. (cid:3) Generalization of Beilinson’s obstruction; proof of Theorem F.
Let K, O K , k be as inthe preceding subsection. Consider a proper scheme X → Spec( O K ) with smooth generic fiber X K and possibly singular special fiber X k . In [Bei14], Beilinson considers more generally the deformationproblem for classes in higher K-theory, and proves: Theorem 4.13 (Beilinson [Bei14]) . Given x ∈ K i ( X k ) Q , there is a natural obstruction class in L r ≥ H r − i dR ( X K /K ) / Fil r H r − i dR ( X K /K ) which vanishes if and only if x lifts to (lim ←− K i ( X/π n )) Q .More precisely, there is a natural equivalence of spectra cofib(K cts ( X ; Q p ) → K( X k ; Q p )) ≃ M r ≥ R Γ dR ( X K /K ) / Fil ≥ r R Γ dR ( X K /K )[2 r ] . In particular, Theorem 4.13 applies for i = 0 and overlaps with the results of [BEK14b], althoughit does not identify the obstruction class with the crystalline Chern character. In this subsection, weobserve that Theorem 4.13 can be extended to essentially arbitrary formal schemes, using comparisonsbetween cyclic and de Rham cohomology as in [Ant19]. This argument will not essentially rely onhaving a fiber square as in Theorem A (versus a fiber sequence), and could be deduced from the resultsof [Bei14]. Theorem 4.14.
Let X be a qcqs p -adic formal scheme with bounded p -power torsion. Given i ∈ Z and a class x ∈ K i ( X ; Q ) there is a natural class c ( x ) ∈ M r ≥ H r − i (cid:16) L Ω X /L Ω ≥ r X (cid:17) Q p with the property that x lifts to K cts i ( X ; Q ) if and only if c ( x ) = 0 . More precisely, there is a naturalequivalence of spectra cofib (cid:0) K cts ( X ; Q ) → K( X ; Q ) (cid:1) ≃ M r ≥ (cid:16) L Ω X /L Ω ≥ r X [2 r ] (cid:17) Q p . (28) Proof of Theorem 4.14.
It clearly suffices to exhibit the natural equivalence (28), and for this we mayassume X = Spf( R ) is affine, since all terms satisfy Zariski descent. Now we have seen that that thecofiber in (28) can be identified with the cofiber of TC( R ; Q p ) → TC(
R/p ; Q p ) , or equivalently with HC( R ; Q p )[2] by Theorem 2.20. We invoke the result of [Ant19] which constructs on HC( R ; Z p )[2] anatural exhaustive decreasing filtration Fil ≥ ⋆ HC( R ; Z p )[2] with graded piecesgr n HC( R ; Z p )[2] ≃ L Ω R /L Ω ≥ nR [2 n ] , where, as before, L Ω R is the p -adic derived de Rham cohomology of R and L Ω ≥ ⋆R is the Hodgefiltration on the derived de Rham cohomology (see specifically the proof of [Ant19, Corollary 4.11]). It follows that on HC( R ; Q p )[2] there is a natural exhaustive decreasing filtration Fil ≥ ⋆ HC( R ; Q p )[2]with graded pieces gr n HC( R ; Q p )[2] ≃ ( L Ω R /L Ω ≥ nR ) Q p [2 n ] . An argument as in [BMS19, Section 9.4] can be used to show that there is an action of Adamsoperations on HC( R ; Z p )[2] where λ ∈ Z × p acts via λ n on gr n HC( R ; Z p ). In particular, these splitHC( R ; Q p )[2] into eigenspaces so that there is a natural decompositionHC( R ; Q p )[2] ≃ M n ( L Ω R /L Ω ≥ nR ) Q p [2 n ] . The result now follows from the Beilinson fiber sequence. (cid:3)
Remark 4.15 (Changing the base ring) . In the above work, Z p was used as the base for cyclic andde Rham cohomology, but often this is not essential. Suppose now that A is a commutative Z -algebrawith [ L A/ Z quasi-isogenous to zero. Then it is not difficult to see that, for formal schemes over A ,we can replace all occurrences of derived de Rham cohomology relative to Z p with such occurrencesrelative to A . 5. The motivic filtration on
TCIn this section (which will not use the Beilinson fiber square), we prove some general structuralresults on topological cyclic homology TC and on the “motivic” filtration constructed by Bhatt–Morrow–Scholze [BMS19].Recall that, according to [BMS19], for R a quasisyntomic ring (see Definition 5.4 below for areview), TC( R ; Z p ) admits a complete descending Z ≥ -indexed filtration Fil ≥ ⋆ TC( R ; Z p ) with associ-ated graded terms given as gr i TC( R ; Z p ) ≃ Z p ( i )( R )[2 i ]. In this section, we will prove some structuralproperties of this filtration. Our main results are as follows. Theorem 5.1 (Connectivity properties) . (1) Let R ∈ QSyn be a quasisyntomic ring. Then Z p ( i )( R ) ∈ D ≤ i +1 ( Z p ) . Consequently, we have Fil ≥ i TC( R ; Z p ) ∈ Sp ≥ i − . (2) The functors R Z p ( i )( R ) and R Fil ≥ i TC( R ; Z p ) are left Kan extended from finitelygenerated p -complete polynomial Z p -algebras. Part (2) was indicated to us by Scholze. In view of it, we can extend the construction of the Z p ( i )to all ( p -complete) rings. Theorem 5.2 (Rigidity) . Let ( R, I ) be a henselian pair where R and R/I are p -complete. Then fib( Z p ( i )( R ) → Z p ( i )( R/I )) ∈ D ≤ i ( Z p ) . In particular, using the known description in characteristic p , we obtain that for any R there is acomplete description of the top cohomology H i +1 ( F p ( i )( R )) and that this vanishes ´etale locally. The work of [Ant19] was essentially motivated by that of [BMS19] which among many other things establishedsuch filtrations for quasisyntomic rings by descent.
N THE BEILINSON FIBER SQUARE 25
Review of [BMS19] . Here we recall some of the major results and techniques of [BMS19].We recall first the quasisyntomic site QSyn (a non-noetherian version of the syntomic site used byFontaine–Messing [FM87]) and the subcategory QRSPerfd ⊂ QSyn of quasiregular semiperfectoidrings.
Definition 5.3 ( p -complete (faithful) flatness and Tor-amplitude, [BMS19, Def. 4.1]) . Let R bea commutative ring. An R -module M is called p -completely flat (resp. p -completely faithfully flat )if M ⊗ L R ( R/p ) ∈ D ( R/p ) is a flat (resp. faithfully flat)
R/p -module concentrated in degree zero.Similarly, an object N ∈ D ( R ) has p -complete Tor -amplitude in [ a, b ] if N ⊗ L R R/p ∈ D ( R/p ) hasTor-amplitude in [ a, b ]. Definition 5.4 (The quasisyntomic site, cf. [BMS19, Sec. 4]) . (1) A commutative ring R is called quasisyntomic if it is p -complete, has bounded p -power torsion, and L R/ Z p has p -complete Tor-amplitude in [ − ,
0] (indexing conventions for the derived category are cohomological). Welet QSyn be the category of quasisyntomic rings, with all ring homomorphisms.(2) The category QSyn (or more precisely its opposite) acquires the structure of a site as follows:a map A → B in QSyn is a cover if A → B is p -completely faithfully flat and if L B/A ∈ D ( B )has p -complete Tor-amplitude in [ − , A → B only assumed p -completely flat (rather than p -completely faithfully flat),a quasisyntomic map .(3) An object R ∈ QSyn is quasiregular semiperfectoid if R admits a map from a perfectoid ringand the Frobenius on R/p is surjective. We let QRSPerfd ⊂ QSyn be the full subcategoryspanned by quasiregular semiperfectoid rings. If R is additionally an F p -algebra, then R iscalled quasiregular semiperfect. For future reference, we will also need the relative versions qSyn A and Q Syn A of the quasisyntomicsites (of which the first is considered in [BMS19]). Definition 5.5 (Relative quasisyntomic sites, cf. [BMS19, Sec. 4.5]) . Fix a quasisyntomic ring A ∈ QSyn. We define the sites qSyn A and Q Syn A as follows.(1) We let Q Syn A denote the category of A -algebras B which are quasisyntomic as underlyingrings and such that L B/A ∈ D ( B ) has p -complete Tor-amplitude in [ − , A ⊂ Q Syn A be the full subcategory spanned by the quasisyntomic A -algebras (i.e., those B suchthat B is additionally p -completely flat over A ).(2) We make Q Syn A and qSyn A into sites by declaring a cover to be a map which is a cover inQSyn.(3) We let Q RSPerfd A (resp. qrsPerfd A ) denote the subcategory of Q Syn A (resp. qSyn A ) spannedby A -algebras whose underlying ring is quasiregular semiperfectoid. Note that if B ∈ Q RSPerfd A ,then the p -completion of L B/A [ −
1] is a p -completely flat, discrete B -module by [BMS19,Lem. 4.7(1)]Note that in the case A = Z p , qSyn Z p is the category of p -torsion free quasisyntomic rings and Q Syn Z p = QSyn. For A = F p , Q Syn F p and qSyn F p are both simply the subcategory of QSyn spannedby those quasisyntomic rings which are F p -algebras [BMS19, Lemma 4.34]; more generally Q Syn A isthe category of A -algebras which are quasisyntomic for any perfectoid ring A .The site QSyn has a basis given by QRSPerfd, and similarly in the relative cases. All the functorsbelow will be sheaves on QSyn; to describe them, it therefore suffices to describe them as sheaves onQRSPerfd [BMS19, Prop. 4.31].We now review the prismatic sheaves on QSyn, constructed via topological Hochschild and cyclichomology. A purely algebraic construction via the prismatic cohomology of Bhatt–Scholze is given in [BS19] (at least for algebras over a base perfectoid ring), which also produces objects before Nygaardcompletion. Definition 5.6 (Prismatic sheaves on QSyn, [BMS19, Sec. 7]) . The objects b ∆ R { i } and N ≥ n b ∆ R definesheaves on QSyn with values in D ( Z p ) ≥ . Each of these sheaves is constructed via descent [BMS19,Prop. 4.31] from QRSPerfd ⊂ QSyn, on which they take discrete values defined via topologicalHochschild homology.(1) For R ∈ QRSPerfd, THH( R ; Z p ) is concentrated in even degrees, so the homotopy fixedpoint and Tate spectral sequences for TC − ( R ; Z p ) and TP( R ; Z p ) degenerate and TP( R ; Z p )is 2-periodic. For R ∈ QRSPerfd, we have b ∆ R = π (TC − ( R ; Z p )) = π (TP( R ; Z p )) . (29)(2) For R ∈ QRSPerfd and n ∈ Z , the ideal N ≥ n b ∆ R ⊂ b ∆ R is the one defined by the homotopyfixed point spectral sequence, i.e., N ≥ n b ∆ R = im (cid:16) π (( τ ≥ n THH( R ; Z p ) hS ) → π (THH( R ; Z p ) hS ) (cid:17) . (30)(3) For i ∈ Z we further have the invertible b ∆ -modules (as sheaves on QSyn) b ∆ { i } , called Breuil–Kisin twists . For R ∈ QRSPerfd, b ∆ R { i } = π i (TP( R ; Z p )) , (31)and by 2-periodicity b ∆ R { i } = b ∆ R { } ⊗ i . We have a natural isomorphism N ≥ i b ∆ R { i } ≃ π i (TC − ( R ; Z p )) . (32)More generally, N ≥ n + i b ∆ R { i } is the image of the injection π i (( τ ≥ n +2 i THH( R ; Z p )) hS ) → π i ((THH( R ; Z p )) hS ) for n ∈ Z .(4) There are two maps of sheaves on QSyn,can , ϕ i : N ≥ i b ∆ R { i } ⇒ b ∆ R { i } (33)arising from the canonical and Frobenius maps TC − ( R ; Z p ) ⇒ TP( R ; Z p ); in particular, weobtain an endomorphism ϕ = ϕ : b ∆ R → b ∆ R .(5) Finally, the map TC − ( R ; Z p ) → R yields a projection map a R : b ∆ R → R , a surjection withkernel N ≥ b ∆ R ⊂ b ∆ R . Example 5.7 (Perfectoid rings, [BMS19, Sec. 6]) . Let R be a perfectoid ring. In this case, wehave Fontaine’s ring A inf ( R ) = W ( R ♭ ) and the surjective map θ : A inf ( R ) → R , whose kernel is aprincipal ideal generated by a nonzerodivisor ξ ∈ A inf ( R ); θ is the universal pro-nilpotent, p -completethickening of R . We have a canonical isomorphism b ∆ R ≃ A inf ( R ) such that the projection map a R : b ∆ R → R is θ . There are also non-canonical isomorphisms b ∆ R { i } ≃ A inf ( R ) for each i . Themap ϕ = ϕ : b ∆ R → b ∆ R is given by the Witt vector Frobenius on A inf ( R ). The Nygaard filtrationon b ∆ R = A inf ( R ) is the ξ -adic filtration. The map ϕ i is injective (and ϕ -semilinear) and its imageis given by ( e ξ − min( i, ) for e ξ = ϕ ( ξ ). Definition 5.8 (The sheaves Z p ( i )) . For i ≥
0, the sheaf Z p ( i ) is defined as the homotopy equalizerof can , ϕ i , i.e., via Z p ( i ) ≃ fib (cid:16) N ≥ i b ∆ R { i } can − ϕ i −−−−−→ b ∆ R { i } (cid:17) . Consequently, since TC is itself a homotopy equalizer, we also have for R ∈ QRSPerfd, Z p ( i )( R ) = ( τ [2 i − , i ] TC( R ; Z p )])[ − i ] . (34) N THE BEILINSON FIBER SQUARE 27
We also define quasisyntomic sheaves F p ( i ) and Q p ( i ) by reducing Z p ( i ) modulo p or inverting p on Z p ( i ), respectively.Using the result [BMS19, Sec. 3] that TC defines a sheaf for the fpqc topology on rings, one sheafifiesthe Postnikov filtration and obtains the following fundamental result. Theorem 5.9 (“Motivic filtrations,” [BMS19, Theorem 1.12]) . Let R ∈ QSyn . Then
THH( R ; Z p ) , TC − ( R ; Z p ) , TP( R ; Z p ) , and TC( R ; Z p ) naturally upgrade to filtered spectra with complete, multiplica-tive descending filtrations Fil ≥ ⋆ THH( R ; Z p ) , Fil ≥ ⋆ TC − ( R ; Z p ) , Fil ≥ ⋆ TP( R ; Z p ) , and Fil ≥ ⋆ TC( R ; Z p ) ,indexed by Z ≥ , Z , Z , and Z ≥ respectively, such that the associated graded pieces are given by (1) gr i THH( R ; Z p ) ≃ N i b ∆ R { i } [2 i ] def = (cid:16) N ≥ i b ∆ R { i } / N ≥ i +1 b ∆ R { i } (cid:17) [2 i ] for all i ≥ ; moreover, inthis case the Breuil–Kisin twists can be trivialized, so also gr i THH( R ; Z p ) ≃ N i b ∆ R [2 i ] , (2) gr i TC − ( R ; Z p ) = N ≥ i b ∆ R { i } [2 i ] for all i ∈ Z , (3) gr i TP( R ; Z p ) = b ∆ R { i } [2 i ] for all i ∈ Z , (4) gr i TC( R ; Z p ) = Z p ( i )( R )[2 i ] for all i ≥ . Remark 5.10 (Comparison with K-theory) . Recall that for p -adic rings, TC and p -adic ´etale K-theory agree in nonnegative degrees. Cf. [GH99] for smooth algebras in characteristic p , and in general[CMM18, CM19]. One may thus expect the filtration of Theorem 5.9 to be the ´etale sheafificationof the filtration on algebraic K-theory with associated graded motivic cohomology, cf. [FS02, Lev08](for smooth schemes over fields). In particular, one expects the Z p ( i ) to be some form of p -adic ´etalemotivic cohomology. This is essentially understood in equal characteristic (already by [BMS19]), aswe review below, but has not yet appeared in mixed characteristic. In mixed characteristic and underfiniteness assumptions (e.g., smooth schemes over a DVR), many authors have studied ´etale motiviccohomology [Gei04] and similar “ p -adic ´etale Tate twists,” e.g., those of [FM87, Sch94, Sat07], thoughthe construction is very different from that of [BMS19]; one ultimately hopes to compare all of them,and we will at least offer some information in this and the next section.We review the discreteness property of the Z p ( i ). By construction, the objects N ≥ i b ∆ R { i } aresheaves on QSyn with values in D ( Z p ) ≥ (recall [Lur18, Cor. 2.1.2.3] that such sheaves form thecoconnective part of the derived ∞ -category of the category of abelian sheaves on QSyn); as objectsof this category, they are in fact discrete , since they take discrete values on the basis QRSPerfd. Adeep result of Bhatt–Scholze (conjectured in [BMS19] and proved in the characteristic p case there) isthat this discreteness also holds for the Z p ( i ), although they in general take values in cohomologicaldegrees [0 ,
1] for rings in QRSPerfd.
Theorem 5.11 (Bhatt–Scholze, [BS19, Theorem 14.1]) . The D ( Z p ) ≥ -valued sheaf Z p ( i ) on QSyn is discrete and torsion free. More precisely, given R ∈ QSyn , there is a cover R → R ′ in QSyn suchthat Z p ( i )( R ′ ) is discrete and torsion free. Finally, we review the prism structure on b ∆ R , for R quasiregular semiperfectoid. For simplicity, wewill assume R to be p -torsion free. Proposition 5.12.
Let R ∈ qrsPerfd Z p . Suppose R is an algebra over the perfectoid ring R , withnotation as in Example 5.7. Then TP( R ; Z p ) / e ξ ≃ THH( R ; Z p ) tC p , and this is concentrated in evendegrees and p -torsion free.Proof. We have that TP( R ; Z p ) /ξ ≃ HP(
R/R ; Z p ) by [BMS19, Theorem 6.7]. Since R is p -torsion free and quasiregular semiperfectoid, we find thatHP( R/R ; Z p ) is concentrated in even degrees and is p -torsion free, where it is given by Hodge-complete derived de Rham cohomology by [BMS19, Prop. 5.15]. In particular, it follows that ( ξ, p )defines a regular sequence on b ∆ R . Since b ∆ R is complete with respect to this ideal, it follows that ( p, ξ )is a regular sequence; since ξ p ≡ e ξ (mod p ), we get that ( p, e ξ ) is a regular sequence, and hence so is( e ξ, p ). Now, the equivalence TP( R ; Z p ) / e ξ ≃ THH( R ; Z p ) tC p is [BMS19, Prop. 6.4], from which theremainder now follows. (cid:3) Construction 5.13 (The prismatic structure on b ∆ R , cf. [BS19, Sec. 13]) . Let R ∈ qrsPerfd Z p .Suppose R is an algebra over the perfectoid ring R . Then the ring b ∆ R = π (TP( R ; Z p )) has thestructure of a prism (in the sense of [BS19]).(1) We have the endomorphism ϕ = ϕ , which is congruent to the Frobenius modulo p , by [BS19,Sec. 13], and thus defines a δ -structure on b ∆ R .(2) We have the ideal I ⊂ b ∆ R given by I = ( e ξ ); I is the kernel of π (TP( R ; Z p )) → π (THH( R ; Z p ) tC p )and therefore does not depend on the choice of R .Finally, there is a natural map η R : R → b ∆ R /I, given via the cyclotomic Frobenius THH( R ; Z p ) → THH( R ; Z p ) tC p = TP( R ; Z p ) / e ξ upon applying π . Remark 5.14.
In fact, by [BS19, Theorem 13.1], b ∆ R is the Nygaard completion of the absoluteprismatic cohomology of R , although we will not need this fact.5.2. Relative
THH and its filtration.
In this subsection and the next, we will prove connectiv-ity bounds for the motivic filtration on THH. We will prove that for any R ∈ QSyn, we haveFil ≥ n THH( R ; Z p ) ∈ Sp ≥ n and N n b ∆ R ∈ D ≤ n ( Z ). It is not difficult to deduce the above connectivitybound in the case R is an algebra over a fixed perfectoid ring, using methods as in [BMS19, Sec. 6–7];see in particular [BMS19, Const. 7.4]. To verify the connectivity bound in the general case, we willuse additionally a fiber sequence which arises from the work of Krause–Nikolaus [KN19], which givesa comparison between relative and absolute THH.Let O K denote a complete discrete valuation ring of mixed characteristic (0 , p ) with perfect residuefield k ; let π ∈ O K be a uniformizer. The primary case of interest is O K = Z p and π = p . Construction 5.15 (Relative topological Hochschild homology) . Let R ∈ Q Syn O K . We considerthe E ∞ -ring S [ z ] and consider R as an S [ z ]-algebra via z π . Using this, we can form the relativetopological Hochschild homology (with p -adic coefficients) THH( R/S [ z ]; Z p ). The construction R THH(
R/S [ z ]; Z p ) defines a sheaf of spectra on Q Syn O K , thanks to [BMS19, Sec. 3] and (36) below.We observe the following two comparisons for relative THH.(1) When we base-change along the map S [ z ] → S where z
0, we find thatTHH(
R/S [ z ]; Z p ) ⊗ O K k ≃ THH( R ⊗ L O K k ; Z p ) . (35)(2) We have an equivalenceTHH( R/S [ z ]; Z p ) ⊗ THH( O K /S [ z ]; Z p ) O K ≃ HH( R/ O K ; Z p ) . (36)Thus, THH( R/S [ z ]; Z p ) is a deformation of Hochschild homology relative to O K .Next, we need an analog of the Hochschild–Kostant–Rosenberg theorem for relative THH (in theabsolute case for algebras over a perfectoid ring, this is [Hes96, Theorem B] and [BMS19, Cor. 6.9]). N THE BEILINSON FIBER SQUARE 29
Proposition 5.16.
Let R be a formally smooth O K -algebra. Then we have a natural isomorphism ofgraded rings THH ∗ ( R/S [ z ]; Z p ) ≃ \ Ω ∗ R/ O K [ σ ] , | σ | = 2 , where \ Ω ∗ R/ O K denotes the p -completion of the de Rham complex of R over O K .Proof. In the case R = O K , this follows from B¨okstedt’s calculation of THH( F p ), cf. [BMS19, Prop.11.10], [AMN18, Theorem 3.5], or [KN19, Theorem 3.1]. Now (36) shows that THH( R/S [ z ]; Z p ) /σ ≃ HH( R/ O K ; Z p ), and the Hochschild–Kostant–Rosenberg theorem yields HH ∗ ( R/ O K ; Z p ) ≃ \ Ω ∗ R/ O K .It remains to show that the induced Bockstein spectral sequence for THH( R/S [ z ]; Z p ) (with respectto taking the cofiber of σ ) degenerates, or equivalently that the map of the HKR isomorphism liftsto a map \ Ω ∗ R/ O K → THH(
R/S [ z ]; Z p ). Indeed, since THH( R/S [ z ]; Z p ) is an E ∞ -algebra withan S -action receiving a map from THH( O K /S [ z ]; Z p ), we obtain the structure of a commutativedifferential graded algebra on THH ∗ ( R/S [ z ]; Z p ), and it receives a map (of cdgas) from O K [ σ ] withtrivial differential. The universal property of the de Rham complex now produces the desired map \ Ω ∗ R/ O K → THH(
R/S [ z ]; Z p ). (cid:3) Left Kan extending from finitely generated polynomial O K -algebras, we obtain the following result,which is proved exactly as in [BMS19, Prop. 7.5]; the key point is that any R ∈ Q RSPerfd O K has theproperty that the p -completion of L R/ O K is the shift of a p -completely flat R -module. Corollary 5.17. If R ∈ Q RSPerfd O K , then THH(
R/S [ z ]; Z p ) is concentrated in even degrees, andeach π n THH(
R/S [ z ]; Z p ) is a p -completely flat R -module. Furthermore, the R -module π n THH(
R/S [ z ]; Z p ) admits a finite increasing filtration with graded pieces the (discrete and p -completely flat) R -modules \ ( V j L R/ O K )[ − j ] for j ≤ n , where \ V j L R/ O K denotes the p -completion of V j L R/ O K . Construction 5.18 (The filtration on relative THH) . Let R ∈ Q Syn O K . In [BMS19, Sec. 11], a mul-tiplicative, convergent Z ≥ -indexed filtration Fil ≥ ⋆ THH(
R/S [ z ]; Z p ) on THH( R/S [ z ]; Z p ) in sheavesof spectra on Q Syn O K is defined. This filtration is defined such that it restricts to the double speedPostnikov filtration for R ∈ Q RSPerfd O K , i.e., Fil ≥ n THH(
R/S [ z ]; Z p ) = τ ≥ n THH(
R/S [ z ]; Z p ) forsuch R . By Corollary 5.17 and [BMS19, Theorem 3.1], the associated gradeds of the Postnikov filtra-tion on THH( − /S [ z ]; Z p ) on Q RSPerfd O K are sheaves; thus, one unfolds and obtains the filtrationfor all R ∈ Q Syn O K . Corollary 5.19.
Let R ∈ Q Syn O K . Then gr n THH(
R/S [ z ]; Z p ) admits a finite increasing filtrationwith associated graded ( \ V j L R/ O K )[2 n − j ] for j ≤ n . In particular, we find that gr n THH(
R/S [ z ]; Z p ) ∈ Sp ≥ n and Fil ≥ n THH(
R/S [ z ]; Z p ) ∈ Sp ≥ n . Furthermore, the constructions R gr n THH(
R/S [ z ]; Z p ) and R Fil ≥ n THH(
R/S [ z ]; Z p ) (as functors on Q Syn O K to p -complete spectra) are left Kan extended from finitely generated p -complete polynomial O K -algebras.Proof. The first assertion follows from Corollary 5.17 by unfolding; the connectivity assertions thenfollow in turn. Since the cotangent complex and its wedge powers are left Kan extended from finitelygenerated polynomial algebras, the last assertion follows too. (cid:3) Actually, in loc. cit , the filtration is defined only on those objects which are flat over O K , but the arguments donot require this. Preliminary connectivity bounds.
We use the spectral sequence of Krause–Nikolaus [KN19]to obtain a relationship between the relative and absolute THH.
Proposition 5.20 (Relative versus absolute THH) . If R ∈ Q RSPerfd O K , then there exist natu-ral surjective maps f n : π n THH(
R/S [ z ]; Z p ) → π n − THH(
R/S [ z ]; Z p ) and natural isomorphisms π n THH( R ; Z p ) ≃ ker( f n ) .Proof. Recall that both THH( R ; Z p ) and THH( R/S [ z ]; Z p ) are concentrated in even degrees since R ∈ Q RSPerfd O K (see [BMS19, Theorem 7.1] and Corollary 5.17). Therefore, the result followsdirectly from [KN19, Prop. 4.1]; the spectral sequence of loc. cit. must degenerate after the firstdifferential, and the maps of the first differential must be surjective or one would have odd degreecontributions to THH( R ; Z p ). (cid:3) The following fiber sequence (37) will be the basic tool in obtaining connectivity bounds on thefiltration on THH and its variants.
Corollary 5.21 (Connectivity of the filtration on THH) . If R ∈ Q Syn O K , then for each n there is anatural fiber sequence gr n THH( R ; Z p ) → gr n THH(
R/S [ z ]; Z p ) → gr n − THH(
R/S [ z ]; Z p )[2] . (37) In particular, we have gr n THH( R ; Z p ) , Fil ≥ n THH( R ; Z p ) ∈ Sp ≥ n for any R ∈ Q Syn O K . Finally,the functors R gr n THH( R ; Z p ) and R Fil ≥ n THH( R ; Z p ) on QSyn are left Kan extended fromfinitely generated p -complete polynomial Z p -algebras, as functors to p -complete spectra.Proof. The fiber sequence follows from Proposition 5.20 by unfolding in R . The connectivity assertionfor gr n THH( R ; Z p ) then follows from Corollary 5.19; the assertion for Fil ≥ n THH( R ; Z p ) then followssince the filtration is complete. The Kan extension assertion for Fil ≥ n THH( R ; Z p ) also follows fromthe one for Fil ≥ n THH(
R/S [ z ]; Z p ) as in Corollary 5.19 (taking O K = Z p ). (cid:3) Corollary 5.22 (Connectivity bounds for N i b ∆ R ) . (1) If R ∈ QSyn , then N n b ∆ R ∈ D ≤ n ( Z p ) . (2) If R → R ′ is a surjective map in QSyn , then fib( N n b ∆ R → N n b ∆ R ′ ) ∈ D ≤ n ( Z p ) .Proof. Part (1) is a special case of Corollary 5.21 (take O K = Z p and π = p ).For part (2), note that the hypothesis implies that R → R ′ induces a surjection on H of p -completed cotangent complexes over Z p , and similarly on any wedge power. It then follows from Corol-lary 5.19 that fib(gr n THH(
R/S [ z ]; Z p )[ − n ] → gr n THH( R ′ /S [ z ]; Z p )[ − n ]) ∈ D ≤ n ( Z p ), whence weconclude by (37). (cid:3) Our main general connectivity bound is Proposition 5.25 below. To formulate it, we need to be ableto twist the ideal I ⊂ b ∆ R from Construction 5.13. First, we observe that this ideal is also trivializedafter base-change along a R . Lemma 5.23.
Let R ∈ qrsPerfd Z p , and let I ⊂ b ∆ R denote the ideal defining the prism structure. Thenthere is a natural isomorphism I ⊗ b ∆ R R ≃ R , i.e., the ideal is naturally trivialized after base-changealong a R : b ∆ R → R .Proof. Observe that the base-change I ⊗ b ∆ R R defines a functorial choice of invertible R -module, forany R ∈ qrsPerfd Z p . By faithfully flat descent, we obtain for any R ∈ qSyn Z p a choice of invertible R -module, which is functorial in R . Choosing a trivialization over R = Z p , we obtain a functorialtrivialization everywhere. (cid:3) N THE BEILINSON FIBER SQUARE 31
Definition 5.24 (Twisting by I ) . For s, i, n ≥
0, we let R I s N ≥ n b ∆ R { i } denote the D ( Z p ) ≥ -valuedsheaf on qSyn Z p defined by unfolding the discrete sheaf on qrsPerfd Z p defined by the aforementionedformula, for I ⊂ b ∆ R the ideal defining the prismatic structure. For R ∈ qrsPerfd Z p , since I defines aCartier divisor in b ∆ R , we have I s N ≥ n b ∆ R { i } ≃ I s ⊗ b ∆ R N ≥ n b ∆ R { i } . Proposition 5.25 (Connectivity of Nygaard quotients) . Let R ∈ qSyn Z p and i, n, s ≥ . Then thecofiber I s b ∆ R { i } /I s N ≥ n b ∆ R { i } belongs to D ≤ n − ( Z p ) . Moreover, this cofiber is left Kan extended fromfinitely generated p -complete polynomial Z p -algebras.Proof. By d´evissage it suffices to show that I s N n b ∆ R { i } ∈ D ≤ n ( R ) for each n ≥ p -complete polynomial Z p -algebras. Here we write I s N n b ∆ R { i } for the unfolding from qrsPerfd Z p of I s ⊗ b ∆ R N n b ∆ R { i } . However, the twists here are trivialized byLemma 5.23 since N n b ∆ R is an R -module, so that I s N n b ∆ R { i } ≃ N n b ∆ R . Thus, the result follows fromCorollary 5.22. (cid:3) We finish this subsection by recording a connectivity bound that depends on the number of gener-ators of the cotangent complex (we will not use this result in the paper, but note that it implies inparticular that b ∆ O K ∈ D ≤ ( Z p )). Lemma 5.26.
Let R be a commutative ring, and let M ∈ D ≤ ( R ) . Suppose H ( M ) is generated by d elements. Then for all j , ( V j M )[ − j ] ∈ D ≤ d ( R ) .Proof. The result is clear if M = R d itself. In general, we have a map R d → M inducing a surjectionon H , so the cofiber F of the map satisfies F ∈ D ( R ) ≤− . It follows that V j ′ F [ − j ′ ] ∈ D ( R ) ≤ forall j ′ by standard connectivity estimates (see [Lur18, Sec. 25.2.4] for an account). Using the naturalfiltration on V j M [ − j ] with associated graded terms V j ′ F [ − j ′ ] ⊗ R V j − j ′ R d [ − ( j − j ′ )], the resulteasily follows. (cid:3) Proposition 5.27.
Let R ∈ Q Syn O K , let n, i ≥ , and suppose that H ( \ L R/ O K ) is generated by d elements. Then N n b ∆ R and N ≥ n b ∆ R { i } lie in D ≤ d +1 ( R ) .Proof. By Corollary 5.19, gr n THH(
R/S [ z ]; Z p ) has a finite filtration with graded pieces \ V j L R/ O K [2 n − j ] for 0 ≤ j ≤ n . By Lemma 5.26, we find that gr n THH(
R/S [ z ]; Z p ) ∈ D ≤ d − n ( R ). Using the fibersequence (37), we find now that gr n THH( R ; Z p ) ∈ D ≤ d − n +1 ( R ). Shifting by 2 n the result nowfollows for N n b ∆ R . The same connectivity bound then follows for each N ≥ n b ∆ R { i } / N ≥ n + r b ∆ R { i } byd´evissage, and then for N ≥ n b ∆ R { i } by passing to the limit. (cid:3) Frobenius nilpotence on b ∆ R /p , and proof of Theorem 5.1 (2). In this subsection, werecord some results about the contracting property of Frobenius on b ∆ R /p and use it to prove partof Theorem 5.1. If R ∈ qrsPerfd Z p is a p -torsion free quasiregular semiperfectoid ring, then both b ∆ R and all graded steps N n b ∆ R of the Nygaard filtration are p -torsion free (e.g., because THH ∗ ( R ; Z p ) is p -torsion free and concentrated in even degrees). For i, r ≥
0, we will consider the mapscan , ϕ i : N ≥ i + r b ∆ R { i } /p → b ∆ R { i } /p. (38)and show that both maps respect the I -adic filtration from Definition 5.24, with ϕ i inducing the zeromap on associated graded pieces in positive degrees (Proposition 5.30). We will show in addition thatcan − ϕ i induces an automorphism of N ≥ i + r b ∆ R { i } /p for r ≫ Proposition 5.28.
Let R ∈ qrsPerfd Z p and i, r ≥ . Then the map ϕ i : N ≥ i b ∆ R { i } → b ∆ R { i } carries N ≥ i + r b ∆ R { i } into I r b ∆ R { i } .Proof. Let R be a perfectoid ring mapping to R and fix ξ, e ξ ∈ A inf ( R ) as usual. Then [BMS19,Sec. 6] we have an isomorphism TC −∗ ( R ; Z p ) ≃ A inf ( R )[ u, v ] / ( uv − ξ ) for | u | = 2 , | v | = −
2. In thiscase, the filtration on N ≥ i b ∆ R { n } ≃ π i (TC − ( R ; Z p )) is the filtration by powers of v : N ≥ i + r b ∆ R { i } = v r π i +2 r TC − ( R ; Z p ) ⊂ π i TC − ( R ; Z p ) . But (as in loc. cit. ) the cyclotomic Frobenius carries v to a multiple of ϕ ( ξ ) = e ξ in π − TP( R ; Z p );recalling that I = ( ˜ ξ ), the result follows. (cid:3) Construction 5.29 (The I -adic filtrations modulo p ) . Let R ∈ qrsPerfd Z p . For each i, s, r ≥
0, themap ϕ i : N ≥ i + r b ∆ R { i } /p → b ∆ R { i } /p is Frobenius semi-linear by Construction 5.13(1), and so carries I s ( N ≥ i + r b ∆ R { i } /p ) to I ps + r ( b ∆ R { i } /p ) by Proposition 5.28. But we have seen in Proposition 5.12that ( p, ˜ ξ ) and ( ˜ ξ, p ) are regular sequences on b ∆ R , whence the canonical maps are isomorphisms I ⊗ b ∆ R b ∆ R /p ≃ I ( b ∆ R /p ) ≃ I/p , and similarly for any power of I and Breuil–Kisin twist of b ∆ R . We thusget maps can , ϕ i : I s ⊗ b ∆ R N ≥ i + r b ∆ R { i } /p → I s ⊗ b ∆ R b ∆ R { i } /p. (39)For convenience, we record what we have proved about the interaction of the Frobenius and the I -adic filtration, as it will be used to prove Proposition 5.35: Proposition 5.30 (The canonical and Frobenius map are I -adically filtered modulo p ) . Let R ∈ qrsPerfd Z p , and let i, r ≥ . The maps (38) upgrade to the structure of filtered maps with respect tothe I -adic filtrations on both sides, i.e., there are compatible maps for each s ≥ , can , ϕ i : I s ⊗ b ∆ R N ≥ i + r b ∆ R { i } /p → I s ⊗ b ∆ R b ∆ R { i } /p. Furthermore, the map ϕ i induces the zero map on associated graded pieces unless s = r = 0 .Proof. In Construction 5.29 we constructed the maps and showed that in fact ϕ i has image in I ps + r ⊗ b ∆ R b ∆ R { i } /p . (cid:3) For the moment we need the following consequence of our arguments.
Corollary 5.31 (The Nygaard filtrations modulo p ) . Let R ∈ qrsPerfd Z p and i ≥ . For r ≫ (in-dependent of R ), the map can − ϕ i : N ≥ i b ∆ R { i } /p → b ∆ R { i } /p carries N ≥ i + r b ∆ R { i } /p isomorphicallyonto itself. Consequently, for such r , one has a natural isomorphism F p ( i )( R ) ≃ fib (cid:16) can − ϕ i : ( N ≥ i b ∆ R { i } / N ≥ i + r b ∆ R { i } ) /p → ( b ∆ R { i } / N ≥ i + r b ∆ R { i } ) /p (cid:17) . (40) Proof.
This is [BMS19, Lemma 7.22]. By Proposition 5.28 above (and choosing as usual a perfectoidring R mapping to R ), we find that ϕ i carries N ≥ i + r b ∆ R { i } /p (equivalently, v r N ≥ i + r b ∆ R { i + r } /p )into multiples of e ξ r b ∆ R { i } /p . Since we are working modulo p , we have e ξ r b ∆ R { i } /p = ξ rp b ∆ R { i } /p ⊂N ≥ rp b ∆ R { i } /p . For r ≫
0, this is contained in N ≥ i + r +1 b ∆ R { i } /p , whence we have shown that ϕ i carries N ≥ i + r b ∆ R { i } /p to N ≥ i + r +1 b ∆ R { i } /p .It follows that can − ϕ i carries N ≥ i + r b ∆ R { i } /p into itself, and it differs from the identity bya topologically nilpotent endomorphism of N ≥ i + r b ∆ R { i } /p with respect to the Nygaard filtration.Therefore it is an isomorphism and the result follows. (cid:3) N THE BEILINSON FIBER SQUARE 33
Proposition 5.32 (A criterion for being left Kan extended) . Let
F, G : QSyn → D ( Z p ) be p -completequasisyntomic sheaves equipped with complete descending Z ≥ -indexed filtrations Fil ≥ ⋆ F and Fil ≥ ⋆ G .Let F → G be a map of functors (not necessarily filtration-preserving). If (1) for R ∈ qrsPerfd Z p , the objects gr r F ( R ) and gr r G ( R ) are discrete, p -complete, and p -torsionfree (and therefore so are F ( R ) , G ( R ) ), (2) each of the associated graded terms gr r F and gr r G is left Kan extended from finitely generated p -complete polynomial Z p -algebras to the p -complete derived category, and (3) there exists N such that for r ≥ N and for R ∈ qrsPerfd Z p , the map F ( R ) /p → G ( R ) /p carries Fil ≥ r F ( R ) /p isomorphically to Fil ≥ r G ( R ) /p ,then fib( F → G ) is p -completely left Kan extended from finitely generated p -complete polynomial Z p -algebras.Proof. It suffices to check that fib( F → G ) /p is left Kan extended from finitely generated p -completepolynomial algebras. Let F ′ , G ′ denote the functors on QSyn obtained by restricting F, G to finitelygenerated p -complete polynomial algebras and then left Kan extending to QSyn, with their left Kanextended filtrations. Our assumptions imply that F, G are the respective completions of F ′ , G ′ withrespect to their filtrations.It suffices to check that the natural map induces an equivalence fib( F ′ → G ′ ) /p ≃ fib( F → G ) /p .We claim that for any R ∈ QSyn, there are natural commutative diagrams, compatible in r ≥ N ,Fil ≥ r F ′ ( R ) /p (cid:15) (cid:15) ≃ / / Fil ≥ r G ′ ( R ) /p (cid:15) (cid:15) F ′ ( R ) /p / / G ′ ( R ) /p. (41)In fact, it suffices to prove this by left Kan extension for R finitely generated p -complete polynomialover Z p , and then we can replace F ′ , G ′ by F, G . By descent for
F, G , we can then reduce to R ∈ qrsPerfd Z p , whence we have the desired diagrams by hypothesis.Using the diagrams (41), we find that there is a natural commutative diagram F ′ ( R ) /p (cid:15) (cid:15) / / G ′ ( R ) /p (cid:15) (cid:15) F ( R ) /p / / G ( R ) /p which is homotopy cartesian (taking the inverse limit over r ). We finally find that fib( F → G ) =fib( F ′ → G ′ ), which is left Kan extended from finitely generated p -complete polynomial algebras asdesired. (cid:3) Proof of Theorem 5.1(2).
We show that R Z p ( i )( R ), as a functor on QSyn, is left Kan ex-tended from finitely generated p -complete polynomial Z p -algebras. Since Z p ( i )( R ) = fib(can − ϕ i : N ≥ i b ∆ R { i } → b ∆ R { i } ), we will apply Proposition 5.32 with F = N ≥ i b ∆ { i } , G = b ∆ { i } using theNygaard filtrations R
7→ N ≥ i + r b ∆ R { i } on QSyn. Indeed, the associated graded terms for the Nygaardfiltration are left Kan extended from finitely generated p -complete polynomial algebras (Proposi-tion 5.25), and they are torsion free on R ∈ qrsPerfd Z p . The last hypothesis follows from Corol-lary 5.31. Then Proposition 5.32 gives that the Z p ( i ) are left Kan extended from finitely generated p -complete polynomial algebras as desired. Since R TC( R ; Z p ) is left Kan extended from finitelygenerated p -complete polynomial rings (by [CMM18, Theorem G] and since TC( − ; Z p ) commutes with geometric realizations on simplicial commutative rings), it follows inductively that the constructions R Fil ≥ i TC( R ; Z p ) are also left Kan extended from finitely generated p -complete Z p -algebras. (cid:3) For future reference, we observe that we can obtain a motivic filtration on TC for any simplicialcommutative ring, by left Kan extension.
Construction 5.33 (Left Kan extending to SCR) . We have seen that the functor which sends R ∈ QSyn to the filtration Fil ≥ ⋆ TC( R ; Z p ) is left Kan extended from finitely generated p -completepolynomial algebras. Thus, we can left Kan extend to all p -complete simplicial commutative ringsto obtain a functor R Fil ≥ ⋆ TC( R ; Z p ) , from SCR to p -complete filtered spectra, which commuteswith sifted colimits. We define functors Z p ( i ) on SCR as Z p ( i )( R ) = gr i TC( R ; Z p )[ − i ], or in otherwords by left Kan extending Z p ( i ) from finitely generated p -complete polynomial algebras.Once we complete the proof of Theorem 5.1 in the next subsection, it will follow that Fil ≥ i TC( R ; Z p )and Z p ( i ) belong to Sp ≥ i − for each i , by left Kan extending the connectivity estimate from thequasisyntomic case.We also emphasize that the above proof of Theorem 5.1(2) shows the following. Given i ≥
0, thereis r ≫ p -complete rings R there is a natural expression F p ( i )( R ) ≃ fib (cid:16) can − ϕ i : ( N ≥ i b ∆ R { i } / N ≥ i + r b ∆ R { i } ) ⊗ L Z F p → ( b ∆ R { i } / N ≥ i + r b ∆ R { i } ) ⊗ L Z F p (cid:17) , (42)where the two Nygaard quotients on the right side are defined by left Kan extension from finitelygenerated p -complete polynomial algebras.5.5. Proofs of the connectivity bounds (Theorem 5.1(1)) for the Z p ( i ) . In this subsection,we complete the proof of Theorem 5.1.
Lemma 5.34 (Connectivity lemma) . Let can , ϕ : Fil ≥ ⋆ M → Fil ≥ ⋆ N be maps of filtered objects in D ( Z ) (with underlying objects M, N ). Suppose that (1) both filtrations are complete, (2) ϕ induces the zero map on associated graded pieces, and (3) there is a fixed r such that, for each s , the induced map can : Fil ≥ s M → Fil ≥ s N has fiber in D ( Z ) ≤ r .Then can − ϕ : M → N has fiber in D ( Z ) ≤ r .Proof. The fiber fib(can − ϕ : M → N ) acquires the natural structure of a filtered spectrum, sincecan , ϕ are filtered maps. On graded pieces, we find gr s fib(can − ϕ : M → N ) ≃ gr s fib(can : M → N )since ϕ vanishes on associated graded terms. In particular, the associated graded terms belong to D ( Z ) ≤ r . Since the filtration on fib(can − ϕ ) is complete, the connectivity assertion on the fiber nowfollows from the analogous assertion on associated graded terms. (cid:3) Proposition 5.35 (The Z p ( i ) connectivity bound for qSyn Z p ) . Let R ∈ qSyn Z p . Then Z p ( i )( R ) ∈ D ≤ i +1 ( Z p ) for each i ≥ .Proof. First, we recall from the proof of Proposition 5.25 that the inclusion N ≥ i +1 b ∆ R { i } → N ≥ i b ∆ R { i } has cofiber in D ≤ i ( Z p ). Using the resulting cofiber sequence, it thus suffices to show that the fiber ofcan − ϕ i : N ≥ i +1 b ∆ R { i } → b ∆ R { i } belongs to D ≤ i +1 ( Z p ). Since everything is p -complete, it suffices tocheck this with mod p coefficients.We consider the two maps can , ϕ i : N ≥ i +1 b ∆ R { i } /p → b ∆ R { i } /p. N THE BEILINSON FIBER SQUARE 35
Unfolding Proposition 5.30 shows that these upgrade to maps can , ϕ i : I s N ≥ i +1 b ∆ R { i } /p → I s b ∆ R { i } /p for all s ≥
0, i.e., of I -adically filtered objects, and that the map ϕ i acts trivially on associated gradedpieces. Furthermore, for each s ≥
0, the fiber of can : I s N ≥ i +1 b ∆ R { i } /p → I s b ∆ R { i } /p belongs to D ≤ i +1 ( Z p ) by Proposition 5.25. Lemma 5.34 (whose hypothesis (1) is satisfied by ˜ ξ -adic completenessin the case of R ∈ qrsPerfd Z p ) now shows that can − ϕ i : N ≥ i +1 b ∆ R { i } → b ∆ R { i } belongs to D ≤ i +1 ( Z p ),as desired. (cid:3) Proof of Theorem 5.1(1).
We wish to show that Z p ( i ) ∈ D ≤ i +1 ( Z p ). But we have already proved part(2) of Theorem 5.1, so the problem reduces to the case of finitely generated p -completely polynomialrings over Z p , which is covered by Proposition 5.35. (cid:3) Rigidity, and proof of Theorem 5.2.
In this subsection, we give a proof of Theorem 5.2. Ourstrategy is to first prove a continuity statement, after which N´eron–Popescu and left Kan extensionarguments reduce the general case to that of a square-zero extension. In that case we use the automaticgradings that exist and argue with the pro-nilpotence of Frobenius. Recall that a ring R is said tobe F-finite if R/p is finitely generated over its subring of p th -powers. The next result is an analog ongraded pieces of [DM17, Th. 4.5]. Proposition 5.36 (Continuity) . Let R be a noetherian, F-finite, p -complete ring, and I ⊆ R anideal such that R is I -adically complete. Then the natural map Z p ( i )( R ) → lim ←− s Z p ( i )( R/I s ) is anequivalence for any i ≥ .Proof. From Definition 5.8 and completeness of the Nygaard filtration, it is enough to prove theanalogous continuity for each N n b ∆ { i } ≃ gr n THH( − ; Z p ). Then Corollary 5.21 reduces us furtherto continuity for each gr n THH( − /S [ z ]; Z p ), and finally the filtration of Corollary 5.19 reduces theproblem to continuity for each \ V n L − / Z p .A cell attachment lemma of Andr´e and Quillen (see [Mor18, Th. 4.4(i)] for a presentation in thiscontext) shows that all cohomology groups of all wedge powers of L ( R/I s ) /R are pro zero in s (except H of V L ). So the transitivity filtration shows that V n L R/ Z p ⊗ LR R/I s → V n L ( R/I s ) / Z p is apro isomorphism on all cohomology groups. This reduces the problem to showing that V n L R/ Z p → lim ←− s V n L R/ Z p ⊗ LR R/I s is an equivalence after p -adic completion. Since R has bounded p -power torsion,this derived p -adic completion may be equivalently computed as lim ←− r ( − ⊗ LR R/p r ). Exchanging thelimits, it is enough to show that M ≃ lim ←− s M ⊗ LR R/I s , where M = V n L R/ Z p ⊗ LR R/p r for any n ≥ r ≥
1. But this follows from the facts that R is noetherian, that M is bounded above(cohomologically), and that its cohomology groups are finitely generated R -modules [DM17]. (cid:3) Remark 5.37.
As usual, one can prove stronger continuity statements when I = ( p ). For ex-ample, given a p -complete ring R with bounded p -power torsion, we claim that Z p ( i )( R ) /p r →{ Z p ( i )( R/p s ) /p r } s is a pro equivalence (i.e., it induces an isomorphism of pro groups on all coho-mology groups) for any fixed r ≥
1. To prove this we reduce to the case r = 1 and appeal to (42)(instead of the completeness of the Nygaard filtration used at beginning of the proof of Proposi-tion 5.36) to again reduce to the analogous assertion for graded pieces of the Nygaard filtration. Thenargue as in the proof of Proposition 5.36 (the assumption that R has bounded p -power torsion impliesthat the ideal ( p ) is pro Tor-unital in the sense of [Mor18], which is needed to verify pro vanishing of V n L ( R/p s ) /R ) to reduce to the analogous statement about ( V n L R/ Z p ) /p → { ( V n L R/ Z p ) /p ⊗ LR R/p s } s ,which follows from boundedness of the p -power torsion in R . Proposition 5.38.
Theorem 5.2 holds in the special case of henselian pairs of the form R = A ⊕ N , I = N , where A is the p -completion of a finitely generated Z p -polynomial algebra, N is a finitelygenerated A -module, and R is the trivial square-zero extension of A by N . The proof of Proposition 5.38 will be given below. Using Proposition 5.38, we explain how todeduce Theorem 5.2.
Proof of Theorem 5.2.
First, note that it is equivalent to prove that the homotopy fiber of Theorem 5.2mod p belongs to D ≤ i ( Z p ), i.e., we may replace Z p ( i ) by F p ( i ) = Z p ( i ) /p .We consider the functor on Z -algebras, R F ( R ) def = F p ( i )( b R p ) (where ˆ R p denotes the derived p -completion of R ). By Theorem 5.1, the functor F commutes with filtered colimits in R . It sufficesto show that if ( R, I ) is a henselian pair, thenfib( F ( R ) → F ( R/I )) ∈ D ≤ i ( F p ) . (43)First, we prove (43) in case I ⊂ R is a nilpotent ideal. By transitivity and an easy induction,it suffices to assume I = 0. Next we apply a standard trick to reduce to the case that R → R/I is split: choose a simplicial resolution P • → R/I by polynomial Z -algebras (possibly in infinitelymany variables), and let Q • be the fiber product along R → R/I . Then each Q j → P j has kernel I and admits a section, since P j is a polynomial algebra. Taking the geometric realization usingTheorem 5.1(2), we thus reduce to proving (43) for each pair ( Q j = P j ⊕ I, I ). But we can write P j as a filtered colimit of polynomial Z -algebras on finitely many variables and I as a filtered colimit offinitely generated modules. Using Theorem 5.1(2) again (which shows that F commutes with filteredcolimits), and Proposition 5.38, we conclude (43) for I ⊂ R nilpotent.Second, we prove (43) in case R is noetherian and F -finite, and R is I -adically complete. Inthis case, Proposition 5.36 shows that F ( R ) ≃ lim ←− F ( R/I n ). We consider the tower (in n ), T n =fib( F ( R/I n ) → F ( R/I )); the fiber of each successive map T n → T n − belongs to D ≤ i ( F p ), and thuswe get that lim ←− n T n = fib( F ( R ) → F ( R/I )) ∈ D ≤ i ( F p ) as desired.Finally, suppose ( R, I ) is a general henselian pair; we prove (43). Since F commutes with filteredcolimits, it suffices to assume that the pair ( R, I ) is the henselization of a finitely generated Z -algebra R along an ideal I ⊂ R . By the previous paragraph, we have fib( F ( ˆ R I ) → F ( R/I )) ∈ D ≤ i ( F p ),since ˆ R I is an F -finite, noetherian ring. Now R is an excellent ring as a finitely generated Z -algebra;since R → R is ind-´etale, R is also excellent [Gre76]. It follows that R → b R I is geometrically regular[Gro65, 7.8.4(v)] and is therefore a filtered colimit of smooth maps by N´eron–Popescu desingularisation[Pop85, Pop86] [Sta19, Tag 07BW]); each of these maps necessarily admits a section. In particular,the map fib( F ( R ) → F ( R/I )) → fib( F ( b R I ) → F ( R/I )) is a filtered colimit of maps, each of whichadmits a section. Since we have just seen that the target of this map belongs to D ≤ i ( F p ), the sourcedoes as well. (cid:3) To complete the proof of Theorem 5.2 by treating the pairs in the statement of Proposition 5.38,we will exploit the presence of the grading induced by the variables.
Remark 5.39 (THH for graded rings) . In the remainder of this subsection we will systematically usegraded objects indexed over a commutative monoid M (which will be Z [1 /p ] ≥ or Z ≥ ): an M -gradedobject in a ( ∞ -)category C is by definition a functor R ⋆ : M → C . When C is symmetric monoidal,then so is the resulting category Fun( M, C ) of M -graded objects, under the Day convolution product,and an M -graded ring is then a ( E ∞ -, etc.) monoid object in M -graded objects.The underlying object R of a graded object R ⋆ is by definition L m ∈ M R m , when it exists. When alldirect sums exist, the functor R ⋆ R is conservative (and faithful when C is an ordinary category).We will be particularly interested in p -complete graded rings , by which we mean a graded ring R ⋆ in the category of p -complete abelian groups; the underlying object R is then the p -completed directsum cL m ∈ M R m , which is itself a p -complete ring. We sometimes abusively identify cL m ∈ M R m with R ⋆ itself; this is a mild abuse of notation given that the functor R ⋆ cL m ∈ M R m is conservative andfaithful. N THE BEILINSON FIBER SQUARE 37 An M -graded commutative ring R ⋆ may of course be viewed as an M -graded E ∞ -ring in spectra,i.e., an E ∞ -monoid in the symmetric monoidal stable ∞ -category Fun( M, Sp). By Appendix A, wemay then form the S -equivariant object THH( R ⋆ ) ∈ Fun( M, Sp) S and the associated homotopyfixed points TC − ( R ⋆ ), homotopy orbits THH( R ⋆ ) hS , and Tate construction TP( R ⋆ ), all of whichare M -graded spectra. Note that the underlying spectrum of THH( R ⋆ ) is the THH of the underlyingring spectrum of R ⋆ because the underlying spectrum functor preserves tensor products and colimits;this is not true for TC − , TP because the underlying spectrum functor need not preserve limits (e.g., S -homotopy fixed points). Construction 5.40 ([BMS19] for graded rings) . Assume that the monoid M is uniquely p -divisible,such as Z [1 /p ] ≥ . Then the main constructions and results of [BMS19] extend to M -graded rings.We will say that a p -complete M -graded ring R ⋆ is quasisyntomic (resp. quasiregular semiperfectoid )if the underlying ring R = cL m ∈ M R m is quasisyntomic (resp. quasiregular semiperfectoid). One hasa natural graded analog of the quasisyntomic site, and similarly quasiregular semiperfectoids form abasis (for example, by extracting p -power roots of homogeneous elements as in [BMS19, Lem. 4.27];this is why p -divisibility of M is required); one obtains an analog of unfolding in this context.For such R ⋆ , we have seen in Remark 5.39 that we have natural M -graded spectra, THH( R ⋆ ; Z p ),TC − ( R ⋆ ; Z p ), THH( R ⋆ ; Z p ) hS , and TP( R ⋆ ; Z p ). Moreover, the latter are naturally filtered objects in M -graded p -complete spectra, by carrying over the construction of the motivic filtration of [BMS19]to the graded context. It follows that we get M -graded p -complete objects b ∆ R ⋆ { i } , N ≥ i b ∆ R ⋆ { i } , etc.In general, the underlying ( p -complete) objects of b ∆ R ⋆ { i } , N ≥ i b ∆ R ⋆ { i } do not agree with those of b ∆ R { i } , N ≥ i b ∆ R { i } , because the underlying object of TC − ( R ⋆ ; Z p ) is not TC − ( R ; Z p ). However, foreach j ≥ i , the underlying p -complete object of N ≥ i b ∆ R ⋆ { i } / N ≥ j b ∆ R ⋆ { i } is N ≥ i b ∆ R { i } / N ≥ j b ∆ R { i } .This follows because the underlying object of THH( R ⋆ ; Z p ) is THH( R ; Z p ) and the forgetful functorfrom M -graded spectra to spectra commutes with finite homotopy limits.Throughout [BMS19], a basic tool is the cotangent complex and its wedge powers; here we im-plicitly use that if R ⋆ is a p -complete M -graded ring, then we have natural p -complete graded ob-jects V i L R ⋆ / Z p (defined in the usual manner as a left derived functor of differential forms). TheHochschild–Kostant–Rosenberg theorem remains valid in the graded context, and from there the re-sults of section 5.2 carry over to this context as well.We now turn to the interaction of the Frobenius with the grading, in the case which interests us. Proposition 5.41 (Frobenius multiplies grading by p ) . Let R ⋆ be a quasisyntomic Z [1 /p ] ≥ - (resp. Z ≥ -) graded ring (with underlying quasisyntomic ring R ). Then we claim that(1) b ∆ R { i } / N ≥ n b ∆ R { i } naturally upgrades to have the structure of a Z [1 /p ] ≥ - (resp. Z ≥ -) gradedobject in the p -complete derived ∞ -category \ D ( Z p ) (2) the Frobenius map modulo p , as in (42) ϕ i : N ≥ i b ∆ R { i } / N i + r b ∆ R { i } ⊗ L Z F p → b ∆ R { i } / N ≥ i + r b ∆ R { i } ⊗ L Z F p multiplies degrees by p .Proof. In the case in which R ⋆ is Z [1 /p ] ≥ -graded, part (1) is covered by the general Construction 5.40:arguing locally on the graded version of the quasisyntomic site, we require a natural graded version ofDefinition 5.6 for any graded quasiregular semiperfectoid ring. But this follows from THH of a gradedring being a graded spectrum with S -action, as explained in Remark 5.39.When R is actually Z ≥ -graded, then we claim that the same is true of b ∆ R { i } / N ≥ n b ∆ R { i } . Byd´evissage, it suffices to prove this for each N t b ∆ R , which in turn follows from the filtrations of Corol-lary 5.19 and Corollary 5.21. Here we use that the p -completed cotangent complex \ L R/ Z p and its wedge powers are naturally Z ≥ -graded, and the filtrations of the aforementioned corollaries respectthese gradings.For part (2) we apply a left Kan extension argument to assume that R is p -torsion-free, andthen argue locally as above to reduce to the case that R is a p -torsion-free, graded, quasiregularsemiperfectoid. Then both sides of ϕ i are discrete, and so the problem reduces to verifying the property that it multiplies degrees by p . But this follows from the treatment of graded cyclotomicspectra in Appendix A. (cid:3) Proof of Proposition 5.38.
Let R = A ⊕ N and I = N be a henselian pair of the form of Proposi-tion 5.38. We view R as being Z ≥ -graded, with A in degree 0 and N in degree one. We must provethat fib( F p ( i )( R ) → F p ( i )( A )) ∈ D ≤ i ( F p ). Choose r ≫ \ V i L R/ Z p identifies with \ V i L A/ Z p , so by d´evissage using Corollary 5.19 andCorollary 5.21 we see that the same is true of the Z ≥ -graded object b ∆ R { i } / N ≥ i + r b ∆ R { i } . Namely,its degree 0 part is b ∆ A { i } / N ≥ i + r b ∆ A { i } . The same is true modulo p , and so fib( F p ( i )( R ) → F p ( i )( A ))identifies with the fiber ofcan − ϕ i : ( N ≥ i b ∆ R { i } / N ≥ i + r b ∆ R { i } ) > ⊗ L Z F p → ( b ∆ R { i } / N ≥ i + r b ∆ R { i } ) > ⊗ L Z F p , where the subscript > Z > -subobject of a Z ≥ -graded object.To complete the proof we must verify the conditions of Lemma 5.42 below. Firstly, the Frobeniusmultiplies degrees by p by Proposition 5.41(2). Next, the fiber of can is ( b ∆ R { i } / N ≥ i b ∆ R { i } ) > [ − D ≤ i ( Z p ) by Proposition 5.25. Finally, to verify condition (2), observe that each coho-mology group of \ V i L R/ Z p is a Z ≥ -graded, finitely generated R -module, so necessarily zero exceptin finitely many degrees of the grading. The same then holds for b ∆ R { i } / N ≥ i + r b ∆ R { i } by anotherd´evissage through Corollary 5.19 and Corollary 5.21. (cid:3) Lemma 5.42.
Let
M, N be Z > -graded objects of D ( F p ) , and i ≥ . Let can : M → N be a map ofgraded objects and let ϕ : M → N be a map which multiplies degrees by p . Suppose that (1) the fiber of can belongs to D ( F p ) ≤ i , (2) for any fixed n , the cohomologies H n ( M ) , H n ( N ) vanish except in finitely many degrees of thegrading.Then fib(can − ϕ : M → N ) belongs to D ( F p ) ≤ i .Proof. By (2), we can replace the direct sums L M i and L N i with the corresponding infinite prod-ucts. Therefore, the result follows from Lemma 5.34. (cid:3) Finally, we use the rigidity theorem to give a description of the top cohomology of the F p ( i ). For B an F p -algebra, let Ω iB be the (underived) module of i -forms on B (relative to Z p , or F p ), and let C − : Ω iB → Ω iB /d Ω i − B be the inverse Cartier operator. Corollary 5.43 (Top cohomology of F p ( i )) . Let R be a p -complete ring. Then there is a naturalisomorphism H i +1 ( F p ( i )( R )) ≃ coker(1 − C − : Ω iR/p → Ω iR/p /d Ω i − R/p ) . (44) In particular, if R is w -strictly local, then H i +1 ( F p ( i )( R )) = H i +1 ( Z p ( i )( R )) = 0 . N THE BEILINSON FIBER SQUARE 39
Proof.
Without loss of generality, we can assume that R is an F p -algebra via Theorem 5.2. By thenpicking a polynomial F p -algebra surjecting onto R and henselizing along the kernel, another use ofTheorem 5.2 reduces us to the case that R is ind-smooth over F p . But then F p ( i )( R ) ≃ fib(Ω iR − C − −−−−→ Ω iR /d Ω i − R )[ − i ] (45)This follows because of the expression of F p ( i ) as the shifted ´etale cohomology of logarithmic formsΩ i log (cf. [BMS19, Cor. 8.21] and reduction modulo p ; we also review this in the next section) and theshort exact sequence of ´etale sheaves (cf. [Ill79, Sec. 2.4] and [Mor, Cor. 4.2])0 → Ω i log → Ω i − C − −−−−→ Ω i /d Ω i − → . Expression (45) implies the claim. (cid:3) The comparison with syntomic cohomology
In this section, we show that the Z p ( i ) for i ≤ p − Q p ( i ) for all i can be describedpurely in terms of derived de Rham (instead of prismatic) cohomology, using a form of syntomiccohomology [FM87, Kat87]. The strategy is to use the description of the Z p ( i ) in equal characteristic p from [BMS19] together with the Beilinson fiber square to relate the Z p ( i ) in mixed and equalcharacteristic. In particular, we prove Theorem F.6.1. Syntomic cohomology.
To begin with, we define another form of syntomic cohomology viathe quasisyntomic site, by descent from quasiregular semiperfectoids.
Definition 6.1 ( p -adic derived de Rham cohomology) . For a map of rings A → R , we let L Ω R/A ∈ D ( A ) denote the p -adic derived de Rham cohomology of R relative to A , see [Bha12]. By definition,when R is a finitely generated polynomial A -algebra, L Ω R/A is given by the p -completed relative deRham complex Ω • R/A , and in general L Ω R/A is defined via p -complete left Kan extension. One canshow [Bha12, Cor. 3.10] that L Ω R/A more generally agrees with the p -completed (underived) relativede Rham complex when R is smooth over A . When A = Z , we omit A from the notation.The p -adic derived de Rham cohomology L Ω R/A is equipped with the derived Hodge filtration L Ω ≥ ⋆R/A , obtained by left Kan extending the naive filtration in the polynomial (or more generallysmooth) case. Example 6.2 (Derived de Rham cohomology and divided powers) . We recall the following basiccalculation: for the map Z [ x ] → Z , we have that L Ω Z / Z [ x ] ≃ d Γ( x ) is the p -complete divided poweralgebra on the class x , and the derived Hodge filtration is the divided power filtration. This is aspecial case of [Bha12, Cor. 3.40]. See also [SZ18, Prop. 3.16] for an account. Definition 6.3 (Derived de Rham–Witt cohomology) . For an F p -algebra S , we let LW Ω S ∈ D ( Z p )denote the p -adic derived de Rham–Witt cohomology or derived crystalline cohomology of S (definedvia p -complete left Kan extension from finitely generated polynomial F p -algebras) [BMS19, Sec. 8]; forind-smooth F p -algebras S this agrees with Illusie’s usual W Ω S . It comes equipped with the derivedNygaard filtration N ≥ ⋆ LW Ω S obtained via left Kan extension of the usual Nygaard filtration inthe finitely generated polynomial case (see [BLM, Sec. 8] for an account); we write \ LW Ω S for thecompletion of LW Ω S with respect to the Nygaard filtration. We have by [BMS19, Lemma 8.2] anidentification of the associated graded terms of the Nygaard filtration N ≥ i LW Ω S / N ≥ i +1 LW Ω S ≃ L (cid:0) τ ≤ i Ω S/ F p (cid:1) with the derived functor of S τ ≤ i Ω S/ F p . The Frobenius ϕ : S → S induces anendomorphism of LW Ω S ; on N ≥ i LW Ω S it becomes divisible by p i , and indeed we have dividedFrobenius maps ϕ i : N ≥ i LW Ω S → LW Ω S . (46) Finally, using the de Rham-to-crystalline comparison, we find that if R is a p -torsion free ring, thenthere is a natural equivalence L Ω R ≃ LW Ω R/p ; (47)in particular, L Ω R naturally carries a Frobenius operator ϕ . Remark 6.4 (Sheaf properties) . The functors S LW Ω S and S \ LW Ω S are sheaves on qSyn F p .For LW Ω S , it suffices to work modulo p , and then use the conjugate filtration on derived de Rhamcohomology [Bha12] and the flat descent for the wedge powers of the cotangent complex [BMS19,Sec. 3]. For the Nygaard-completed \ LW Ω S , this follows since the associated graded terms have thisproperty, by a similar argument. Similarly, the filtration pieces S
7→ N ≥ i LW Ω S and S
7→ N ≥ i \ LW Ω S are sheaves on qSyn F p . Construction 6.5 (Derived de Rham–Witt cohomology of quasiregular semiperfect rings, cf. [BMS19,Sec. 8.2]) . Let S ∈ qrsPerfd F p be a quasiregular semiperfect F p -algebra. In this case, one forms thering A crys ( S ) (defined by Fontaine [Fon94]), which is the universal p -adically complete divided powerthickening of S , with divided powers compatible with those on ( p ) ⊂ Z p ; quasiregularity ensures thatit is p -torsion free. Then, one has a natural identification LW Ω S = A crys ( S ) , and the Nygaard filtration becomes the filtration N ≥ i A crys ( S ) = (cid:8) x ∈ A crys ( S ) : ϕ ( x ) ∈ p i A crys ( S ) (cid:9) .Here ϕ : A crys ( S ) → A crys ( S ) denotes the endomorphism induced by the Frobenius on S ; it has thefurther property that ϕ ( x ) ≡ x p (mod p ) for x ∈ A crys ( S ), i.e., ϕ defines the structure of a δ -ring onthe p -torsion free ring A crys ( S ). Construction 6.6 (Derived de Rham and de Rham–Witt cohomology of qrsp rings) . Let R ∈ qrsPerfd Z p . We consider the following rings.(1) The derived de Rham–Witt cohomology LW Ω R/p of R/p . Since the ring
R/p is a quasiregularsemiperfect F p -algebra, it follows from Construction 6.5 that there is an isomorphism LW Ω R/p ≃ A crys ( R/p ) . (2) The ( p -adic) derived de Rham cohomology L Ω R . Here L Ω R is a discrete, p -complete and p -torsion free ring; this follows from reduction modulo p and the conjugate filtration on L Ω ( R/p ) / F p . The ring L Ω R is also equipped with the multiplicative, descending Hodge fil-tration L Ω ≥ ⋆R .Via the de Rham-to-crystalline comparison, we have equivalences L Ω R ≃ LW Ω R/p ≃ A crys ( R/p ) . Inparticular, we find via (1) and (2) above that the p -complete, p -torsion free ring L Ω R is equipped withboth a Frobenius operator and a Hodge filtration. Lemma 6.7.
Let r ≥ be an integer. (1) The p -adic valuation of ( pr )! r ! is equal to r . (2) The p -adic valuation of p r r ! is at least min( r, p − .Proof. Both assertions follow from Legendre’s formula, v p ( n !) = P j> ⌊ n/p j ⌋ for v p the p -adic valua-tion. (cid:3) Proposition 6.8 (Divisibility of Frobenius, cf. also [Tsu99, Lem. A1.4]) . Let R ∈ qrsPerfd Z p . Thenfor i ≤ p − , the Frobenius ϕ : L Ω R → L Ω R carries L Ω ≥ iR into p i L Ω R , or in other words the deRham-to-crystalline comparison carries L Ω ≥ iR into N ≥ i A crys ( R/p ) . N THE BEILINSON FIBER SQUARE 41
Proof.
For any R ∈ qrsPerfd Z p , we can write R = W ( A ) /I , where A is a perfect F p -algebra and I ⊂ W ( A ) is an ideal. We have an identification of the p -complete cotangent complex, \ L R/ Z p ≃ d I/I [1].We first verify the assertion when the ideal I as above can be written as I = ( f ), for f anonzerodivisor, so R = W ( A ) / ( f ). In this case, in view of Example 6.2 and base change, wefind that L Ω R = L Ω R/W ( A ) is the p -completion of the divided power envelope of the regular ideal( f ), i.e., the ring W ( A )[ f n /n !] n ≥ ; furthermore, for each i , the Hodge filtered piece L Ω ≥ iR identifieswith the corresponding divided power filtration, i.e., the ideal ( f j /j !) j ≥ i . Now the Frobenius ϕ on L Ω R ≃ LW Ω R/p ≃ A crys ( R/p ) is a Frobenius lift coming from a δ -structure, so ϕ (cid:18) f j j ! (cid:19) = ( f p + pδ ( f )) j j ! = P ≤ l ≤ j (cid:0) jl (cid:1) f pl p j − l δ ( f ) j − l j ! . (48)The l th term in the sum above is divisible (in the ring L Ω R ) by f pl p j − l l !( j − l )! = f pl ( pl )! ( pl )! p j − l l !( j − l )! , where weuse the divided powers on ( f ) to see f pl ( pl )! ∈ L Ω R . Now the p -adic valuation of ( pl )! p j − l l !( j − l )! is at least l + min( j − l, p −
1) thanks to Lemma 6.7. So if i ≤ p −
1, then it follows that ϕ carries L Ω ≥ iR into p i L Ω R .Now suppose R is a p -complete tensor product over α ∈ A of rings of the form W ( A α ) / ( f α ), for A α perfect F p -algebras and f α ∈ W ( A α ) regular elements. In this case, we have an isomorphism(after p -completion) of filtered rings L Ω ≥ ⋆R ≃ N α ∈A L Ω ≥ ⋆W ( A α ) /f α by the K¨unneth formula, which iscompatible with the Frobenius operators. The assertion ϕ ( L Ω ≥ iR ) ⊂ p i L Ω R for i ≤ p − R thus follows by taking tensor products.Finally, let R ∈ qrsPerfd Z p be arbitrary and write R = W ( A ) /I for A a perfect F p -algebra. Toprove the claim ϕ ( L Ω ≥ iR ) ⊂ p i L Ω R for i ≤ p −
1, we will reduce to the previous cases, following thestrategy of [BMS19, Theorem 8.14]. Let { x t } t ∈ T be a system of generators for the ideal I and foreach t , we write x t = P i ≥ p i [ y t,i ] for some y t,i ∈ A . For each t ∈ T , we have a map W ( F p [ u , u , . . . , ] perf ) / ([ u ] + p [ u ] + . . . ) → W ( A ) /I = R (49)sending [ u i ] [ y t,i ]; note that the source belongs to qrsPerfd Z p , and its cotangent complex is theshift of a free of rank 1 module. The map (49) has image on p -completed cotangent complexes givenby the class of x t .We consider the p -completed tensor product R ′ def = W ( A ) ˆ ⊗ O t ∈ T ( W ( F p [ u , u , . . . , ] perf ) / [ u ] + p [ u ] + . . . ) , which maps surjectively to R (via the above maps) and induces a surjection on H ( \ L − / Z p [ − p and using the Hodge and conjugate filtrations on derived deRham cohomology, we find that L Ω ≥ iR ′ → L Ω ≥ iR is a surjection for each i . Since the previous discussionshows that ϕ ( L Ω ≥ iR ′ ) ⊂ p i L Ω R ′ for i ≤ p −
1, we can now conclude the claim for R by naturality, asdesired. (cid:3) Using the divisibility property of Frobenius, we can define, for R ∈ qrsPerfd Z p and i ≤ p −
1, a divided Frobenius ϕ/p i : L Ω ≥ iR → L Ω R (of discrete, p -torsion free abelian groups). Using the dividedFrobenius, we now define syntomic cohomology; this definition is based on the ideas of [FM87, Kat87](and can be compared with it using the comparison between derived de Rham and crystalline coho-mology in the lci case, cf. [Bha12, Sec. 3]). Definition 6.9 (Syntomic cohomology) . We define sheaves Z p ( i ) FM for 0 ≤ i ≤ p −
2, and Q p ( i ) FM for i ≥
0, on qrsPerfd Z p via Z p ( i ) FM ( R ) = fib( ϕ/p i − L Ω ≥ iR → L Ω R ) , (50) Q p ( i ) FM ( R ) = fib( ϕ − p i : L Ω ≥ iR → L Ω R ) Q p . (51)These are sheaves on qrsPerfd Z p because R L Ω ≥ iR is a sheaf. Unfolding, we obtain sheaves Z p ( i ) FM for 0 ≤ i ≤ p − Q p ( i ) FM for all i ≥ Z p . Remark 6.10.
While one could define Z p ( p − FM ( R ) via the same formula, this does not give thecorrect integral theory in weight ( p − The Z p ( i ) in equal characteristic p . In equal characteristic p , the Z p ( i ) can be determinedvia the theory of the de Rham–Witt complex and its derived versions, cf. [Ill72, Sec. VIII.2], [Bha12],and in particular [BMS19, Sec. 8]. We next review this.
Theorem 6.11 ( Z p ( i ) in equal characteristic p , cf. [BMS19, Sec. 8]) . Suppose S is a quasisyntomic F p -algebra. Then, for each i , (1) b ∆ S { i } is the Nygaard-completed derived de Rham–Witt cohomology \ LW Ω S of S and (2) the Nygaard filtration N ≥ i b ∆ S { i } identifies with the de Rham–Witt Nygaard filtration N ≥ i \ LW Ω S ,and the prismatic Frobenius ϕ i identifies with the divided Frobenius (46) .Consequently, Z p ( i )( S ) = fib( ϕ i − can : N ≥ i \ LW Ω S → \ LW Ω S ) , (52) where ϕ i : N ≥ i \ LW Ω S → \ LW Ω S is the divided Frobenius operator, so that p i ϕ i is the Frobenius. Remark 6.12.
The Nygaard completion is redundant in the formula (52) for Z p ( i )( S ). This followseasily from the fact that ϕ i acts by zero on N ≥ i +1 \ LW Ω S /p . In particular, we can write F p ( i )( S ) = fib( ϕ i − can : ( N ≥ i LW Ω S / N ≥ i +1 LW Ω S ) ⊗ L Z p F p → ( LW Ω S / N ≥ i +1 LW Ω S ) ⊗ L Z p F p ) . Example 6.13. If S is a quasiregular semiperfect F p -algebra, then for i > Z p ( i )( S ) is discrete, p -torsion free, and there is a natural isomorphism of abelian groups Z p ( i )( S ) ≃ ker( ϕ − p i : A crys ( S ) → A crys ( S )) , (53)where ϕ is induced by the Frobenius. For i = 0, we should instead take the homotopy fiber of ϕ − crys ( S ), so it may have terms in cohomological degree 1.In the ind-smooth case, one has an identification with logarithmic de Rham–Witt forms. Definition 6.14 (Logarithmic de Rham–Witt forms) . For S an ind-smooth F p -algebra, we let W Ω • S, log denote the graded subring of the de Rham-Witt complex W Ω • S consisting of fixed points for F . When S is local, one knows that W Ω • S, log is generated, modulo any power of p , in degree 1 by elements ofthe form d [ x ] / [ x ], for x ∈ S × and [ x ] ∈ W ( S ) the Teichm¨uller representative, cf. [Ill79, Th. 5.7.2]which proves this ´etale locally and [Mor, Theorem 0.10] for a very general Zariski local result. Notethat for each i , W Ω i − , log defines a pro-´etale sheaf on Spec( S ). Theorem 6.15 (Cf. [BMS19, Cor. 8.21]) . Let S be an ind-smooth F p -algebra. Then there are naturalidentifications Z p ( i )( S ) ≃ R Γ proet (Spec( S ) , W Ω i − , log )[ − i ] . See also [GL00, GH99] for the identification with with p -adic ´etale motivic cohomology. N THE BEILINSON FIBER SQUARE 43
The Beilinson fiber square on graded pieces.
Our goal is to relate the Z p ( i ) in mixed andin equal characteristic. We use the Beilinson fiber sequence to prove a basic fiber square which givesa version of Theorem A on associated graded terms for the motivic filtrations. Construction 6.16 (The trace on graded pieces) . Let R be any commutative ring. Then we havethe trace maps K( R ; Z p ) → TC( R ; Z p ) → HC − ( R ; Z p ) . When R ∈ qrsPerfd Z p , then HC − ( R ; Z p ) is concentrated in even degrees and π i is given by c L Ω ≥ iR ,cf. [BMS19, Sec. 5] and [Ant19]. Unfolding we conclude that, on graded pieces, we obtain a naturalmap Z p ( i )( R ) → c L Ω ≥ iR for R ∈ qSyn Z p . This naturally factors through L Ω ≥ iR since R Z p ( i )( R ) isleft Kan extended from p -complete polynomial algebras (Theorem 5.1). Theorem 6.17 (The Beilinson fiber square on graded terms) . Let R ∈ qSyn Z p . Then, for each i ≥ ,there exists a natural map χ i : Q p ( i )( R/p ) → ( L Ω R ) Q p and a functorial pullback square Q p ( i )( R ) (cid:15) (cid:15) / / Q p ( i )( R/p ) χ i (cid:15) (cid:15) ( L Ω ≥ iR ) Q p / / ( L Ω R ) Q p (54) in the derived ∞ -category D ( Q p ) . The map χ i arises from a natural map Z p ( i )( R/p ) → p − N L Ω R forsome N ≫ (depending only on i ), fitting into an analogous commutative diagram.Furthermore, the associated fiber sequence holds up to isogeny: cofib( Z p ( i )( R ) → Z p ( i )( R/p )) and L Ω R /L Ω ≥ iR are naturally isogenous to each other. Finally, for i ≤ p − , we have natural equivalencesfor qSyn Z p , fib( Z p ( i )( R ) → Z p ( i )( R/p )) ≃ fib (cid:16) L Ω R /L Ω ≥ iR → L Ω R/p /L Ω ≥ iR/p (cid:17) [ − . (55) Proof.
Note that the hypothesis that R ∈ qSyn Z p ensures that R/p ∈ QSyn. The square (54) willbe constructed in the ∞ -category of D ( Q p ) ≥ -valued sheaves on qSyn Z p . It suffices to construct theabove pullback square for R ∈ qrsPerfd Z p , by unfolding. For R ∈ qrsPerfd Z p , we have a pullbacksquare by Theorem A, TC( R ; Q p ) (cid:15) (cid:15) / / TC(
R/p ; Q p ) (cid:15) (cid:15) HC − ( R ; Q p ) / / HP( R ; Q p ) . (56)Note since R ∈ qrsPerfd Z p , the terms in the bottom row of the above fiber square are concentrated ineven degrees [BMS19, Th. 7.1 and Prop. 8.20]. Consequently, for each i , we can apply τ [2 i − , i ] andstill obtain a fiber square. By definition of the Q p ( i ) and by the corresponding description of derivedde Rham cohomology, as in [BMS19, Th. 1.17] (or using the filtration of [Ant19]), we obtain (54) for R (after a shift), albeit with a Hodge completion. In particular, instead of χ i , we obtain a completedversion ˆ χ i : Q p ( i )( R/p ) → ( c L Ω R ) Q p , as well as a Hodge-completed version of the fiber square (54).We can refine ˆ χ i to χ i (and obtain (54)) as follows. First, by construction of these maps via theBeilinson fiber square, a multiple of ˆ χ i lifts to a map Z p ( i )( R/p ) → c L Ω R . Since the source is leftKan extended (as a functor to the p -complete derived ∞ -category) from finitely generated p -complete polynomial algebras, we can restrict and left Kan extend to obtain that a multiple of ˆ χ i lifts to Z p ( i )( R/p ) → L Ω R . Inverting p , we obtain that ˆ χ i factors through a map χ i : Q p ( i )( R/p ) → ( L Ω R ) Q p .From the quasi-isogeny between TC( R, ( p ); Z p ) and ΣHC( R, ( p ); Z p ) as in Theorem 2.20, we obtainthe isogeny claim.Finally, we verify (55); again, we can assume that R ∈ qrsPerfd Z p by unfolding. Since everythingis a pro-´etale sheaf, we can even assume that R is w -strictly local in the sense of [BS15], so that π − TC( R ; Z p ) = coker( F − W ( R ) → W ( R )) (by [HM97b, Theorem F]) vanishes. Recall we havean equivalence τ ≤ p − TC( R, ( p ); Z p ) ≃ τ ≤ p − ΣHC( R, ( p ); Z p ) by Theorem 2.20. It follows that for i ≤ p −
2, we have an equivalence τ [2 i − , i ] TC( R, ( p ); Z p ) ≃ τ [2 i − , i ] ΣHC( R, ( p ); Z p ) . Now TC(
R/p ; Z p ), HC( R ; Z p ), and HC( R/p ; Z p ) are concentrated in even degrees since R ∈ qrsPerfd Z p .For the first claim see [BMS19, Proposition 8.20]. The second and third follow from the filtrationsconstructed in [Ant19] and [BMS19, Sec. 5].It follows from the above definitions that τ [2 i − , i ] TC( R, ( p ); Z p ) ≃ fib( Z p ( i )( R ) → Z p ( i )( R/p ))[2 i ] , and from [BMS19, Sec. 5] and [Ant19] that τ [2 i − , i ] ΣHC( R, ( p ); Z p ) ≃ fib( L Ω R /L Ω ≥ iR → L Ω R/p /L Ω ≥ iR/p )[2 i − . Using these identifications, we deduce (55). (cid:3)
We next identify the p -adic Chern character χ i : Q p ( i )( R/p ) → ( L Ω R ) Q p on graded pieces (fromTheorem 6.17) more explicitly. To this end, we prove the following basic result: Proposition 6.18 (The image of χ i ) . Let R ∈ qrsPerfd Z p . Then for each i > , the map (of Q p -vector spaces) χ i : Q p ( i )( R/p ) → ( L Ω R ) Q p = A crys ( R/p ) Q p is injective, and has image given by the ϕ = p i eigenspace. The main issue is the following: both the source and target of χ i are functors of R/p , thanksto de Rham–Witt theory. We have seen that the p -adic Chern character χ i induces a natural map Z p ( i )( R/p ) → p − N L Ω R for R ∈ qSyn Z p for some N . However, it is not a priori obvious that the map χ i arises from a natural transformation of functors on F p -algebras (which would force it to commutewith Frobenius operators, for example). Our first goal is to verify this. Lemma 6.19 (The Frobenius action on Z p ( i )( R )) . For any R ∈ qSyn F p , the Frobenius on R acts asmultiplication by p i on Z p ( i )( R ) .Proof. This reduces to the case of a quasiregular semiperfect F p -algebra by descent. But in this case,the identification of Example 6.13 clearly proves the claim. (cid:3) Corollary 6.20.
The natural map χ i : Z p ( i )( R/p ) → p − N ( L Ω R ) → ( L Ω R ) Q p , for R ∈ qSyn Z p arises (by precomposition with reduction mod p ) from a unique natural transformation χ i : Z p ( i ) → p − N ′ LW Ω ( − ) on qSyn F p for some N ′ ≥ N .Proof. This follows from Corollary B.4 and Lemma 6.19 (the latter shows that the hypotheses of theformer are satisfied), and then left Kan extension from finitely generated p -complete polynomial rings.Uniqueness follows since these sheaves are torsion free. (cid:3) Next, we consider the sheaf of graded E ∞ -rings L ∞ i =0 Z p ( i ) on qSyn F p . For each N ≥
0, we canalso truncate to obtain a sheaf of graded E ∞ -rings L Ni =0 Z p ( i ). N THE BEILINSON FIBER SQUARE 45
Proposition 6.21.
Let f : L Ni =0 Z p ( i ) → L Ni =0 Z p ( i ) be a natural map of sheaves of graded E ∞ ringson qSyn F p . Then there exists λ ∈ Z p such that in degree i , f is given by multiplication by λ i .Proof. We first observe that the only endomorphisms of Z p (1) (as a functor on qSyn F p ) are given byscalars. It suffices to verify this on quasiregular semiperfect algebras, and there Z p (1) is corepresentableby F p [ x /p ∞ ] / ( x − Z p (1)( F p [ x /p ∞ ] / ( x − ≃ Z p . So the endomorphism f is given by ascalar action at least on Z p (1).Note that all these functors are left Kan extended from smooth algebras (to the p -complete cate-gory), so f is determined by the values on smooth F p -algebras. Furthermore, the map f is determinedby its values modulo p n for each n . However, classes in H i ( Z /p n ( i )) are ´etale locally written as sumsof products of classes in H ( Z /p n (1)) (thanks to Theorem 6.15), so the value of f on Z p (1) determinesthe value of f in general. The result now follows because on smooth algebras, Z /p n ( i ) is concentratedin cohomological degree i ´etale locally. (cid:3) Proof of Proposition 6.18.
Recall that the map χ i is actually a special case of a map Z p ( i )( R ) → p − N ′ ( LW Ω R ) defined on R ∈ qSyn F p , by Corollary 6.20. For R quasiregular semiperfect, we knowthat the Frobenius acts as p i on Z p ( i )( R ), so we obtain a natural, multiplicative map Q p ( i )( R ) → (A crys ( R ) ϕ = p i ) Q p . We wish to see that these maps are isomorphisms.Now we know independently that Q p ( i )( R ) is identified (for i >
0) with A crys ( R ) ϕ = p i Q p via thetheory of topological cyclic homology (Theorem 6.11, following [BMS19, Sec. 8]). Thus, we actuallyobtain natural, multiplicative (in i ) maps Q p ( i )( R ) → Q p ( i )( R ) for R ∈ qSyn F p , and we wish to seethat these are isomorphisms. Up to rescaling by a power of p , furthermore, they carry Z p ( i )( R ) into Z p ( i )( R ). As we saw in Proposition 6.21, these maps are necessarily all given by scalar multiplicationby some λ i in degree i , for some λ ∈ Z p ; we know that λ = 0 (by comparing with i = 1, say), so theresult now follows. (cid:3) Comparison of the Z p ( i ) FM and Z p ( i ) . Our main result is the following comparison, whichestablishes Theorem F.
Theorem 6.22.
For R ∈ qSyn Z p , there are natural, multiplicative identifications Z p ( i ) FM ( R ) ≃ Z p ( i )( R ) for i ≤ p − and Q p ( i ) FM ( R ) ≃ Q p ( i )( R ) for all i ≥ . By [Gei04, Theorem 1.3], for i ≤ p − p -adic ´etale motivic cohomology. Proof of the rational case of Theorem 6.22.
Fix i ≥
0. It is enough to prove the equivalences forall R ∈ qrsPerfd Z p . Thanks to the odd vanishing conjecture proved in [BS19, Sec. 14], we maymoreover assume that R ∈ qrsPerfd Z p is such that Z p ( i )( R ) is concentrated in degree zero. Inthe homotopy cartesian square of Theorem 6.17, the terms Q p ( i )( R ), ( L Ω ≥ iR ) Q p , and ( L Ω R ) Q p areall concentrated in degree 0, whence the same is true of the remaining term Q p ( i )( R/p ) (that is, ϕ − p i : ( L Ω R ) Q p → ( L Ω R ) Q p is surjective) and the fiber square is simply a cartesian and cocartesiansquare of abelian groups Q p ( i )( R ) (cid:15) (cid:15) / / Q p ( i )( R/p ) (cid:15) (cid:15) ( L Ω ≥ iR ) Q p / / ( L Ω R ) Q p . Note that all the arrows are injections: the bottom since it is inclusion of the Hodge filtration, theright by Proposition 6.18), and the others since the diagram is cartesian.
We claim that the map ϕ − p i : ( L Ω ≥ iR ) Q p → ( L Ω R ) Q p is surjective. Indeed, given x ∈ ( L Ω R ) Q p , wecan write x = ( ϕ − p i )( x ′ ) for some x ′ ∈ ( L Ω R ) Q p ; as we noted above, ϕ − p i : ( L Ω R ) Q p → ( L Ω R ) Q p is surjective. Using Proposition 6.18 to identify the image of the vertical map, the diagram beingcocartesian means that ( L Ω R ) ϕ = p i Q p ⊕ ( L Ω ≥ iR ) Q p → ( L Ω R ) Q p = A crys ( R/p ) Q p is surjective. So we canwrite x ′ = y ′ + z ′ for y ′ ∈ ker( ϕ − p i ) and z ′ ∈ ( L Ω R ) ≥ i Q p . Applying ϕ − p i , we get that x = ( ϕ − p i )( z ′ ),proving the claim as desired.Combining these observations, we have established a natural identification Q p ( i )( R ) = ( L Ω R ) ϕ = p i Q p ∩ ( L Ω ≥ iR ) Q p ≃ fib( ϕ − p i : ( L Ω ≥ iR ) Q p → ( L Ω R ) Q p ) , as desired. (cid:3) Corollary 6.23 (A description of TC( R ; Q p )) . Let R be any simplicial commutative ring. Then thereis a natural equivalence TC( R ; Q p ) ≃ M i ≥ fib( ϕ − p i : L Ω ≥ iR → L Ω R ) Q p . Proof.
The map from R to its derived p -adic completion induces an equivalence on all the termsappearing in the statement: for derived de Rham cohomology and its Hodge filtration this followsfrom base change, while it holds for THH( − ; Z p ) (and hence TC( − ; Q p )) by [CMM18, Lem. 5.2]. Wemay therefore assume R is p -complete, at which point we know from Construction 5.33 that TC( R ; Q p )admits a complete descending filtration with associated graded given by Q p ( i )( R )[2 i ], for i ≥
0. UsingAdams operations on TC as in [BMS19, Sec. 9.4], we can split the filtration functorially. Combiningwith the rational part of Theorem 6.22 (or more precisely its left Kan extension to p -complete simplicialcommutative rings), the claim follows. (cid:3) Next, we will prove the integral case of Theorem 6.22. The main step is to show that the Z p ( i ) FM ( − )for i ≤ p − Z p ; this is the analog of Theorem 5.11, i.e., of the oddvanishing conjecture. This will essentially follow the proof in [BS19, Sec. 14], and will take somesteps. First, we identify the syntomic cohomology of perfectoids, starting with the case of absolutelyintegrally closed valuation rings. Construction 6.24.
Assume p >
2. For any R ∈ qrsPerfd Z p , we have a natural map T p ( R × ) → H ( Z p (1) FM ( R )) given as follows. Given a compatible system ( ǫ m ) = (1 , ǫ , ǫ , . . . ) of p -power rootsof unity in R , we obtain an element ǫ ∈ R ♭ . The construction yields a natural map T p ( R × ) → L Ω R ( ǫ m ) log([ ǫ ]) = ∞ X n =1 ( − n +1 ([ ǫ ] − n n ∈ L Ω R = A crys ( R/p ) , which is easily seen to be well-defined since [ ǫ ] − crys ( R/p ). This map landsin L Ω ≥ R since all the terms in the logarithm lie in the first step of the divided power filtration onA crys ( R/p ), as well as in the ϕ = p eigenspace by functoriality (note that the map is actually definedon the larger group 1 + ker( R ♭ → R/p ) ⊇ T p ( R × ), where ϕ raises elements to the p th -power). Thuswe have indeed defined a map T p ( R × ) → H ( Z p (1) FM ( R )). Proposition 6.25 ([Fon94, Prop. 5.3.6] and [Tsu99, Theorem A3.26]) . Assume p > . Let V be a p -complete, rank , absolutely integrally closed valuation ring of mixed characteristic. Then for each ≤ i ≤ p − , Z p ( i ) FM ( V ) is a free Z p -module of rank . Moreover, the map T p ( V × ) → Z p (1) FM ( V ) and the natural maps Z p (1) FM ( V ) ⊗ i → Z p ( i ) FM ( V ) are isomorphisms. N THE BEILINSON FIBER SQUARE 47
Next we consider perfect F p -algebras. To do so, we extend Z p ( i ) FM from qSyn Z p to all p -completesimplicial commutative Z p -algebras, by left Kan extension from p -complete finitely generated polyno-mial Z p -algebras. This left Kan extension does not change the value of Z p ( i ) FM on qSyn Z p , by (50). Proposition 6.26.
Assume p > . Let V be a perfect F p -algebra. Then Z p ( i ) FM ( V ) = 0 for < i ≤ p − .Proof. We consider the filtered ring L Ω ≥ ⋆V . By Example 6.2, we can identify this with the p -completionof the tensor product, W ( V ) ⊗ Z [ x ] Γ( x ), where x p ∈ W ( V ); the induced filtration is the dividedpower filtration. In other words, L Ω V is the p -adic completion of the divided power envelope of( p ) ⊂ W ( V ).We recall that the canonical map of rings W ( V ) → L Ω V is actually split [Bha12, Cor. 8.6] since W ( V ) already admits divided powers along ( p ): namely, we send x i /i ! to p i /i !, defining a surjection L Ω V → W ( V ). The kernel I ⊂ L Ω V of this surjection lies inside the p th step of the divided powerfiltration L Ω ≥ pV , since the splitting is easily checked to induce L Ω ≥ iV /L Ω ≥ i +1 V ≃ p i W ( V ) /p i +1 W ( V )for i ≤ p −
1. The splitting is moreover natural in V , and so in particular compatible with Frobeniusmaps. Since W ( V ) is p -torsion free, it is therefore also compatible with divided Frobenii.The assertion of the proposition is that ϕ/p i − L Ω ≥ iV → L Ω V is an isomorphism for 0 < i ≤ p − ≤ i ≤ p −
1, the previous paragraph shows that ϕ/p i restricts to I ; so if 0 ≤ i ≤ p − ϕ/p i = pϕ/p i +1 is a contracting operator on the p -adically complete group I , whence ϕ/p i − I . So to prove the proposition it is equivalent to show that ϕ/p i − L Ω ≥ iV /I → L Ω V /I is an isomorphism, or equivalently that ϕ/p i − p i W ( V ) → W ( V ) is an isomorphism. Butthis is the same as showing that 1 − p i ϕ − : W ( V ) → W ( V ) is an isomorphism, which holds when i > (cid:3) We next show that syntomic cohomology recovers the ´etale cohomology of the generic fiber forperfectoids. Proposition 6.25 already proves this for absolutely integrally closed valuation rings. Toextend to arbitrary perfectoid rings, we will use the arc-topology, as in [BS19, Sec. 8] and [BM18],and reduce to this case. See also [CS19, Sec. 2.2.1] for a treatment.
Definition 6.27 (The arc ˆ p -topology on perfectoid rings) . Let Perfd denote the category of perfectoidrings. We define a map R → R ′ in Perfd to be an arc ˆ p -cover if, for every rank ≤ V which is p -complete and map R → V , there exists an extension of valuation rings V → W such that R → V → W extends over R ′ (note that W may itself be taken to be p -complete and of rank ≤ p -completing [BM18, Prop. 2.1]). This defines the arc ˆ p -topology on Perfd op , so that we have thenotion of an arc ˆ p -sheaf. Proposition 6.28.
The following functors from
Perfd to D ( Z p ) ≥ are arc ˆ p -sheaves: (1) R R ; (2) R L Ω ≥ iR , for each i ≥ ; (3) R Z p ( i ) FM ( R ) , for ≤ i ≤ p − ; (4) R Z p ( i )( R ) , for i ≥ .Proof. Part (1) is [BS19, Prop. 8.9]; see also [Sch17, Prop. 8.5] or [BS17, Th. 11.2] for earlier versionsof this result. For part (2), we observe that if R is perfectoid, then \ V i L R/ Z p is the shift by i of aninvertible (indeed, rank 1 and free) R -module. It then follows by (1) that R L Ω R /L Ω ≥ iR is anarc ˆ p -sheaf, so it suffices to prove that R L Ω R is an arc ˆ p -sheaf. But L Ω R ⊗ L Z F p = L Ω R ⊗ L Z F p / F p .Using the conjugate filtration on the latter, one sees that R L Ω R ⊗ L Z F p is an arc ˆ p -sheaf, so that R L Ω R is one by p -completeness, as desired. Now, (3) follows because arc ˆ p -sheaves are closedunder taking fibers. For part (4), observe first that R π i THH( R ; Z p ) is an arc ˆ p -sheaf for each i ≥ R -module. It then follows that R b ∆ R { i } and N ≥ i b ∆ R { i } are also arc ˆ p -sheaves,as they admit complete filtrations with graded pieces given by π j THH( R ; Z p ) for various j . Thedefinition of Z p ( i )( R ) as the homotopy equalizer of can , ϕ i (Definition 5.8) now shows that it is alsoan arc ˆ p -sheaf. (cid:3) The next proposition reduces the study of the Z p ( i ) of perfectoids to the case of weight one. Proposition 6.29 (Syntomic cohomology of perfectoids) . Assume p > . Let O be a p -complete,rank , absolutely integrally closed valuation ring of mixed characteristic, and let R be a perfectoid O -algebra. Then the canonical map Z p (1) FM ( R ) ⊗ Z p Z p ( i − FM ( O ) → Z p ( i ) FM ( R ) is an equivalencefor ≤ i ≤ p − .Proof. In light of Proposition 6.25, we may let t ∈ Z p (1) FM ( O ) be a generator of this rank 1, free Z p -module, and it is equivalent to check that the natural cup product map × t i − : Z p (1) FM ( R ) → Z p ( i ) FM ( R ) is an equivalence. It suffices to check that the cup product induces an equivalence modulo p , when both sides preserve filtered colimits. But both sides are sheaves for the arc ˆ p -topology, byProposition 6.28 so, by [BM18, Prop. 3.28], it suffices to check that the natural map induces anequivalence on p -complete absolutely integrally closed valuation rings V . However, this follows fromProposition 6.25 and Proposition 6.26. (cid:3) Now we treat the weight one case. For this, we need the following lemma, which essentially is avariant of [BM18, Prop. 3.28].
Proposition 6.30.
Let F : Perfd → D ( F p ) ≥ be an arc ˆ p -sheaf; suppose that as a functor, it preservesfiltered colimits. Suppose that for every absolutely integrally closed valuation ring O ∈
Perfd , we have F ( O ) is discrete. Then F is discrete as an arc ˆ p -sheaf. More precisely, given R ∈ Perfd , there existsan arc ˆ p -cover R → R ′ with F ( R ′ ) discrete.Proof. We can always find an arc ˆ p -cover R → R ′ such that R ′ is a product of copies of p -completerank ≤ R ′ , we have F ( R ′ ) discrete. But this follows via an ultraproduct argument as in the proof of [BM18, Cor. 3.15]. (cid:3) Proposition 6.31.
There is a natural isomorphism Z p (1) FM ( R ) ≃ Z p (1)( R ) for R ∈ Perfd .Proof.
We know that both are D ( Z ) ≥ -valued sheaves for the arc ˆ p -topology on Perfd. Moreover,we claim that both are discrete with respect to the natural t -structure on D ( Z )-valued arc ˆ p -sheaves.Indeed, by Proposition 6.30 and reducing modulo p , it suffices to show that Z p (1) FM ( O ) and Z p (1)( O )are discrete and p -torsion free when O is a p -complete absolutely integrally closed valuation ring. Thisfollows from Proposition 6.25 and Proposition 6.26 for Z p (1) FM ( O ). For Z p (1)( O ), it follows from[BMS19, Prop. 7.17] (and the beginning of its proof).Thus, it suffices to produce a natural isomorphism of abelian groups H ( Z p (1) FM ( R )) ≃ H ( Z p (1)( R )).However, the right-hand-side is given by the construction R T p ( R × ) [BMS19, Prop. 7.17]. We havea map T p ( R × ) → H ( Z p (1) FM ( R )) thanks to Construction 6.24. To see that it is an isomorphism ofarc-sheaves, it suffices to reduce modulo p and, since both sides commute with filtered colimits, onemay check on absolutely integrally closed valuation rings (of rank 1), cf. [BM18, Prop. 3.28] for theanalogous argument. Therefore, the result follows from Proposition 6.25 and Proposition 6.26. (cid:3) The discreteness in the previous proposition showed that higher degree classes in Z p ( i ) FM could bekilled by passage to an arc cover; now we show that quasisyntomic covers suffice. Corollary 6.32.
Let R ∈ Perfd . Then there exists a quasisyntomic cover R → R ′ with R ′ perfectoidand such that for each i ≤ p − , Z p ( i ) FM ( R ′ ) is discrete and p -torsion free. N THE BEILINSON FIBER SQUARE 49
Proof.
Without loss of generality, we can assume R is an O C -algebra. In particular, it follows (viaProposition 6.29) that for any perfectoid R -algebra R ′ , the Z p ( i ) FM ( R ′ ) are all isomorphic for 0
2. Therefore, it suffices to find such an R ′ with Z p ( i ) FM ( R ′ ) discrete for i = 0 ,
1. We havethat Z p (0) FM ( R ) = R Γ proet (Spec( R ) , Z p ) = Z p (0)( R ) (as one checks by reducing modulo p and usingthe map L Ω R → R → R/p ) and Z p (1) FM ( R ) ≃ Z p (1)( R ) (Proposition 6.31).Then the result follows as in the proof of [BS19, Th. 14.1]. Indeed, we can arrange R ′ so that R ′ isabsolutely integrally closed [Sta19, Tag 0DCK] by Andr´e’s lemma in the form of [BS19, Theorem 7.12].Then by the Artin–Schreier sequence, we find that H i et (Spec( R ) , F p ) = 0 for i >
0, so F p (0)( R ) = R Γ et (Spec( R ) , F p ) is discrete. For i = 1, we must show that (cf. [BS19, Theorem 9.4]) Z p (1)( R ′ ) ≃ R Γ proet ( R ′ [1 /p ] , Z p ) is all concentrated in degree zero. Since R [1 /p ] is absolutely integrally closedas well, this follows via the Kummer sequence and the unique p -divisibility of Pic( R [1 /p ]) [BS19,Cor. 9.5], as well as the fact that Spec( R [1 /p ]) has mod p cohomological dimension ≤ (cid:3) The main step in the proof of the integral part of Theorem 6.22 is the following surjectivity result;this is much easier than the analog in [BS19, Sec. 14], which relies on q -de Rham complexes. Proposition 6.33.
Let R be a p -torsion free perfectoid ring. Let R ∞ denote the p -completion of R [ x /p ∞ , . . . , x /p ∞ n ] and let S ∞ = R ∞ / ( x , . . . , x n ) . (1) For ≤ i ≤ p − , the map Z p ( i ) FM ( R ∞ ) → Z p ( i ) FM ( S ∞ ) induces a surjection on H . (2) For all i , the map Q p ( i ) FM ( R ∞ ) → Q p ( i ) FM ( S ∞ ) induces a surjection on H . Recall that p -adic derived de Rham cohomology commutes with tensor products. Therefore L Ω R ∞ =A crys ( R ∞ /p ) is the p -adic completion of L Ω nc R ∞ := A crys ( R/p )[ x /p ∞ , . . . , x /p ∞ n ] and, using Ex-ample 6.2, L Ω S ∞ = A crys ( S ∞ /p ) is the p -completion of the divided power envelope L Ω nc S ∞ := L Ω nc R ∞ [ x j /j ! , · · · , x jn /j ! : j ≥ L Ω R ∞ → L Ω S ∞ and ϕ/p i − L Ω ≥ iS ∞ → L Ω S ∞ are jointlysurjective when 0 ≤ i ≤ p −
2, and similarly for arbitrary i ≥ p and replacing ϕ/p i − ϕ − p i .Fix i ≥ I ⊆ L Ω nc S ∞ be the ideal generated by the elements x j /j ! , · · · , x jn /j ! for j > i . Itis clear that the composition L Ω nc R ∞ → L Ω nc S ∞ /I is surjective when i ≤ p −
1, and that for arbitrary i their joint cokernel is killed by a power of p (depending on i ).We note also that that L Ω nc S ∞ /I is p -adically separated and p -torsion free (indeed, it is free as anA crys ( R/p )-module); so we may p -adically complete the short exact sequence 0 → I → L Ω nc S ∞ → L Ω nc S ∞ /I → → b I → L Ω S ∞ → \ L Ω nc S ∞ /I →
0. By the previousparagraph, the final term in the sequence is surjected onto by L Ω R ∞ when i ≤ p − p -adic completion), respectively has cokernel killed by a power of p for general i .The crystalline Frobenius on L Ω S ∞ is induced by the Frobenius ϕ on L Ω nc S ∞ which is given by theusual Frobenius on A crys ( R/p ) and raises the variables to their p th -powers. In particular, for any j > i , the divided Frobenius ϕ/p i sends x jr j ! to p j − i x pjr ( pj )! ( pj )! j ! p j , where the final fraction is a p -adic unitby Lemma 6.7. Therefore the divided Frobenius ϕ/p i : I → I is well-defined and even has image in pI ; after p -adic completion it follows that ϕ/p i − b I → b I is an isomorphism.Combining the final assertions of the previous two paragraphs, we have shown that the maps L Ω R ∞ → L Ω S ∞ and ϕ/p i − b I → L Ω S ∞ are jointly surjective when i ≤ p −
1, and that for arbitrary i their joint cokernel is killed by a power of p . It remains only to observe that b I ⊆ L Ω ≥ iS ∞ , as beforecompletion I lies inside the i th step of the divided power filtration. Proposition 6.34. As D ( Z p ) ≥ -valued sheaves on qSyn Z p , (1) Z p ( i ) FM ( · ) is discrete and torsion free for ≤ i ≤ p − and (2) Q p ( i ) FM ( · ) is discrete.Proof. The proof entirely follows [BS19, Sec. 14]. Without loss of generality, we can assume that R ∈ qrsPerfd Z p . We can write R as the quotient of a p -torsion free perfectoid ring R ′ , via a map R ′ → R with kernel given by a quasiregular ideal I .Now, we are allowed to enlarge R ′ via p -completely faithfully flat maps of perfectoids. Let { x s , s ∈ S } be a system of generators of I/I . Using a version of Andr´e’s lemma [BS19, Theorem7.12], we can assume that all the x s admit p -power roots, so that there is a surjection of qrsp rings R ′ D x /p ∞ s E s ∈ S / ( x s ) → R. which additionally induces a surjection on π \ L − / Z p . It follows that for i ≤ p −
2, the maps H ( Z p ( i ) FM ( R ′ D x /p ∞ s E )) → H ( Z p ( i ) FM ( R ′ D x /p ∞ s E / ( x s ))) → H ( Z p ( i ) FM ( R ))are surjective, the first claim by Proposition 6.33 and the second by the above. However, aftermaking a faithfully flat map of perfectoid rings R ′ D x /p ∞ s E → R ′′ , we can annihilate all classes in H ( Z p ( i ) FM ( · )) in view of Corollary 6.32. The same argument works for Q p ( i ) FM (though alternativelywe can use the odd vanishing conjecture, since the equivalence Q p ( i ) FM ≃ Q p ( i ) has already beenproved). (cid:3) Proof of Theorem 6.22 for i ≤ p − . Fix i ≤ p − R ∈ qrsPerfd Z p such that Z p ( i )( R ) and Z p ( i ) FM ( R ) are concentrated in degree zero for all i ≤ p −
2; we can do this by the odd vanishing con-jecture and Proposition 6.34. In this case, the map of discrete abelian groups Z p ( i )( R ) → Z p ( i )( R/p )is injective and has torsion free cokernel, thanks to the equivalence (55). So from the Beilinson fibersquare on graded pieces (Theorem 6.17) and the description of the image of χ i (Proposition 6.18), wefind that Z p ( i )( R ) ⊂ Z p ( i )( R/p ) = ( L Ω R ) ϕ = p i is the submodule consisting of those elements such thatthe image in L Ω R belongs to L Ω ≥ iR . In particular, it is precisely the kernel of ϕ/p i − L Ω ≥ iR → L Ω R .Since this map is surjective, we get the natural equivalence Z p ( i )( R ) ≃ Z p ( i )( R ) FM as desired. (cid:3) Examples -theory of p -adic fields. Let F be a complete discretely valued field of characteristic 0 withring of integers O F ⊂ F and perfect residue field k of characteristic p . In this subsection, we willuse the Beilinson fiber sequence to recover various calculations of the p -adic K-theory of F . All theseresults are previously known at least in the case of F local; see [Wei05, Theorem 61] for a detailedsurvey. Theorem 7.1.
The homotopy groups of K( F ; Q p ) are given (as Q p -vector spaces) as follows:(1) K s ( F ; Q p ) = 0 for s > ;(2) there is a natural isomorphism K s − ( F ; Q p ) ≃ F for each s > ;(3) there is a natural short exact sequence → F → K ( F ; Q p ) → Q p → .Proof. Since k is perfect, we have that K i ( k ; Z p ) = Z p for i = 0 and 0 otherwise, cf. [Hil81, Th. 5.4] and[Kra78, Cor. 5.5]. Taking the d´evissage cofiber sequence K( k ) → K( O F ) → K( F ) with Z p -coefficientsshows that K i ( O F ; Z p ) ∼ = K i ( F ; Z p ) for i = 1 and that there is an exact sequence0 → K ( O F ; Z p ) → K ( F ; Z p ) → Z p → , where the map K ( F ; Z p ) ∼ = F × ⊗ Z Z p → Z p is induced by the p -adic valuation. N THE BEILINSON FIBER SQUARE 51
Next, since K( O F /p ; Q p ) ≃ K( k ; Q p ) ≃ Q p , the Beilinson fiber square (Theorem A) for O F yieldsa fiber sequence ΣHC( O F ; Q p ) → K( O F ; Q p ) → Q p . But note that the cyclic homology term may be equivalently written as ΣHC(
F/F ), where F := W ( k )[ p ]; indeed, the vanishing of the p -adic completion of the cotangent complex L W ( k ) / Z impliesthat HC( O F ; Z p ) ≃ HC( O F / O F ; Z p ), but HC( O F / O F ) is already derived p -adically complete sinceits homology groups are all finitely generated O F -modules. The Beilinson fiber sequence thereforeimplies that τ ≥ ΣHC(
F/F ) ≃ τ ≥ K( F ; Q p )and HC ( F/F ) ≃ K ( O F ; Q p ) . The proof is completed by noting that, since F is an ´etale F -algebra, its cyclic homology is given byHC i ( F/F ) = F for i ≥ (cid:3) Example 7.2 (Local fields) . If F is a finite extension of Q p , of degree d , the theorem shows thatdim Q p K s − ( F ; Q p ) = ( d + 1 if s = 1, d otherwise. (57)This dimension calculation is a classical result, arising from Wagoner’s [Wag76] calculation of theranks of the continuous K-groups and Panin’s [Pan87] proof of an early case of the p -adic continuityof K-theory.In addition, the dimension calculation (57) is in accordance with the Beilinson–Lichtenbaum con-jecture for F . Recall that the Beilinson–Lichtenbaum conjecture, prior to its general proof by Rost–Voevodsky for all fields, was proved by Hesselholt–Madsen [HM03] in this case when p > F has cohomological dimension 2, the Beilinson–Lichtenbaum conjec-ture predicts K s − ( F ; Q p ) ≃ H ( F, Q p ( s )) and K s − ( F ; Q p ) ≃ H ( F, Q p ( s )) for s >
0. Then thedimensions in (57) agree with the dimensions of the Q p -cohomology of F , as computed via Tate’slocal duality and Euler characteristic formula (see [NSW08, VII.3] for an account). Example 7.3 (Integral calculation, unramified case) . Assume in this example that p > ≤ i ≤ p −
5, the p -adic K-groupsof W ( k ) are given by K i ( W ( k ); Z p ) ≃ ( W ( k ) if i = 2 s − , i is even. (58)Note that for k finite, the calculation of the entire homotopy type of K( W ( k ); Z p ) is carried out byB¨okstedt–Madsen [BM95] at odd primes and Rognes [Rog99] at p = 2 (for k = F ); again, see [Wei05,Theorem 61] for a survey for all of these results.The integral form of the Beilinson fiber sequence (Corollary B) takes the form of a natural fibersequence τ ≤ p − ΣHC( W ( k ) , ( p ); Z p ) → τ ≤ p − K( W ( k ); Z p ) → Z p . As in the proof of Theorem 7.1, the cyclic homology term may be replaced by the cyclic homology τ ≤ p − ΣHC( W ( k ) , ( p ) /W ( k )) over W ( k ).A standard calculation of derived de Rham cohomology with divided powers (as in Proposition 6.26)gives L Ω k /L Ω ≥ sk ≃ W ( k ) /p s for s ≤ p −
1; in general, L Ω k /L Ω ≥ sk is discrete for all s . Using the filtrations of [Ant19], we conclude for i ≤ p − π i HC( k/W ( k )) ≃ ( W ( k ) /p s +1 if i = 2 s ≥ i ( W ( k ); Z p ) ∼ = HC i ( W ( k ) /W ( k )) ∼ = W ( k )for i ≥ ≤ i ≤ p −
2, we deduce that π i HC(( W ( k ) , ( p ) /W ( k )) ≃ ( p s +1 W ( k ) if i = 2 s, i is oddThis completes the proof. Remark 7.4 (Integral calculation, ramified case) . Assume now that F is ramified so that O F /p ∼ = k [ x ] / ( x e ), where e is the absolute ramification degree of F . Corollary B implies, by rewriting the p -adiccyclic homology as non-completed cyclic homology with respect to W ( k ), that τ ≤ p − ΣHC( O F , ( p ) /W ( k )) ≃ τ ≤ p − K( O F , ( p ); Z p ).We now appeal to the fact that the algebraic K-theory of truncated polynomial rings over fieldsis known [HM97a, Spe19]. The positive even p -adic K-groups of k [ x ] / ( x e ) vanish so that we get asurjection π p − ΣHC( O F , ( p ) /W ( k )) → K p − ( O F ; Z p )and hence 5-term exact sequences0 → π s − ΣHC( O F , ( p ) /W ( k )) → K s − ( O F ; Z p ) → W se ( k ) /V e W s ( k ) → π s − ΣHC( O F , ( p ) /W ( k )) → K s − ( O F ; Z p ) → ≤ s ≤ p −
2. In low degrees, this gives a computation of the integral p -adic K-groups of O F which is independent of [HM03]; on the other hand, using this calculation and [HM03], we can viewthe computation as giving information about the low-degree ´etale cohomology of F .We can also carry out previous types of calculations for the syntomic complexes Z p ( i ) rather thanK-theory. Theorem 7.5 (Syntomic cohomology of DVRs) . Let O F be a complete discrete valuation ring ofmixed characteristic (0 , p ) with perfect residue field k . (1) We have natural identifications Q p ( i )( O F ) ≃ ( R Γ proet (Spec( k ) , Q p ) if i = 0 ,F [ − if i > If O F is unramified, Z p ( i )( W ( k )) ≃ ( R Γ proet (Spec( k ) , Z p ) if i = 0 W ( k )[ − if < i ≤ p − . (61) Proof.
For (1), we have (in view of Lemma 6.19 to compare with the perfect residue field) Q p ( i )( O F /p ) ≃ Q p ( i )( k ) ≃ ( R Γ proet (Spec( k ) , Q p ) if i = 0,0 otherwise.Therefore, the fiber sequence from Theorem 6.17 yields equivalences Q p ( i )( O F ) ≃ R Γ proet (Spec( k ) , Q p ) if i = 0 , (cid:16) L Ω O F /L Ω ≥ i O F (cid:17) Q p [ −
1] if i > . N THE BEILINSON FIBER SQUARE 53
As in the proof of Theorem 7.1, the latter truncated p -adic derived de Rham cohomologies may becomputed as the analogous un-completed derived de Rham cohomologies for F → F ; since L F/F ≃ Q p ( i )( O F ) ≃ F [ −
1] for i > Z p (0)( W ( k )) = fib( ϕ − W ( k ) → W ( k )) ≃ R Γ proet (Spec( k ) , Z p ). For i >
0, we get Z p ( i )( W ( k )) = fib( ϕ/p i − → W ( k )),so the claim follows. (cid:3) Perfectoid rings.
In this section, we apply the Beilinson fiber square to a perfectoid ring. Themain result (which was indicated to us by Scholze) is that it recovers the fundamental exact sequencein p -adic Hodge theory.Let R be a perfectoid ring. We review the period rings associated to R and their interpretation viaderived de Rham theory, cf. [Bei12, Bha12]. Construction 7.6 (Period rings) . Let R be a perfectoid ring.(1) As before, we have Fontaine’s ring A inf ( R ) equipped with the canonical map θ : A inf ( R ) → R with kernel ( ξ ). Here A inf ( R ) is also the prismatic cohomology b ∆ R ; the Nygaard filtration isthe ξ -adic filtration.(2) We have A crys ( R ) = A crys ( R/p ), the p -adic completion of the divided power envelope of( ξ ) ⊂ A inf ( R ); we have A crys ( R ) ≃ L Ω R is the derived de Rham cohomology of R . The Hodgefiltration is given by the divided power filtration. We let B +crys ( R ) = A crys ( R )[1 /p ] = ( L Ω R ) Q p ;the ring B +crys ( R ) also inherits a Frobenius operator ϕ .(3) We have B +dR ( R ) = lim ←− (A inf ( R ) /ξ n [1 /p ]). The ring B +dR ( R ) can also be obtained as theHodge completion of ( L Ω R ) Q p . The Hodge filtration yields a filtration (the ξ -adic filtration)on B +dR ( R ).Our goal is now to recover the following result in p -adic Hodge theory, cf. [Fon94, Theorem 5.3.7]and [FF18, Theorem 6.4.1]. Theorem 7.7 (The fundamental exact sequence) . For any R ∈ Perfd and i > , there is a naturalpullback square in D ( Q p ) , RΓ proet (Spec( R [1 /p ]) , Q p ( i )) (cid:15) (cid:15) / / B +crys ( R ) ϕ = p i (cid:15) (cid:15) Fil ≥ i B +dR ( R ) / / B +dR ( R ) . (62) Example 7.8.
When R = O C , where C is a complete algebraically closed nonarchimedean field, thefundamental exact sequence is often written as the exact sequence0 → Q p ( i ) → B +crys ( O C ) ϕ = p i → B +dR ( O C ) / Fil ≥ i B +dR ( O C ) → Proof of Theorem 7.7.
We apply Theorem 6.17 to the perfectoid ring R and obtain a fiber square Q p ( i )( R ) (cid:15) (cid:15) / / Q p ( i )( R/p ) (cid:15) (cid:15) ( L Ω ≥ iR ) Q p / / ( L Ω R ) Q p . (63)By [BS19, Theorem 9.4], the first term is identified with RΓ proet (Spec( R [1 /p ]) , Q p ( i )). The ring R/p is quasiregular semiperfect, so we have Q p ( i )( R/p ) ≃ B +crys ( R ) ϕ = p i [BMS19, Sec. 8]. Note that we can replace the square (63) by Hodge completing the bottom row and it will stillremain cartesian, since the homotopy fibers do not change. This yields a new homotopy cartesiansquare where one identifies the rings as Construction 7.6, and then the result follows. (cid:3)
Remark 7.9 (Identifying the maps) . Unfortunately, in general we do not know a good way of iden-tifying the map K( O C /p ; Q p ) → HP( O C ; Q p ) with the usual map in the fundamental exact sequence.However, we can argue that it has to match with the usual map, at least for C = C p , by appealingto some general results. For simplicity, in this example, we drop the argument of the perfectoid ring,i.e., we write B +dR for B +dR ( O C p ), etc.Our first goal is to identify the map obtained from π in (62),(B +crys ) ϕ = p → B +dR ; (64)by construction, it is Gal( Q p )-equivariant. Now we have a (Galois-equivariant) short exact sequence0 → Q p (1) → (B +crys ) ϕ = p → C p → Q p (1) ⊂ (B +crys ) ϕ = p is essen-tially determined: it is given by the dlog map to derived de Rham cohomology as it comes from theusual Chern character K( O C ; Q p ) → HC − ( O C ; Q p ), for O C . As is proved in [Fon82, Prop. 2.17], theimage of a generator of Q p (1) gives a uniformizer of B +dR (which is a DVR).Recall that B +dR has a complete, exhaustive filtration (via powers of the augmentation ideal) withassociated graded given by C p , C p (1) , C p (2) , . . . . Moreover, there are no Gal( Q p )-equivariant maps C p → B +dR (see [Fon94, Rem. 1.5.8]). Thus there is at most one (and hence exactly one, by construc-tion) Galois-equivariant map (B +crys ) ϕ = p → B +dR which extends the dlog map. This shows that themap (64) is actually completely determined by its behavior on Q p (1).Now by a deep result of Fargues–Fontaine, the graded ring L i ≥ B +crys ( O C ) ϕ = p i is generated indegree one [FF18, Th. 6.2.1], so the maps for higher i are determined by their behavior for i = 1by multiplicativity. In particular, these observations show that the maps in the fundamental exactsequence, although here they are produced by topological means, are entirely determined by theirvalue on Q p (1), as long as they are Galois-equivariant.7.3. Application to p -adic nearby cycles. Let C be an algebraically closed, complete nonar-chimedean field of mixed characteristic (0 , p ). In [BMS19, Sec. 10], an explicit description of the Z p ( i ) sheaves is given for smooth formal schemes over O C . Using this, we can recover some cases ofcomparison results of Colmez–Nizio l [CN17]. Definition 7.10 ( p -adic nearby cycles) . Let X be a formal scheme over O C . We consider the pro-´etalesite X proet of X (equivalently, of its special fiber), cf. [BS15].For each i , we consider the sheaf of p -adic nearby cycles Rψ ∗ ( Z p ( i )), which is a D ( Z p ) ≥ -valuedsheaf on X proet . Explicitly, given an affine pro-´etale open Spf A → X , we have that R Γ(Spf
A, Rψ ∗ ( Z p ( i ))) ≃ R Γ proet (Spec A [1 /p ] , Z p ( i )) is the pro-´etale cohomology of A [1 /p ] with values in the (usual) sheaf Z p ( i ). Theorem 7.11 (Bhatt–Morrow–Scholze [BMS19]) . Let R be a formally smooth O C -algebra and let X = Spf( R ) . Then, as sheaves on X proet , we have a natural equivalence Z p ( i ) ≃ τ ≤ i Rψ ∗ ( Z p ( i )) . Inparticular, it follows that Z p ( i )( R ) ≃ R Γ( X proet , τ ≤ i Rψ ∗ ( Z p ( i ))) . The functor ψ here should refer to the generic fiber functor, but we do not define it here to avoid technicalities. Here we can consider either the scheme Spec( A [1 /p ]) or the rigid analytic generic fiber by the affinoid comparisontheorem [Hub96, Cor. 3.2.2]. N THE BEILINSON FIBER SQUARE 55
To apply this, let K be a discretely valued field with perfect residue field k , and suppose K ⊂ C (e.g., we could take C = b K ). Let X be a smooth proper formal scheme over O K with generic fiber X , a smooth proper rigid space over K . Construction 7.12 (de Rham cohomology of formal schemes and rigid spaces) . We let L Ω X / O K denote the ( p -adic) derived de Rham cohomology of X over O K equipped with its Hodge filtration.In fact, L Ω X / O K is also the p -complete usual de Rham complex since X is formally smooth over O K , cf. [Bha12], and the Hodge filtration is a finite filtration. We let (by a slight abuse of notation)Ω X /K = ( L Ω X / O K ) Q p denote the rationalization, which we can interpret as the de Rham cohomologyof the rigid generic fiber X . Note that L Ω X / O K is a perfect O K -module, and Ω X /K is a perfect K -module.We also consider the ring B +dR = B +dR ( O C ) with its ξ -adic filtration. Together, it follows thatΩ X /K ⊗ K B +dR admits a filtration in the derived ∞ -category D ( K ). Then one has the followingresult, a special case of results of [CN17] in the case of good reduction; note that [CN17] treatsthe more general semistable case, which we do not consider here. In the following, all references toA crys , B +dR , etc. will implicitly be with respect to the perfectoid ring O C . Theorem 7.13 (Cf. Colmez–Nizio l [CN17]) . Let X / O K be a smooth proper formal scheme. Let X denote the base change of X to O C and X its reduction modulo π . For each i ≥ , we have a naturalpullback square in D ( Q p ) , R Γ( X proet , τ ≤ i Rψ ∗ ( Q p ( i ))) (cid:15) (cid:15) / / (A crys ⊗ W ( k ) R Γ crys ( X /W ( k ))) ϕ = p i [1 /p ] (cid:15) (cid:15) Fil ≥ i (Ω X /K ⊗ K B +dR ) / / (Ω X /K ⊗ K B +dR ) . Proof.
We claim that this follows from Theorem 6.17, applied to X . Let π ∈ O K be a uniformizer.Note that (54) gives a fiber square Q p ( i )( X ) (cid:15) (cid:15) / / Q p ( i )( X /π ) (cid:15) (cid:15) ( L Ω ≥ i X / O K ) Q p / / ( L Ω X / O K ) Q p (65)in fact, Q p ( i )( X /π ) → Q p ( i )( X /p ) is an isomorphism thanks to Lemma 6.19.The top left term in (65) is identified via Theorem 7.11. For the top right, we observe that thereis a natural equivalence X ⊗ O C O C /π ≃ X ⊗ k O C /π. For F p -algebras, the construction LW Ω ( − ) satisfies a K¨unneth formula, so we get LW Ω X /π ≃ LW Ω X ⊗ W ( k ) A crys . Note that we do not need to p -complete again, since X is smooth and proper. Taking Frobenius fixedpoints, we identify the top right term now rationally, thanks to (53).We can replace the bottom row of (65) with its completion with respect to the (rationalized) Hodgefiltration. Recall that p -adic derived de Rham cohomology together with its Hodge filtration (so as afiltered E ∞ -algebra) satisfies a K¨unneth formula. Therefore, we have L Ω X / O K ≃ L Ω X / O K ⊗ O K L Ω O C / O K , since the filtration on L Ω X / O K is finite, and it is by perfect O K -modules. It follows that the Hodgecompletion of the rationalization of L Ω X / O K is equivalent, in the filtered derived ∞ -category of K , toΩ X /K ⊗ K B +dR , where we use Construction 7.6 for the identification with B +dR . (cid:3) Appendix A. Twisted Tate diagonals
In this section we investigate under which conditions Hochschild homology in a general symmetricmonoidal ∞ -category admits a (twisted) cyclotomic structure. The main result is Corollary A.9 andthe fact that it applies to graded and filtered THH as recorded in Examples A.10 and A.11.As usual, we fix a prime p . Let C be a presentably symmetric monoidal ∞ -category and L : C → C be a symmetric monoidal, left adjoint functor.
Definition A.1. An L -twisted diagonal on C is a symmetric monoidal natural transformation∆ : L ( C ) → ( C ⊗ · · · ⊗ C ) hC p of lax symmetric monoidal functors C → C . Assume C is additionally stable; then an L -twisted Tatediagonal is a lax symmetric monoidal natural transformation∆ : L ( C ) → T p ( C ) := ( C ⊗ · · · ⊗ C ) tC p . Example A.2. (1) The ∞ -category of spaces admits a (unique) id-twisted diagonal and the ∞ -category of spectra admits a (unique) id-twisted Tate diagonal, see [NS18, Section III.1].(2) More generally, if C admits the cartesian symmetric monoidal structure then it admits acanonical id-twisted diagonal induced by the actual diagonal. Example A.3.
Suppose R is an E ∞ -ring and C = Mod R . Then every left adjoint, symmetric monoidalfunctor L is given by an induction along an E ∞ -map l : R → R and we will prove below that the datumof an L -twisted Tate diagonal on C is equivalent to an E ∞ -homotopy between the composition R l −→ R triv −−→ R tC p and the Tate-valued Frobenius ϕ : R → R tC p of R , [NS18, IV.1]. We shall refer to an E ∞ -ring R withsuch a datum as a cyclotomic base. An example is R = S [ z ], see Example A.12 below. Proof.
To see that the datum of a twisted Tate diagonal is equivalent to such an equivalence, wefirst note that a symmetric monoidal natural transformation L → T p of functors Mod R → Mod R isdetermined by its restriction to the perfect modules Mod ωR ⊂ Mod R since L preserves filtered colimits.Since T p is lax symmetric monoidal we get a factorizationMod R T p −→ Mod T p ( R ) res triv −−−−→ Mod R as lax symmetric monoidal functors. Upon restriction to Mod ωR the first functor is given by base-changealong the Tate-valued Frobenius ϕ : R → R tC p such that we get a factorization T p | Mod ωR = res triv ◦ ind ϕ .Now a symmetric monoidal transformation L | Mod ωR = ind l → res triv ◦ ind ϕ is by adjunction equivalent to a natural transformationind triv ◦ l = ind triv ind l → ind ϕ . In fact, semiadditive (so that the Tate construction is defined) suffices.
N THE BEILINSON FIBER SQUARE 57
But every object in Mod ωR is dualizable and both functors ind triv ◦ l and ind ϕ are symmetric monoidal.Thus every such symmetric monoidal transformation is necessarily an equivalence and thus inducedby an equivalence of maps of E ∞ -rings R → R tC p . This shows the claim. (cid:3) Remark A.4.
Note that a general symmetric monoidal, stable ∞ -category C does not admit L -twisted Tate-diagonals for arbitrary L . For example if we consider C = D ( Z ) ≃ Mod H Z then byExample (3) above a twisted Tate diagonal would be the same as a factorization of the Tate-valuedFrobenius H Z → ( H Z ) tC p through the triv-map H Z → H Z tC p . These two maps however differ by Steenrod operations as shownin [NS18, IV.1]. But any twist would be the identity.In the following, we use the notation HH A to denote the Hochschild homology object of an algebraobject A ∈ Alg( C ) internal to C ; e.g., if C = Sp, this recovers THH. Proposition A.5.
Assume that C is equipped with an L -twisted diagonal resp. Tate diagonal. Thenwe get for each algebra object A ∈ Alg( C ) an induced S -equivariant map L (HH A ) → (HH A ) hC p resp. L (HH A ) → (HH A ) tC p functorial and lax symmetric monoidal in A .Proof. We closely follow the construction of the cyclotomic structure on THH given in [NS18, Sec. III.2].We will mostly indicate the necessary changes and thus recommend that the reader take a look at theconstruction there first. We treat the case of the Tate diagonal which is the only case that we willneed in this paper, but the case of the diagonal works exactly the same.We first recall that HH A is the geometric realization of the cyclic object in C informally written as · · · / / / / / / / / A ⊗ A ⊗ A C (cid:6) (cid:6) / / / / / / A ⊗ A C (cid:6) (cid:6) / / / / A .
Thus L (HH A ) is the geometric realization of the cyclic object · · · / / / / / / / / L ( A ⊗ A ⊗ A ) C (cid:7) (cid:7) / / / / / / L ( A ⊗ A ) C (cid:7) (cid:7) / / / / L ( A ) . For a given L -twisted Tate diagonal∆ : L ( C ) → ( C ⊗ ... ⊗ C ) tC p = T p ( C )we want to construct a natural map of cyclic objects · · · / / / / / / / / L ( A ⊗ ) ∆ (cid:15) (cid:15) C (cid:8) (cid:8) / / / / / / / / L ( A ⊗ ) ∆ (cid:15) (cid:15) C (cid:8) (cid:8) / / / / LA ∆ (cid:15) (cid:15) · · · / / / / / / / / (cid:0) A ⊗ p (cid:1) tC p C U U / / / / / / / / (cid:0) A ⊗ p (cid:1) tC p C U U / / / / (cid:0) A ⊗ p (cid:1) tC p , and obtain the desired map L (HH A ) → (HH A ) tC p as the geometric realization of this map of cyclicobjects followed by the canonical interchange map from the realization of the Tate constructions tothe Tate construction of the realization. In order to construct such a natural transformation of cyclic objects, we proceed as in [NS18]:we eventually need to show that we can extend the lax symmetric monoidal natural transformation∆ : L → T p of functors C → C to a BC p -equivariant lax symmetric monoidal natural transformationof functors from the functor˜ L : N (Free C p ) × N (Fin) C ⊗ act pr −→ C ⊗ act ⊗ −→ C L −→ C given by ( S, ( X s ∈ S = S/C p )) L (cid:16) O s ∈ S X s (cid:17) to the functor ˜ T p : N (Free C p ) × N (Fin) Sp ⊗ act → ( C ⊗ act ) BC p ⊗ −→ C BC p − tCp −−−→ C given by ( S, ( X s ∈ S = S/C p )) ( O s ∈ S X s ) tC p . Here Free C p is the category of finite free C p -sets equipped with the cocartesian symmetric monoidalstructure. The group object BC p -acts on this category in the obvious way and acts trivially on C .For a precise construction of these functors we refer to [NS18, Section III.3], specifically PropositionIII.3.6 and the construction around that.The inclusionFun ⊗ (cid:0) N (Free C p ) × N (Fin) C ⊗ act , C (cid:1) ⊆ Fun lax (cid:0) N (Free C p ) × N (Fin) C ⊗ act , C (cid:1) admits a right adjoint by Lemma III.3.3 resp. Remark III.3.5 in [NS18]. Using the same constructionand argument as in the proof of [NS18, Lemma III.3] we see that the ∞ -category Fun ⊗ (cid:0) N (Free C p ) × N (Fin) C ⊗ act , C (cid:1) is equivalent to the ∞ -category Fun( N Tor C p , Fun lax ( C , C )) where Tor C p denotes the category of C p -torsors. Under this equivalence the right adjoint to the inclusion is given by restricting a functor inFun lax (cid:0) N (Free C p ) × N (Fin) C ⊗ act , C (cid:1) to N Tor C p × C ⊆ N (Free C p ) × N (Fin) C ⊗ act and forming the adjunct.Now the functor ˜ L is symmetric monoidal rather than lax symmetric monoidal. Thus to construct amap from ˜ L to ˜ T p is by adjunction equivalent to constructing a transformation in Fun( N Tor C p , Fun lax ( C , C ))between the respective restrictions. Moreover BC p -acts on all those categories, i.e., to construct a BC p -equivariant transformation between ˜ L and ˜ T p is equivalent to construct a transformation inFun BC p ( N Tor C p , Fun lax ( C , C )) . Now the category Tor C p is in fact equivalent to BC p . Since the BC p -action on Fun lax ( C , C ) is trivialit follows that the above ∞ -category of BC p -equivariant functors is equivalent to Fun lax ( C , C ).Taking everything together we see that there is a unique lax symmetric monoidal transformation˜ L → ˜ T p extending the transformation ∆ : L → T p . Together with the constructions above this finishesthe proof. (cid:3) We shall refer to the map L (HH A ) → (HH A ) tC p as a twisted cyclotomic structure on HH A . Thusthe last result shows that for ∞ -categories with a twisted Tate diagonal we find that Hochschildhomology admits a twisted cyclotomic structure. Lemma A.6.
For a given L -twisted diagonal on C , the stabiliziation Sp( C ) admits a canonical induced Sp( L ) -twisted Tate diagonal.Proof. We would like to construct a symmetric monoidal natural transformationSp( L )( C ) → ( C ⊗ · · · ⊗ C ) tC p = T p ( C ) . N THE BEILINSON FIBER SQUARE 59
Such a transformation is by adjunction the same as a symmetric monoidal transformationid → R ′ T p where R ′ : Sp( C ) → Sp( C ) is the right adjoint to Sp( L ). We now use that the functor Ω ∞ induces anequivalence Fun Exlax (Sp( C ) , Sp( C )) → Fun
Exlax (Sp( C ) , C )by [Nik16]. It follows that it suffices to construct a symmetric monoidal transformationΩ ∞ → Ω ∞ R ′ T p of functors Sp( C ) → C . We denote by R : C → C the right adjoint to the functor L : C → C . Then wehave an equivalence Ω ∞ R ′ ≃ R Ω ∞ of lax symmetric monoidal functors which follows from the factthat the left adjoint diagram C L / / Σ ∞ (cid:15) (cid:15) C Σ ∞ (cid:15) (cid:15) Sp( C ) Sp( L ) / / Sp( C ) . commutes (up to symmetric monoidal equivalence). As a result we need to construct a symmetricmonoidal natural transformation Ω ∞ → R Ω ∞ T p . (66)Now we use that we have canonical symmetric monoidal transformations γ : (Ω ∞ C ⊗ · · · ⊗ Ω ∞ C ) hC p → Ω ∞ (cid:0) ( C ⊗ · · · ⊗ C ) hC p (cid:1) → Ω ∞ (cid:0) ( C ⊗ · · · ⊗ C ) tC p (cid:1) where the first one is induced by the lax symmetric monoidal structure of Ω ∞ together with the factthat it commutes with limits and the second by the canonical map from homotopy fixed points to theTate construction.Now we use the unstable diagonal on C to get as the adjoint a lax symmetric monoidal, naturaltransformation Ω ∞ C → R (Ω ∞ C ⊗ · · · ⊗ Ω ∞ C ) hC p and compose it with the map R ( γ ) above to get a symmetric monoidal transformation as in (66). (cid:3) For every symmetric monoidal ∞ -category I we consider the symmetric monoidal functor l p : I → I given by sending i to i ⊗ p . We let L p : Fun( I, S ) → Fun( I, S )be left Kan extension along l p . We equip the category Fun( I, Sp) with the Day convolution symmetricmonoidal structure. Then the left Kan extension L p becomes symmetric monoidal. Lemma A.7.
Assume that the ∞ -category I has the following property: for every pair of objects i, j ∈ I we have that the canonical forgetful map Map I ( i ⊗ p , j ) hC p → Map I ( i ⊗ p , j ) (67) is an equivalence of spaces. Then the inverse of the map (67) induces a canonical L p -twisted diagonalon Fun( I, S ) . Note that an equivalent way of stating this condition is to say that the homotopy orbits ( i ⊗ p ) hC p exist in I andthe map i ⊗ p → ( i ⊗ p ) hC p is an equivalence. Proof.
We consider the symmetric monoidal (co)Yoneda embedding I op → Fun( I, S ) . Then symmetric monoidal transformations L p ( C ) → ( C ⊗ · · · ⊗ C ) hC p as functors Fun( I, S ) → Fun( I, S ) are the same as symmetric monoidal transformations between therestrictions of the functors along the Yoneda embedding. The restricted functors I op → Fun( I, S ) aregiven by the lax symmetric monoidal assignments i ( j Map I ( i ⊗ p , j )) and i ( j Map I ( i ⊗ p , j ) hC p ) . The canonical map Map I ( i ⊗ p , j ) hC p → Map I ( i ⊗ p , j ) is a lax symmetric monoidal transformation. Byassumption it is an equivalence so that the inverse induces the required transformation. (cid:3) Remark A.8.
For a general symmetric monoidal ∞ -category I the category Fun( I, S ) does not admitan L p -twisted diagonal. As an example consider any cocartesian symmetric monoidal ∞ -category I .Then the day convolution structure on Fun( I, Sp) is cartesian. Thus an L p -twist Tate diagonalwould amount to a natural symmetric monoidal transformation F × p → ( F × p ) hC p which does not exist.But note that this category admits an id-twisted diagonal. This raises the question if for everysymmetric monoidal ∞ -category I there is a twist on Fun( I, S ) and a twisted diagonal. The answer tothis question is also ‘no’ in general but we will not go into the intricacies of concrete counterexampleshere. Corollary A.9. If I is a symmetric monoidal ∞ -category satisfying the condition of Lemma A.7then we have for every algebra A in Fun( I, Sp) a twisted cyclotomic structure on HH A , i.e. an S -equivariant map L p (HH A ) → (HH A ) tC p . This map is natural and symmetric monoidal in A .Proof. Combine Proposition A.5 with Lemma A.6 and Lemma A.7. (cid:3)
Example A.10.
We consider the category I = Z ds ≥ . Then Fun( I, Sp) is the ∞ -category of gradedspectra. The category I obviously satisfies the condition of Lemma A.7. Thus we get that for agraded ring R • we get that graded THH admits an L p -twisted cyclotomic structure or equivalently asequence of S -equivariant maps THH( R ) i → THH( R ) tC p pi . The same logic applies to spectra graded over any discrete monoid in place of Z ds ≥ . Example A.11.
Consider the ∞ -category I = Z op ≥ associated to the poset of positive integers. Thenthis also satisfies the condition of Lemma A.7. The category of functors Fun( I, Sp) is given by filteredspectra and thus filtered THH of a filtered ring spectrum R admits a filtered cyclotomic structure,i.e. S -equivariant maps Fil ≥ i THH( R ) → (Fil ≥ pi THH( R )) tC p . This follows from the fact that generally Day convolution for a cocartesian source is given by the pointwise tensorproduct, which in our case happens to agree with the cartesian product.
N THE BEILINSON FIBER SQUARE 61
Example A.12.
Consider the category I = B Z ≥ . This category also obviously satisfies the conditionof Lemma A.7. Thus the category Fun( I, Sp) ≃ Mod S [ z ] admits a twisted Tate diagonal and thus relative THH admits a (twisted) cyclotomic structure, as isused in [BMS19, Sec. 11]. The twist L p corresponds to the map l : S [ z ] → S [ z ] sending z to z p .We want to end this section by remarking some functorialities of the twisted cyclotomic structures. Definition A.13. A symmetric monoidal category with (Tate) diagonals consists of a triple ( C , L, ∆)as in Definition A.1. A map of symmetric monoidal categories with (Tate) diagonals( C , L, ∆) → ( C ′ , L ′ , ∆ ′ )is given by a left adjoint symmetric monoidal functor F : C → C ′ together with a symmetric monoidalequivalence L ′ ◦ F ≃ F ◦ L and a natural symmetric monoidal equivalence between the two maps L ′ ( F X ) → ( F X ⊗ · · · ⊗
F X ) tC p induced from ∆ and ∆ ′ (both sides considered as lax symmetric monoidal functors C → C ′ ).Form the construction of the twisted cyclotomic structure in Proposition A.5 we see immediatelythat for such a map of symmetric monoidal ∞ -categories with Tate diagonals we get an equivalenceof twisted cyclotomic objects F (HH A ) ≃ HH(
F A )for every algebra A in C . Here the first object F (HH A ) is twisted cyclotomic by the composition LF (HH A ) ≃ −→ F L (HH A ) F ϕ −−→ F (HH A tC p ) → F (HH A ) tC p . We also have a relative analogue of Lemma A.6: every map of symmetric monoidal ∞ -categories withdiagonals induces upon stabilization a map of symmetric monoidal ∞ -categories with Tate diagonals.This is straightforward to prove. Finally there is also an analogue of Lemma A.7 which we will stateand prove now. Lemma A.14.
Assume that f : I → I ′ is a symmetric monoidal functor such that I and I ′ satisfythe condition of Lemma A.7. Then left Kan extension along f induces a map of symmetric monoidal ∞ -categories with diagonals (Fun( I, S ) , L p , ∆) → (Fun( I ′ , S ) , L ′ p , ∆ ′ ) . where L p , L ′ p , ∆ and ∆ ′ are as in Lemma A.7.Proof. We have a commutative square I f / / l p (cid:15) (cid:15) I ′ l ′ p (cid:15) (cid:15) I f / / I ′ for the functors l p ( i ) = i ⊗ p and l ′ p ( j ) = j ⊗ p . Thus we get an induced square of the left Kan extensionsFun( I, S ) F / / L p (cid:15) (cid:15) Fun( I ′ , S ) L ′ p (cid:15) (cid:15) Fun( I, S ) F / / Fun( I ′ , S ) . This provides the first part of the datum of a map of symmetric monoidal ∞ -categories with Tatediagonals. We now also have to provide an equivalence of two different natural transformationsbetween two functors Fun( I, S ) → Fun( I ′ , S ) . Such a transformation is determined by its restriction to I op ⊆ Fun( I, Sp) and there the functors aregiven by i (cid:0) j Map I ′ ( f ( i ) ⊗ p , j ) (cid:1) and i (cid:0) j Map I ′ ( f ( i ) ⊗ p , j ) hC p (cid:1) . Unravelling the constructions we see that both of the two transformations are given by the inverse ofthe canonical forgetful map Map I ′ ( f ( i ) ⊗ p , j ) hC p → Map I ′ ( f ( i ) ⊗ p , j )and thus are canonically equivalent. (cid:3) From these statements together we can deduce the following corollary:
Corollary A.15.
Assume that f : I → I ′ is a symmetric monoidal functor such that I and I ′ satisfythe condition of Lemma A.7. Then for every algebra A ∈ Fun( I, Sp) we have an equivalence of L ′ p twisted cyclotomic objects F (HH A ) ≃ HH(
F A ) where F is left Kan extension along f . Example A.16.
For a graded ring spectrum R • we have that the direct sum M i THH( R • ) i is equivalent to THH( L i R ) as cyclotomic spectra. Similarly, for a filtered ring spectrum R we havethat the filtered cyclotomic structure refines the cyclotomic structure on THH( R ). Example A.17.
We finally note that one can also look at the functorev : Fun( Z ds ≥ , Sp) → Spgiven by restriction to the 0-th component. We claim that this also refines to a map of symmetricmonoidal ∞ -categories with Tate diagonals. This can be seen by verifying the corresponding unstablestatement which is straightforward using an argument similar to the one in the proof of CorollaryA.15. This then shows that the cyclotomic structure on the 0-th graded component THH( R • ) agreeswith the one on THH( R ) for every graded ring spectrum R • . Appendix B. Categorical lemmas
Construction B.1 (Left Kan extensions) . Let R be a ring, and let Poly R be the category offinitely generated polynomial R -algebras. Given a presentable ∞ -category C and an accessible functor f : Poly R → C , we can left Kan extend to obtain a functor Lf : SCR R → C which commutes withgeometric realizations, for SCR R the ∞ -category of simplicial commutative R -algebras. Compare[Lur09, Sec. 5.5.8] and [Lur11, Sec. 4.2].Let ( L , R ) : C ⇄ D be an adjunction of ∞ -categories. Then for any ∞ -category E , we obtain anadjunction ( R ∗ , L ∗ ) = ( f f ◦ R , f ′ f ◦ L ) : Fun( C , E ) ⇄ Fun( D , E ) . (68) N THE BEILINSON FIBER SQUARE 63
Remark B.2.
Let f , f : D → E be functors. Suppose that for any x ∈ D , the natural map f ( LR x ) → f ( x ) is an equivalence. Then we findHom Fun( D , E ) ( f , f ) ≃ Hom
Fun( C , E ) ( f ◦ L , f ◦ L ) . (69)This follows from the adjunction (68).Now we specialize to the case where C = SCR is the ∞ -category of simplicial commutative ringsand D = SCR F p is the ∞ -category of simplicial commutative F p -algebras. We have an adjunction( L , R ) : SCR ⇄ SCR F p , where the left adjoint is R R ⊗ L Z F p and the right adjoint is simply theforgetful functor.For any R ∈ SCR F p , we have a canonical endomorphism ϕ : R → R , the Frobenius. Lemma B.3.
Let R ∈ SCR F p . There is a natural map f : R → R ⊗ L Z F p in SCR F p such thatthe composites R f → R ⊗ L Z F p → R and R ⊗ L Z F p → R → R ⊗ L Z F p are the respective Frobeniusendomorphisms.Proof. It suffices to assume that R is discrete (even a finitely generated polynomial ring) via left Kanextension. In this case, R ⊗ L Z F p is concentrated in homological degrees zero and one (with π = R itself), and one knows that the Frobenius endomorphism annihilates π , cf. [BS17, Prop. 11.6]. Thus,the Frobenius map R ⊗ L Z F p → R ⊗ L Z F p factors canonically through the truncation map R ⊗ L Z F p → π ( R ⊗ L Z F p ) ∼ = R . This gives the map f as desired. (cid:3) Corollary B.4.
Let F , F : SCR F p → D ( Z ) be two functors. Suppose F has the property thatthe natural map F ( R ) → F ( R ) given by Frobenius is multiplication by p i . Then for any naturaltransformation u : F ( − ⊗ L Z F p ) → F ( − ⊗ L Z F p ) of functors SCR Z → D ( Z ) , we have that p i u arisesfrom a natural transformation F → F . In fact, we have Hom
Fun(SCR Z ,D ( Z )) ( F ( − ⊗ L Z p F p ) , F ( − ⊗ L Z p F p ))[1 /p ] ≃ Hom
Fun(SCR F p ,D ( Z )) ( F , F )[1 /p ] . Proof.
This follows from (68) in the case of the adjunction ( L , R ) : SCR Z ⇄ SCR F p . By construction,we are given a map L ∗ F → L ∗ F of functors SCR F p → D ( Z ), or equivalently by adjointness a map R ∗ L ∗ F → F of functors SCR Z → D ( Z ). Now we have a natural map F → R ∗ L ∗ F given bythe natural map f : R → R ⊗ L Z p F p of Lemma B.3; it has the property that the composites in eitherorder with the adjunction map R ∗ L ∗ F → F are given by multiplication by p i . The composition F → R ∗ L ∗ F → F defines the desired map F → F . This argument also proves the displayedequation. (cid:3) Let K be a complete discretely valued field with ring of integers O K ⊂ K and residue field k ; let π ∈ O K be a uniformizer. Let FSmooth O K denote the category of topologically finitely generated,formally smooth O K -algebras and let Smooth k denote the category of smooth k -algebras.For the next result, we will argue similarly, but with a smaller set of ∞ -categories. For thesefiniteness conditions, see [Lur17, Sec. 7.2] (in the slightly more complicated E ∞ -case). Definition B.5. (1) Let SCR afp k denote the ∞ -category of simplicial commutative k -algebras R which are almost finitely presented: equivalently, π ( R ) is finitely generated as a k -algebraand each π i ( R ) is a finitely generated π ( R )-module. Equivalently, R belongs to SCR afp k ifand only if R can be written as the geometric realization of a simplicial diagram of finitelygenerated polynomial k -algebras.(2) Similarly, we define [ SCR afp O K be the ∞ -category of π -complete simplicial commutative O K -algebras R such that π ( R ) is topologically finitely generated over O K (i.e., a quotient of a π -completed polynomial ring) and each π i ( R ) is finitely generated over R . Equivalently, R belongs to [ SCR afp O K if and only if R can be written as the geometric realization of a simplicialdiagram of π -completed finitely generated polynomial O K -algebras. Yet another characteriza-tion is that R should be almost finitely presented over the π -completion of a finitely generatedpolynomial algebra over O K . Corollary B.6.
Let E be an ∞ -category admitting sifted colimits. Let F , F : SCR afp k → E be functors.If (1) F commutes with geometric realizations and (2) F ( R ) ≃ F ( π R ) for R ∈ SCR afp k ,then Hom
Fun(Smooth k , E ) ( F , F ) ≃ Hom
Fun(FSmooth O K , E ) ( F ( − ⊗ O K k ) , F ( − ⊗ O K k )) . Proof.
Since F is left Kan extended from smooth (even finite type polynomial) k -algebras as itcommutes with geometric realizations, we haveHom Fun(Smooth k , E ) ( F , F ) ≃ Hom
Fun(SCR afp k , E ) ( F , F ) . Similarly,Hom
Fun(FSmooth O K , E ) ( F ( − ⊗ O K k ) , F ( − ⊗ O K k )) ≃ Hom
Fun( [ SCR afp O K , E ) ( F ( − ⊗ O K k ) , F ( − ⊗ O K k )) , because F ( −⊗ O K k ) : [ SCR afp O K → E is left Kan extended from FSmooth O K . Now we have an adjunction [ SCR afp O K ⇄ SCR afp k given by base-change and restriction of scalars. Thus, the result follows as inRemark B.2. (cid:3) References [AMN18] Benjamin Antieau, Akhil Mathew, and Thomas Nikolaus,
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