aa r X i v : . [ m a t h . A T ] J a n SOME RECENT ADVANCES IN TOPOLOGICAL HOCHSCHILD HOMOLOGY
AKHIL MATHEW
Abstract.
We give an account of the construction of the Bhatt–Morrow–Scholze motivic filtra-tion on topological cyclic homology and related invariants, focusing on the case of equal charac-teristic p and the connections to crystalline and de Rham–Witt theory. Introduction
Let X be a smooth quasi-projective scheme over a field k . In this case, one has the algebraic K -theory spectrum K ( X ) of X , defined by Quillen [Qui73] using the exact category of vector bundleson X (in general, one should use perfect complexes as in [TT90]). One can think of K ( X ) as a typeof “cohomology theory” for the scheme X , analogously to the topological K -theory of a compacttopological space. With this in mind, the following fundamental result gives an analog of the classicalAtiyah–Hirzebruch spectral sequence relating topological K -theory to singular cohomology. Theorem 1.1 (The motivic filtration on algebraic K -theory, [FS02, Lev08]) . There is a func-torial, convergent, decreasing multiplicative filtration
Fil ≥∗ K ( X ) and identifications gr i K ( X ) ≃ Z ( i ) mot ( X )[2 i ] for i ≥ . Here the Z ( i ) mot ( X ) , called the motivic cohomology of X , are explicit cochain complexes intro-duced by Bloch [Blo86] in terms of algebraic cycles on X × A nk for n ≥ . In particular, we havethat H i ( X, Z ( i )) def = H i ( Z ( i ) mot ( X )) = CH i ( X ) is given by the Chow group CH i ( X ) of codimension i cycles on X modulo rational equivalence. The Z ( i ) mot ( X ) (considered as objects of the derived category of abelian groups) can also be describedas maps in the A -motivic stable homotopy category into motivic Eilenberg–MacLane spectra.Theorem 1.1 gives substantial information about algebraic K -theory, especially after profi-nite completion. After reducing modulo a prime l which is different from the characteristic, theBeilinson–Lichtenbaum conjecture proved by Voevodsky–Rost [Voe03, Voe11] identifies mod l mo-tivic cohomology as Zariski (or Nisnevich) sheaves,(1) Z /l ( i ) mot ≃ τ ≤ i ( Rν ∗ µ ⊗ il ) , for ν the pushforward from the étale to the Zariski topology. For example, Theorem 1.1 implies thatin high degrees, we can compute mod l algebraic K -theory of a variety over C as the topological K -theory of the space of C -points. (By contrast, K with mod l coefficients can be enormous, cf. forinstance [Sch02, RS10, Tot16].) Date : January 5, 2021. One difference in this analogy is that algebraic K -theory historically preceded motivic cohomology, whereassingular cohomology preceded topological K -theory. After p -adic completion when the ground field k has characteristic p , the analog of the Beilinson–Lichtenbaum conjecture (1) is given by the theorem of Geisser–Levine [GL00] and Bloch–Kato–Gabber [BK86], Z /p ( i ) mot ≃ Ω i log [ − i ] , identifying the object Z /p ( i ) mot (which lives in the derived category of Zariski or Nisnevich sheaveson X ) with the − i -shift of the subsheaf Ω i log ⊂ Ω i of differential i -forms generated by dx x ∧ · · · ∧ dx i x i ,for the x i local units. For example, this implies that if X is a smooth variety over a perfect field ofcharacteristic p , then the mod p K -theory K ∗ ( X ; Z /p ) vanishes in degrees > dim( X ) .However, the construction of Theorem 1.1 (and the definition of motivic cohomology) reliesheavily on the smoothness assumption on X . The higher Chow groups of a singular variety X overa field k give an analog of Borel–Moore homology rather than cohomology; in particular, they arenilinvariant (while algebraic K -theory is far from nilinvariant) and lack a product structure. Theexisting constructions of the motivic filtration use the A -invariance of K -theory (valid only in theregular case), and giving a general notion of the motivic filtration (or of motivic cohomology) on K -theory applicable to singular rings appears to be an open problem. Even the setting of regularrings in mixed characteristic is not fully understood (but see [Lev06]). Question 1.2.
Is there a motivic filtration on K ( R ) for any ring R which extends Theorem 1.1when R is smooth over a field?In the study of the algebraic K -theory of singular rings, the main new tool is the theory of tracemethods. Trace methods provide maps from algebraic K -theory to more computable invariants builtfrom Hochschild homology, and the basic tools are relative comparison results to the effect thatthe homotopy fiber of such maps satisfy excision and nilinvariance. The most general and powerfulform of these results uses topological cyclic homology TC , introduced by Bökstedt–Hsiang–Madsen[BHM93] in the p -complete case (see [DGM13] for the integral version), which compares to K -theoryvia the cyclotomic trace map(2) K ( R ) → TC( R ) . Theorem 1.3.
The homotopy fiber F of (2) has the following properties: (1) (Dundas–Goodwillie–McCarthy [DGM13] ) F is nilinvariant. (2) (Land–Tamme [LT19] ) F satisfies excision, i.e., given a pullback square of rings with thevertical arrows surjective, then F carries this to a pullback of spectra. (3) (Clausen–Mathew–Morrow [CMM18] ) The profinite completion of the variant F ′ ( R ) =fib( K ≥ ( R ) → TC( R )) is rigid for henselian pairs, i.e., if ( R, I ) is a henselian pair, then F ′ ( R ) /n ∼ −→ F ′ ( R/I ) /n for any integer n > . When R is a Q -algebra, then TC( R ) agrees with the negative cyclic homology HC − ( R/ Q ) ,which is closely related to the de Rham cohomology of R , and parts (1) and (2) of the result aredue respectively to Goodwillie [Goo86] and to Cortiñas [Cn06]. For example, compare [GRW89] forapplications of these results (at the time conjectural) to the calculation of the K -theory of singularcurves over Q (e.g., rings such as Q [ x, y ] / ( xy ) ). Part (3) in this case, or more generally when n isinvertible on R , is the Gabber rigidity theorem [Gab92].In this survey, we concentrate on the situation after p -adic completion for p -adic rings, in whichcase parts (1) and (2) are due to McCarthy [McC97] and Geisser–Hesselholt [GH06a]. In this case,it is known that the map (2) is not only useful for detecting “infinitesimal” behavior, but is also an See also [Ras18] for an account using the approach to TC of [NS18]. OME RECENT ADVANCES IN TOPOLOGICAL HOCHSCHILD HOMOLOGY 3 absolute approximation to p -adic K -theory R : specifically, it is p -adic étale (connective) K -theory;moreover, it is an equivalence in high enough degrees depending on the ring, under mild hypotheses.This follows from the work of Geisser–Levine [GL00] on the p -adic K -theory of smooth algebrasin characteristic p , Geisser–Hesselholt [GH99] on TC of such rings, extended to the more generalsituation using the rigidity result of [CMM18] (see also [CM19]). Theorem 1.4. (1)
For p -adic rings R , the trace (2) exhibits TC( R ; Z p ) as the p -completion ofthe étale K -theory of R . (2) If R is p -complete, R/p has finite Krull dimension and d = sup x ∈ Spec(
R/p ) log p [ k ( x ) : k ( x ) p ] ,then the map K ( R ; Z p ) → TC( R ; Z p ) is an equivalence in degrees ≥ max( d, . The definition of TC is much more elaborate than that of K -theory and (in particular in the p -adicsetting) requires significant homotopical foundations, classically approached through equivariantstable homotopy theory [BM15], which were recently dramatically reworked (and simplified) byNikolaus–Scholze [NS18] (see also [AMGR17, BG16] for ∞ -categorical accounts of the theory ofcyclotomic spectra). We refer to [HN19] for a modern survey of topological Hochschild and cyclichomology.There is a sense, however, in which topological cyclic homology is a structurally simpler theory:while the building blocks of algebraic K -theory come from algebraic cycles, the building blocksfor topological cyclic homology are the cotangent complex and its wedge powers. In practice,this means that the formal properties of TC are somewhat simpler (e.g., TC has much betterdescent properties), and that TC is easier to compute in practice. There is an extensive literaturecalculating various instances of the p -adic K -theory of p -adic rings using TC , cf. for instance[HM97b, HM97a, HM03, HM04, GH06b] for some examples.The work of Bhatt–Morrow–Scholze [BMS19] constructs analogs of the motivic filtration fortopological Hochschild homology and its variants in great generality; the associated graded objectsof this filtration are objects of deep interest in arithmetic geometry and especially p -adic Hodgetheory, and have now been constructed purely algebraically using the prismatic theory [BS19]. Herewe state first the analog for TC . Theorem 1.5 (Bhatt–Morrow–Scholze [BMS19]) . Let R be any p -complete ring. Then there existsa natural multiplicative, Z op ≥ -indexed convergent filtration Fil ≥∗ TC( R ; Z p ) with associated gradedterms gr i TC( R ; Z p ) ≃ Z p ( i )( R )[2 i ] , for the Z p ( i )( R ) natural objects of the p -complete derived ∞ -category. The constructions Z p ( i ) (as functors to the derived ∞ -category D ( Z p ) ) satisfy flat descent,and for regular F p -algebras reproduce the objects R Γ proet ( − , W Ω i log )[ − i ] . Theorem 1.5 is supposed to be an analog of Theorem 1.1 for TC . It is not entirely clear ifthis analogy can be made precise (i.e., if both filtrations can be realized as instances of a commonconstruction). However, it is at least known in the case where R is a smooth algebra over a field k of characteristic p (so that Theorem 1.1 is in effect), the filtration TC( R ; Z p ) is the p -completion ofthe étale sheafification of the motivic filtration on K ( R ) (indeed, both are Postnikov towers in the(pro-)Nisnevich and (pro-)étale topologies). The construction of Theorem 1.5 has the advantage ofbeing very direct: the filtration is the Postnikov filtration when these invariants are considered assheaves in the quasisyntomic topology (Section 4). Strictly speaking, this refers to the étale sheafification of the connective K -theory of R . The work [BMS19] only treats the case where R is quasisyntomic; it is shown in [AMMN20, Sec. 5] that theconstruction naturally extends (via left Kan extension) to all p -adic rings. AKHIL MATHEW
The Z p ( i ) in their most generality are supposed to be a general version of p -adic étale motiviccohomology for p -adic rings. They arise as a type of filtered Frobenius eigenspaces on prismaticcohomology, a new p -adic cohomology theory for p -adic formal schemes introduced by Bhatt–Scholze [BS19] of deep interest in integral p -adic Hodge theory and constructed in some cases in[BMS18, BMS19]. The case of “absolute” prismatic cohomology was originally constructed usingtopological Hochschild homology. For the formulation of the next result, we write THH for topo-logical Hochschild homology equipped with its natural T -action. Theorem 1.6 ([BMS19]) . Let R be a formally smooth algebra over a perfectoid ring R . Thenthere is a complete, exhaustive Z -indexed filtration on TP( R ; Z p ) def = THH( R ; Z p ) t T such that gr i TP( R ; Z p ) ≃ b ∆ R [2 i ] . In fact, the above filtration is constructed using the similar descent techniques, which gives aconstruction of prismatic cohomology, independent of the prismatic site of [BS19]. When R is thering of integers O C in a complete, algebraically closed nonarchimedean field C , then the b ∆ R recoverthe A inf -cohomology of [BMS18]: in particular, they specialize both to the de Rham cohomologyof the formal scheme Spf( R ) and the p -adic étale cohomology of the generic fiber.We will not attempt to do justice to the new landscapes of integral p -adic Hodge theory. Inthis survey article, we will work through the characteristic p situation in some detail, in particular,constructing the filtration on TP( R ; Z p ) for R a smooth (or more generally quasisyntomic, e.g.,lci) F p -algebra R and identifying the associated graded pieces in terms of crystalline cohomology.In equal characteristic p , absolute prismatic cohomology in this context reduces to crystalline co-homology, constructed using the (quasi-)syntomic site instead of the crystalline site (an approachthat goes back to [FM87]). Finally, we will circle back to the motivation of algebraic K -theory, andexplain how one can recover the calculations of the algebraic K -theory of the dual numbers over aperfect field [HM97b, Spe20]. It would be interesting to revisit other such calculations. Acknowledgments.
It is a pleasure to thank Benjamin Antieau, Alexander Beilinson, BhargavBhatt, Dustin Clausen, Vladimir Drinfeld, Lars Hesselholt, Jacob Lurie, Matthew Morrow, ThomasNikolaus, Nick Rozenblyum, and Peter Scholze for numerous helpful conversations related to thissubject over the past few years, and Hélène Esnault for comments on an earlier version. I also thankLars Hesselholt and Shuji Saito for the invitation to a workshop on this material in Hara-mura, andMike Hopkins and Jacob Lurie for organizing the Thursday seminar at Harvard in 2015–2016 onthis subject. This work was done while the author was a Clay Research Fellow.
Notation.
We let T denote the circle group. We denote by Sp the ∞ -category of spectra, withthe smash product ⊗ , and S the sphere spectrum. For a ring R , we let D ( R ) denote the derived ∞ -category of R .We will freely use the language of higher algebra, and in particular the theory of E ∞ -ring spectra.We refer to [Gep19] for a modern survey and introduction.Throughout the paper, we fix a prime p . We will occasionally use the theory of δ -rings, but onlyin the p -torsionfree case; a p -torsionfree δ -ring consists of a commutative ring R equipped with anendomorphism ϕ : R → R which lifts the Frobenius modulo p . We refer to [BS19, Sec. 2] for anaccount of the theory of δ -rings in general.We will often drop the notation of p -completions, since we will almost exclusively be workingwith p -complete objects. OME RECENT ADVANCES IN TOPOLOGICAL HOCHSCHILD HOMOLOGY 5 Topological Hochschild homology
Let R be a commutative ring. Definition 2.1.
The topological Hochschild homology
THH( R ) is the universal E ∞ -algebra equippedwith a T -action and a map R → THH( R ) . As an E ∞ -ring, there is an identification THH( R ) = R ⊗ R ⊗ S R R. Definition 2.1 (which works equally for an E ∞ -ring R ) is not the most flexible definition of THH ,since
THH is more generally defined for stable ∞ -categories (it is a localizing invariant in the senseof [BGT13], like algebraic K -theory); then THH of a commutative ring is defined as
THH of its ∞ -category of perfect complexes. For example, THH can be defined using factorization homologyover the circle [AMGR17]. This perspective, while extremely important in the foundations of thetheory (in particular, in producing the cyclotomic trace), plays less of a role in the work of [BMS19],which focuses on commutative rings. The above formulation for E ∞ -rings is due to [MSV97].Before describing some of the features of THH , we begin by reviewing the simpler algebraicanalog of Hochschild homology.
Variant 2.2 (Classical Hochschild homology) . Let R be a commutative k -algebra, for k a basering. Then the Hochschild homology
HH(
R/k ) is the universal E ∞ -algebra under k equipped witha T -action and a map R → HH(
R/k ) of E ∞ - k -algebras. As an E ∞ - k -algebra, one has HH(
R/k ) = R ⊗ LR ⊗ Lk R R. When k = Z , we will sometimes drop the k in the above notation.Hochschild homology over k is a very controllable construction, because of the classical Hochschild–Kostant–Rosenberg theorem. Since T acts on the E ∞ - k -algebra HH(
R/k ) , one obtains a commu-tative differential graded algebra structure on HH ∗ ( R/k ) (with the differential arising from the T -action). The Hochschild–Kostant–Rosenberg theorem gives a natural isomorphism of commuta-tive differential graded algebras for R smooth over k ,(3) HH ∗ ( R/k ) ≃ (Ω ∗ R/k , d ) , where d is the de Rham differential.Topological Hochschild homology is a much richer theory than classical Hochschild homologyfor p -adic rings (whereas for Q -algebras, it reduces to Hochschild homology relative to Q ). TakingHochschild homology over the base S leads to extra symmetries in the theory which are not availablewith an ordinary ring (e.g., F p or Z ) as the base; moreover, it leads to Bökstedt’s computation of THH( F p ) . We begin by reviewing these aspects, following Nikolaus–Scholze [NS18]; see also thesurvey [HN19] for a more detailed overview. Construction 2.3 (The cyclotomic Frobenius on
THH , cf. [NS18, Sec. IV.2]) . Given a commuta-tive ring R , one has a natural T -equivariant map(4) ϕ : THH( R ) → THH( R ) tC p , using the natural embedding C p ⊂ T and the T ≃ T /C p -action on the right-hand-side. To construct(4), we use the universal property of THH( R ) to construct a map of E ∞ -rings R → THH( R ) tC p One could also formulate the universal property in terms of animated (or simplicial) commutative k -algebras,without using the language of E ∞ -rings. AKHIL MATHEW and then extend it canonically to a T -equivariant map as in (4). This in turn comes from the Tatediagonal [NS18, Sec. III.1] R → ( R ⊗ S · · · ⊗ S R ) tC p = ( ⊗ C p S R ) tC p , (which exists for every spectrum) followed by the map ( ⊗ C p S R ) tC p → THH( R ) tC p obtained fromthe inclusion C p ⊂ T .The map ϕ is called the cyclotomic Frobenius and plays a central role in the theory. Its con-struction depends crucially on working over the sphere spectrum, and is thus a feature of THH that does not exist for ordinary Hochschild homology: by universal properties of the ∞ -categoryof spectra, one shows [NS18, Sec. III.1] that there is a canonical, lax symmetric monoidal naturaltransformation, called the Tate diagonal ,(5) X → ( X ⊗ S X ⊗ S · · · ⊗ S X ) tC p for any spectrum X . Indeed, the Tate diagonal is roughly analogous to (and refines) the mapdefined for every abelian group A , A → H ( C p , A ⊗ p ) = ( A ⊗ p ) C p / norms , given by the formula a a ⊗ a ⊗ · · · ⊗ a . The analog of the map (5) does not exist in D ( Z ) , thederived ∞ -category of the integers Z , which lacks the analogous universal property, and is a keyreason why THH yields a richer theory.In the work [NS18], it is shown that ϕ is enough to study the so-called ( p -typical) “cyclotomicstructure” on the p -completion THH( R ; Z p ) ; in particular, it can be used to define the topologicalcyclic homology. Construction 2.4 (Topological cyclic homology) . Let R be a ring (or more generally a connective E ∞ -ring). We define TC − ( R ) = THH( R ) h T and TP( R ) = THH( R ) t T to be the T -homotopy fixedpoints and Tate construction, respectively. We have two maps can , ϕ : TC − ( R ; Z p ) → TP( R ; Z p ) , where can is the canonical map from T -invariants to the T -Tate construction, and ϕ is obtainedby taking T -invariants from (4) and using the identification TP( R ; Z p ) = (cid:0) THH( R ; Z p ) tC p (cid:1) h T /C p ,cf. [NS18, Lem. II.4.2]. In particular, since can identifies π TC − ( R ; Z p ) and π TP( R ; Z p ) , we canregard ϕ as an endomorphism of the ring π TC − ( R ; Z p ) . The spectrum TC( R ; Z p ) is the homotopyequalizer(6) TC( R ; Z p ) = fib( ϕ − can) : TC − ( R ; Z p ) → TP( R ; Z p ) . The expression (6) is very different from ones that appear in the more classical approach to
THH .It plays an essential role in the work [BMS19], and has many applications both structural andcomputational. For example, it implies the following basic structural feature of TC : for connectivering spectra, the construction TC /p commutes with filtered colimits [CMM18, Theorem G].By contrast, in the classical approach to THH and cyclotomic spectra via equivariant stablehomotopy theory (cf. [Mad94] for a survey), the objects TC − , TP do not play a direct role. Oneconstructs the structure of a genuine C p n -spectrum on THH( R ) , which enables one to form variousfixed points THH( R ) C pn , n ≥ , together with maps R, F : THH( R ) C pn → THH( R ) C pn − , V : THH( R ) C pn − → THH( R ) C pn . OME RECENT ADVANCES IN TOPOLOGICAL HOCHSCHILD HOMOLOGY 7
The various fixed points are related to each other inductively using the cofiber sequences
THH( R ) hC pn → THH( R ) C pn R → THH( R ) C pn − . In particular, one forms the inverse limit
TR( R ) = lim ←− R THH( R ) C pn , which is a connective E ∞ -ring. The maps F, V act on
TR( R ) , and TC( R ; Z p ) is the equalizer fib( F − R ; Z p ) → TR( R ; Z p )) . A major advance of the work [NS18] is the insight thatmuch of the “glueing” data that leads to the construction of the fixed points THH( R ) C pn is actuallyredundant (and is not needed to construct TC in particular). On the other hand, TR has recentlybeen used to give an entirely new formulation (and “decompletion”) of the theory of cyclotomicspectra via the theory of topological Cartier modules due to Antieau–Nikolaus [AN18]; this hasmany advantages, including that it yields a natural t -structure on cyclotomic spectra.The most fundamental calculation in topological Hochschild homology is that of F p . Theorem 2.5 (Bökstedt) . We have
THH ∗ ( F p ) = F p [ σ ] with | σ | = 2 . For a discussion of a proof of Theorem 2.5 (and in particular the multiplicative structure, whichis crucial to everything that follows), cf. [HN19]. In particular, Theorem 2.5 is closely related tothe result of Hopkins–Mahowald that the free E -algebra over S with p = 0 is H F p . Moreover,using Bökstedt’s theorem, one can in fact give a complete description of THH( F p ) as a cyclotomicspectrum, cf. [NS18, Sec. IV-2].It is instructive to compare Bökstedt’s theorem with the calculation of HH( F p / Z ) . One sees easily(e.g., using the Hochschild–Kostant–Rosenberg theorem, cf. [NS18, Prop. IV.4.3]) that HH ∗ ( F p / Z ) is the divided power algebra Γ ∗ [ σ ] (for the same class σ in degree two); in particular, replacing Z by S replaces the divided power algebra by the polynomial algebra.For any F p -algebra R , one has the formula THH( R ) ⊗ THH( F p ) F p ≃ HH( R/ F p ) . Given the above description of
THH ∗ ( F p ) , one may view THH( R ) as a “one-parameter deformation”of HH( R/ F p ) along σ . Upon taking circle-fixed points and passing to associated gradeds, thisobservation is ultimately connected to the fact that crystalline cohomology gives a one-parameterdeformation of de Rham cohomology in characteristic p (along the parameter given by “ p ”).Connections between THH and arithmetic have been explored in [Hes96, HM03, HM04, GH06b],which relate the homotopy groups of the fixed points of
THH to (various forms) of the de Rham–Witt complex. The first such result, in equal characteristic, gives a complete calculation of TR inthe case of a regular F p -algebra: Theorem 2.6 (Hesselholt [Hes96]) . Let R be a regular F p -algebra. Then there is an isomorphism TR( R ; Z p ) ∗ ≃ W Ω ∗ R , where W Ω ∗ R is the de Rham–Witt complex of Bloch–Deligne–Illusie [Ill79] .This isomorphism carries the operators F, V on TR( R ; Z p ) to the similarly named operator F, V on W Ω ∗ R . In particular, one obtains a complete calculation of TC using the fixed points of operator F onthe de Rham–Witt forms. In mixed characteristic, [HM04] introduces analogs of the de Rham–Wittcomplex, and [HM03, HM04, GH06b] discuss the connections between TR of smooth algebras inmixed characteristic and the de Rham–Witt complex. This in particular was used in loc. cit. toverify the Lichtenbaum–Quillen conjecture for certain p -adic fields (prior to the general proof by AKHIL MATHEW
Voevodsky–Rost). See also [LW20] for a new approach to this calculation using the methods of[BMS19]. 3.
The cotangent complex and its wedge powers
The building blocks of all the constructions involved are the cotangent complex and its wedgepowers; these are the “animations” (for our purposes, left Kan extensions) of the usual differentialforms functors. We begin with a brief review. Fix a base ring k . Construction 3.1 (Left Kan extension) . Let C be an ∞ -category admitting sifted colimits. Let Poly k be the category of finitely generated polynomial k -algebras, and let F : Poly k → C be afunctor. Then one can construct the left Kan extension or left derived functor LF : Ring k → C ,extending the functor F to Ring k . Explicitly, we define LF on all polynomial k -algebras (possiblyon infinitely many variables) by forcing LF to commute with filtered colimits. Given an arbitrary k -algebra R , we can choose a simplicial resolution P • → R where each P i is a polynomial k -algebra,and then LF ( R ) = | F ( P • ) | .The above construction (in various forms classical, going back to Quillen) is a type of nonabelianleft derived functor [Lur09, Sec. 5.5.8]. More generally, we can express the above constructionusing the theory of animated rings. Let Ani(Ring k ) be the ∞ -category of animated k -algebras (alsocalled simplicial commutative k -algebras; we refer to [ČS19] for a discussion of this terminology).Then with C as above one has an equivalence of ∞ -categories Fun Σ (Ani(Ring k ) , C ) ≃ Fun(Poly k , C ) between sifted-colimit preserving functors Ani(Ring k ) → C and functors Poly k → C . Definition 3.2 (The cotangent complex and its wedge powers) . Let R be a k -algebra. Then the cotangent complex L R/k ∈ D ( R ) is defined as the left derived functor of the functor R Ω R/k of Kähler differentials. Similarly, the wedge power V i L R/k ∈ D ( R ) (for i ≥ ) is defined as thederived functor of the functor V iR Ω R/k of differential i -forms; this can also be defined using theDold–Puppe nonabelian derived exterior powers V i : D ( R ) ≤ → D ( R ) ≤ (cf. [Lur18, Sec. 25.2.1]for a modern account) applied to the cotangent complex.We refer to [Sta20, Tag 08P5] for a comprehensive treatment of the cotangent complex. A basicfact about the cotangent complex is that it agrees with ordinary differential forms not only forpolynomial k -algebras, but more generally for smooth k -algebras. The other fundamental tools arethe transitivity sequence for a sequence of ring maps A → B → C , which yields a cofiber sequencein D ( C ) ,(7) L B/C ⊗ LB C → L C/A → L C/B , and the base-change property L B/A ⊗ A A ′ = L B ⊗ A A ′ /A ′ for a map A → A ′ of k -algebras such that B, A ′ are Tor -independent over A (if they are not Tor -independent, one has to consider the derivedtensor product B ⊗ LA A ′ as an animated ring itself). Example 3.3.
Using the cofiber sequence and base-change, we find that if B = A/r for r ∈ A anonzerodivisor, then L B/A ≃ ( r ) / ( r )[1] . Example 3.4.
More generally, suppose B = A/I for I ⊂ A an ideal generated by a regular se-quence; this (and its generalizations) will be one of the primary examples for us. Then L B/A = In principle, the functor L · /k takes values in D ( k ) as the argument varies, but with more effort (e.g., using the ∞ -category of pairs of an animated ring and a module over it) one can construct the functor as stated. OME RECENT ADVANCES IN TOPOLOGICAL HOCHSCHILD HOMOLOGY 9
I/I [1] , which is the suspension of a free B -module (cf. [Sta20, Tag 08SH], [Ill71, III.3.2]) A conse-quence is that V i L B/A = Γ i ( I/I )[ i ] , for Γ i the i th divided power functor on flat B -modules. Thisis a consequence of the décalage isomorphism of [Ill71, Sec. I.4.3.2] between V i ( M [1]) = (Γ i M )[ i ] for any M ∈ D ( B ) ≤ . Proposition 3.5.
Let R be a perfect F p -algebra. Then L R/ F p = 0 .Proof. For every F p -algebra S , the Frobenius S → S induces zero on L S/ F p ; this follows by in-spection in the case of a polynomial F p -algebra and then follows in general by taking simplicialresolutions. The claim follows. (cid:3) The key structural result in [BMS19] used in defining the motivic filtrations is the flat descentfor the cotangent complex and its wedge powers. This was originally observed by Bhatt [Bha12a]and is treated further in [BMS19, Sec. 3].
Theorem 3.6 ([Bha12a, BMS19]) . Let k be a base ring. Then the construction A V i L A/k , asa functor from k -algebras to D ( k ) , satisfies flat descent. More generally, for any k -module M , theconstruction A V i L A/k ⊗ k M satisfies flat descent. This result is proved using the transitivity cofiber sequence for the cotangent complex, flatdescent for modules, and an inductive argument on the degree i . As pointed out in loc. cit. , itremains open whether the above functors are flat hypersheaves.4. Quasisyntomic rings
A key insight in [BMS19] is that to understand invariants such as
THH , etc. it is extremelyclarifying to work with very “large” (e.g., perfectoid) rings. In particular, one should make highlyramified extensions by adding lots of p -power roots. This strategy is expressed using the quasisyn-tomic topology, a key construction of [BMS19, Sec. 4]; all the filtrations of loc. cit. are defined onquasisyntomic rings, and probably cannot be defined more generally (without sacrificing conver-gence properties). This class of rings is also extremely useful in other contexts, including in theprismatic Dieudonné theory of Anschütz–Le Bras [AB19]. Definition 4.1 (Quasisyntomic rings) . A ring R is quasisyntomic if:(1) R is p -complete, and the p -power torsion in R is bounded, i.e., annihilated by p N for N ≫ .(2) The cotangent complex L R/ Z p ∈ D ( R ) has the property that L R/ Z p ⊗ LR ( R/p ) ∈ D ( R/p ) has Tor -amplitude in [ − , .We let QSyn denote the category of quasisyntomic rings.
Example 4.2 (Complete intersections) . Let ( R, m ) be a p -complete local noetherian ring with p ∈ m . Then R is quasisyntomic if R is a complete intersection, i.e., if for any (or one) surjection f : A ։ b R with A complete regular local, the kernel of f is generated by a regular sequence.Indeed, a result of Avramov [Avr99] states that R is a complete intersection if and only if L R/ Z has Tor -amplitude in [ − , . Therefore, if R is a complete intersection then R is clearly quasisyntomic.The idea is that quasisyntomic rings are those which behave like local complete intersectionsat the level of the cotangent complex (which is enough to control all Hochschild-type invariants).However, this class of rings includes many highly non-noetherian examples. Example 4.3 (Perfect rings) . Any perfect F p -algebra R (i.e., one where the Frobenius is an iso-morphism) is quasisyntomic. In fact, we have that L R/ F p = 0 as we saw in Proposition 3.5; thetransitivity cofiber sequence applied to Z p → F p → R thus implies that L R/ Z p is the suspension ofa rank free R -module. Example 4.4 (Witt vectors of perfect rings) . If R is a perfect F p -algebra, then the ring of Wittvectors W ( R ) is quasisyntomic. In fact, W ( R ) is p -torsionfree and W ( R ) /p ≃ R ; one thus obtainsthat L W ( R ) / Z p vanishes p -adically, whence the claim.The class of perfect F p -algebras admits a remarkable generalization to mixed characteristic,namely the class of integral perfectoid rings [BMS18, Sec. 3.2] (based on the notion of perfectoidTate ring introduced in [Sch12, KL15, Fon13]). Definition 4.5 (Perfectoid rings) . A p -adically complete ring R is called perfectoid if R can beexpressed as the quotient W ( R ′ ) /ξ , where R ′ is a perfect F p -algebra, and ξ ∈ W ( R ′ ) is an elementof the form [ a ] + pu where u ∈ W ( R ′ ) is a unit and a ∈ R ′ .In the above, by replacing R ′ by its a -adic completion, which does not change the quotient W ( R ′ ) / ([ a ]+ pu ) , we may in fact assume that R ′ is a -adically complete. Note that this in particularimplies that R/ [ a ] is an F p -algebra, and the Frobenius induces an isomorphism of F p -algebras, R/ [ a ] /p ∼ −→ R/ [ a ] . This is in fact the essential feature of perfectoid rings: Proposition 4.6 ([BMS18, Lem. 3.10]) . Let R be a ring such that there exists a nonzerodivisor ω ∈ R such that: (1) R is ω -adically complete. (2) ω p | p . (3) The Frobenius induces an isomorphism
R/ω ∼ −→ R/ω p .Then R is perfectoid. Conversely, if R is perfectoid, then there exists an element ω such that ω p | p ,and for any such element, the Frobenius map R/ω → R/ω p is an isomorphism. Remark 4.7.
Let R = W ( R ′ ) /ξ be a perfectoid ring with ξ = [ a ] + pu . By a -adically completing R ′ if necessary, we may assume that R ′ is a -adically complete. In this case, R ′ is the tilt R ♭ of R ,namely, R ′ = lim ←− ϕ R/p . In fact,
R/p = R ′ /a , and for any perfect F p -algebra S which is x -adicallycomplete, we see that S agrees with the inverse limit perfection of S/x . Remark 4.8.
A perfectoid ring R is quasisyntomic. Indeed, the p -complete cotangent complex L R/ Z p = L R/W ( R ♭ ) is the suspension of a free R -module of rank , since W ( R ♭ ) → R is the quotientby a nonzerodivisor. Example 4.9.
The p -adic completion of the ring Z p [ p /p ∞ ] is perfectoid. In fact, this ring can bewritten as the quotient of W ( F p [ t /p ∞ ]) / ([ t ] − p ) = (cid:16) Z p [ u /p ∞ ] (cid:17) ˆ p / ( u − p ) . More generally, let R be a p -torsionfree, p -adically complete Z p [ p /p ∞ ] -algebra. Then R is perfectoidif and only if the Frobenius induces an isomorphism ϕ : R/p /p ∼ −→ R/p . Example 4.10.
The ring Z p [ ζ p ∞ ] ˆ p is perfectoid. In this case, we can form the ring Z p [ q /p ∞ ] \ ( p,q − = W ( F p [ ǫ /p ∞ ] \ ( ǫ − ) (via q = [ ǫ ] ) and the element [ p ] q := q p − q − = 1+ q + · · · + q p − . By considering themap F p [ ǫ /p ∞ ] \ ( ǫ − → F p , ǫ , one checks that the coefficient of p in the Teichmüller expansionis a unit. OME RECENT ADVANCES IN TOPOLOGICAL HOCHSCHILD HOMOLOGY 11
The original definition of a perfectoid field was given in [Sch12]: a perfectoid field K is a completenonarchimedean field K with ring of integers O K ⊂ K such that the valuation of K is nondiscrete, p is topologically nilpotent, and the Frobenius on O K /p is surjective. This implies that there existsa nonzero topologically nilpotent element ω ∈ O K with ω p | p and such that the Frobenius inducesan isomorphism O K /ω ∼ −→ O K /ω p . The datum of a perfectoid field is equivalent to the datum of acomplete, rank valuation ring which is perfectoid. The above two examples arise in this manner.To obtain more examples of perfectoid rings, note that if R is a perfectoid ring, then the p -completion R (cid:10) t /p ∞ (cid:11) of R [ t /p ∞ ] is perfectoid. More subtly, there is the construction of the “per-fectoidization” of a semiperfectoid ring. If R is a perfectoid ring and I ⊂ R is a p -complete ideal,then there is a p -complete ideal J ⊃ I such that R/J is perfectoid, and is the universal perfectoidring to which
R/I maps (cf. [BS19, Th. 7.4]). This construction is not easy to describe explicitlyin general. But if I = ( f ) for f ∈ R admitting a system of p -power roots { f /p n } n ≥ , then J is the p -completion of S n ≥ ( f /p n ) .We next review the quasisyntomic topology on QSyn , cf. [BMS19, Def. 4.1, Cor. 4.8]. This isa non-noetherian version of the syntomic topology, cf. [FM87] or [Sta20, Tag 0224], and in the p -complete context. Strictly speaking, it is QSyn op that has the structure of a site. Definition 4.11 (The quasisyntomic site) . A map R → R ′ in QSyn is a cover if:(1)
R/p n → R ′ /p n is faithfully flat for all n ≥ .(2) L R ′ /R ⊗ LR ′ R ′ /p ∈ D ( R ′ /p ) has Tor -amplitude in [ − , .The condition (1) is called p -complete faithful flatness , and is the appropriate replacement forfaithful flatness in this (highly non-noetherian) setup. Note that if R is noetherian, then thecondition (1) is simply faithful flatness thanks to [Yek18]. Example 4.12 (Adding systems of p -power roots) . Given a collection of elements { x t ∈ R } , thering R ′ obtained as the p -completion of R [ u /p ∞ t , t ∈ T ] / ( u t − x t ) , i.e., obtained by p -completelyadding a system of p -power roots of the elements x t , gives a cover of R in the quasisyntomictopology. Iterating this construction transfinitely many times, one sees that every object of QSyn can be covered by an object where all elements admit compatible systems of p -power roots. Example 4.13 (Covers of regular rings) . Let R be a p -complete, regular noetherian ring. Thenthere is a quasisyntomic cover R → R ∞ , with R ∞ perfectoid. Conversely, a p -complete noetherianring admitting such a cover is regular. This is proved in [BIM19]. Example 4.14 ( p -complete valuation rings are quasisyntomic) . This follows by a result of Gabber–Ramero [GR03, Th. 6.5.8]. Moreover, if V is a valuation ring over F p , then L V/ F p is a flat V -module.An important general structural result for perfectoid rings, formulated in terms of the quasisyn-tomic site, is that locally one can add solutions to polynomial equations. This is highly non-trivial,since there is no obvious way to add such solutions while retaining the perfectoid property. For thisresult, compare [And18, Sec. 2.5], [GR04, Th. 16.9.17], [BS19, Th. 7.12], and [ČS19, Th. 2.3.4]. Theorem 4.15 (André’s lemma) . Let R be a perfectoid ring. Then there exists a map of perfectoidrings R → R ∞ such that: (1) R ∞ is absolutely integrally closed. (2) R → R ∞ is a cover in QSyn . In fact, R → R ∞ can be taken to be the p -completion of anind-syntomic map. Definition 4.16 (Quasiregular semiperfectoid rings) . A quasisyntomic ring R is said to be quasireg-ular semiperfectoid (or quasiregular semiperfect if R is additionally an F p -algebra) if either of thefollowing equivalent conditions hold:(1) R receives a surjection from a perfectoid ring.(2) R/p is semiperfect (i.e., the Frobenius is surjective), and R receives a map from a perfectoidring R .To see that (2) implies (1), consider the map θ : W ( R ♭ ) → R for R ♭ the inverse limit perfec-tion of R/p ; this is surjective modulo p , hence surjective since R is p -complete. The extension R ˆ ⊗ Z p W ( R ♭ ) → R is also therefore surjective, and the source is perfectoid, whence (1).We denote by QRSPerfd the category of quasiregular semiperfectoid rings, equipped with theinduced site structure. The subcategory
QRSPerfd ⊂ QSyn is a basis for the quasisyntomic site:any object of
QSyn admits a cover by an object of
QRSPerfd . Moreover, the tensor product of tworings in
QRSPerfd remains in
QRSPerfd . Remark 4.17.
Suppose that R is a quasiregular semiperfectoid ring. In this case, L R/ Z p is thesuspension of a p -completely flat R -module.Heuristically quasisyntomic rings are those which behave like lci rings, at least at the level of thecotangent complex (and after p -completion). We also discuss a class of F p -algebras which behavemore like smooth algebras. Definition 4.18 (Cartier smooth F p -algebras, [KM18]) . Let R be an F p -algebra. We say that R is Cartier smooth if:(1) The cotangent complex L R/ F p is a flat R -module in degree zero.(2) For each i , the inverse Cartier operator C − : Ω iR/ F p → H ∗ (Ω ∗ R/ F p ) is an isomorphism. Herethe inverse Cartier operator is the unique map of graded algebras Ω ∗ R/ F p → H ∗ (Ω ∗ R/ F p ) carrying r ∈ R to the class of r p and ds, s ∈ R to the class of s p − ds . Compare [BLM18,Prop. 3.3.4]. Example 4.19. (1) Any smooth algebra over a perfect field (or more generally a perfect F p -algebra) is Cartier smooth, thanks to the classical Cartier isomorphism (cf. [Kat70, Th. 7.2]for an account).(2) Any regular noetherian F p -algebra is Cartier smooth. Indeed, this follows because Cartiersmooth algebras are closed under filtered colimits and any regular noetherian F p -algebra isind-smooth by Néron–Popescu desingularization. However, one can also prove this claimdirectly, [BLM18, Sec. 9.5].(3) Any valuation ring over F p is Cartier smooth. This follows from results of Gabber–Ramero[GR03, Th. 6.5.8] and Gabber [KST18, App. A]. Conjecturally (by local uniformization,a weak form of resolution of singularities) valuation rings over F p are ind-smooth, whichwould imply Cartier smoothness, but local uniformization is not known in general.(4) A collection of elements { x i } i ∈ I in an F p -algebra R is a p -basis if the elements Q i ∈ I x a i i ∈ R ,as { a i } i ∈ I ranges over all finitely supported functions I → { , , . . . , p − } , forms a basisfor R as a module over itself via the Frobenius map. If R admits a p -basis, then R is Cartiersmooth, cf. [BLM18, Th. 9.5.21] and its proof.We refer to [KM18, KST18] for some applications of the theory of Cartier smooth algebras.In particular, loc. cit. it is shown that the calculation [GL00, GH99] of the p -adic K -theory and OME RECENT ADVANCES IN TOPOLOGICAL HOCHSCHILD HOMOLOGY 13 topological cyclic homology of regular local F p -algebras also generalizes to local Cartier smooth F p -algebras (e.g., valuation rings). Remark 4.20.
We do not know if the condition of Cartier smoothness guarantees that the Frobe-nius endomorphism is flat. By a classical theorem of Kunz (see [Kun69] or [Sta20, Tag 0EC0]), anoetherian F p -algebra is regular if and only if the Frobenius endomorphism is flat. All the aboveexamples of Cartier smooth algebras have the property that the Frobenius is flat.On the other hand, condition (2) in Cartier smoothness is definitely not implied by condition(1). For instance, there exist semiperfect F p -algebras R such that L R/ F p = 0 but such that R isnot perfect; compare [Bha] for an example. In this case, the inverse Cartier operator reproducesthe Frobenius ϕ : R → R in degree zero, which is not an isomorphism. Question 4.21.
Is there an analog of Cartier smoothness for arbitrary quasisyntomic rings?5.
Some quasisyntomic sheaves
Throughout, we use the language of sheaves of spectra [Lur18, Sec. 1.3] (this was implicitlyused in the formulation of Theorem 3.6). Note that this is slightly more general than the theoryintroduced by Jardine [Jar87], which correspond to hypercomplete sheaves, cf. [DHI04]. However,all the sheaves used in the constructions of [BMS19] will be shown to be hypercomplete (this is aconvenient feature of the quasisyntomic site), so the distinction does not play a significant role in[BMS19].
Definition 5.1 (Sheaves on
QSyn ) . A spectrum-valued sheaf on
QSyn is a functor F : QSyn → Sp such that(1) F preserves finite products.(2) If A → B is a cover in QSyn , then the natural map F ( A ) → lim ←− (cid:16) F ( B ) ⇒ F ( B ˆ ⊗ A B ) →→→ . . . (cid:17) is an equivalence.We let Shv(QSyn , Sp) denote the ∞ -category of sheaves of spectra. Definition 5.2 (Hypercomplete sheaves) . We will say that a sheaf of spectra F ∈ Shv(QSyn , Sp) is hypercomplete if F satisfies descent for hypercovers in the quasisyntomic topology (rather thanonly for Čech covers as above). We let Shv hyp (QSyn , Sp) ⊂ Shv(QSyn , Sp) denote the subcategoryof hypercomplete sheaves; this inclusion is the right adjoint of a Bousfield localization ( − ) h :Shv(QSyn , Sp) → Shv hyp (QSyn , Sp) called hypercompletion.The presentable, stable ∞ -category Shv(QSyn , Sp) admits a canonical t -structure (as sheaves onany site do). A Sp -valued sheaf F on QSyn is connective if for every A ∈ QSyn and x ∈ π j ( F ( A )) for j < , there exists a quasisyntomic cover A → B such that x is carried to zero in π j ( F ( B )) .Similarly, F is coconnective if it takes values in coconnective spectra. The t -structure restricts to a t -structure on the hypercomplete sheaves, and every bounded-above sheaf is automatically hyper-complete. With respect to this t -structure, the heart of Shv(QSyn , Sp) is the ordinary category ofsheaves of abelian groups on
QSyn . Construction 5.3 (Postnikov towers) . Given any
F ∈
Shv(QSyn , Sp) , we have its Postnikov tower {F ≤ n } n ∈ Z with respect to the above t -structure. The limit of this Postnikov tower is given by itshypercompletion F h . This is a consequence of the fact that the quasisyntomic site is “replete” in the The hypercomplete sheaves are those sheaves which receive no maps from ∞ -connected sheaves. sense of [BS15, Sec. 3]; compare [Mat20, Prop. A.10]. In particular, if F is already hypercomplete,then F is the limit of its Postnikov tower.A basic tool for working with sheaves on QSyn is restriction to the basis
QRSPerfd ⊂ QSyn .In general, given a Grothendieck site, then it is a classical result [AGV72, Exp. III, Th. 4.1] thatsheaves of sets or abelian groups are equivalent to sheaves on any basis of the site. The analog neednot hold for sheaves of spaces or spectra, but it at least holds for hypercomplete sheaves in general,cf. [Aok20, App. A] or [BGH18, Prop. 3.12.11]. In the case of
QRSPerfd ⊂ QSyn , it is actuallytrue that arbitrary sheaves on
QSyn identify with sheaves on the basis
QRSPerfd , cf. [BMS19,Prop. 4.31] or [Hoy14, Lem. C.3] (for a more general statement); the main point is that a pushout B b ⊗ A C in QSyn along quasisyntomic covers with
B, C ∈ QRSPerfd belongs to
QRSPerfd . Notethat this strategy of restricting to
QRSPerfd ⊂ QSyn is useful precisely because we are workingwith such “infinitary” sites; it would be much less useful if we worked with more classical sites suchas the syntomic or fppf site.Now we discuss some examples of sheaves on
QSyn . Example 5.4.
For each i ≥ , the construction R V i L R/ Z p [ − i ] defines a sheaf of spectra on QSyn (thanks to Theorem 3.6, with a slight modification since we are working with p -completelyfaithful flatness). This sheaf belongs to the heart (so thus corresponds to a sheaf of ordinary abeliangroups). In fact, this follows because it takes discrete values on the quasiregular semiperfectoid rings. Theorem 5.5.
The functors
HH( − ; Z p ) , HH( − ; Z p ) h T , HH( − ; Z p ) t T , THH( − ; Z p ) , TC − ( − ; Z p ) , TP( − ; Z p ) , TC( − ; Z p ) ,etc. all define hypercomplete sheaves on QSyn . In fact, all of these functors define (a priori not hypercomplete) sheaves on the ( p -completely) flattopology on all rings, as in [BMS19, Sec. 3]; for quasisyntomic rings the argument shows that theyare hypersheaves. One uses the Hochschild–Kostant–Rosenberg filtration [NS18, Prop. IV.4.1] toprove that HH( − ; Z p ) is a hypercomplete sheaf on QSyn starting from the fact that the p -completecotangent complex and its wedge powers are sheaves on QSyn . Taking homotopy fixed points,we find that
HH( − ; Z p ) h T is a hypercomplete sheaf. Similarly, using THH( − ; Z p ) ⊗ THH( Z ) Z =HH( − ; Z p ) and taking the limit of the Postnikov tower of THH( Z ) , one bootstraps to THH( − ; Z p ) and the invariants defined from it.6. The motivic filtrations of [BMS19]To begin with, we describe the Hochschild–Kostant–Rosenberg filtration on Hochschild homologyusing the quasisyntomic site.
Construction 6.1 (The Hochschild–Kostant–Rosenberg filtration) . For a ring R and any R -algebra A , there is a functorial, complete multiplicative Z ≥ -indexed descending filtration Fil ≥∗ HKR
HH(
A/R ) on HH(
A/R ) with gr i HH(
A/R ) = V i L A/R [ i ] . This filtration is the Postnikov filtration when A is a polynomial algebra over R (using the Hochschild–Kostant–Rosenberg theorem to identify thegraded pieces), and is more generally defined via left Kan extension, cf. [NS18, Prop. IV.4.1]. Auniversal property of this filtration has been given by Raksit, [Rak20].The Hochschild–Kostant–Rosenberg filtration is the prototype of the motivic filtrations of [BMS19].However, the strategy is to define the filtration by descent from quasiregular semiperfectoids, i.e.,by a right Kan extension process rather than a left Kan extension process. These filtrations willgenerally be more complicated to construct directly for polynomial algebras. To begin with, weshow that the HKR filtration can be obtained for quasisyntomic rings in such a fashion, after p -completion. OME RECENT ADVANCES IN TOPOLOGICAL HOCHSCHILD HOMOLOGY 15
Construction 6.2 (The Hochschild–Kostant–Rosenberg filtration as a Postnikov filtration) . Sup-pose R is a quasisyntomic ring. On the category of quasisyntomic R -algebras, we consider thefunctor A HH(
A/R ; Z p ) , which defines a hypercomplete sheaf of spectra. We claim that the ho-motopy group sheaves are concentrated in even degrees, and that A Fil ≥∗ HKR
HH(
A/R ; Z p ) definesthe double-speed Postnikov tower. In other words, Fil ≥ i HKR is the i th connective cover in quasisyn-tomic sheaves. This follows easily from the observation that if A/R is such that the p -completion L A/R is the suspension of a p -completely flat R -module, then gr i HH(
A/R ; Z p ) is concentrated indegree i , and the HKR filtration reduces to the double-speed Postnikov filtration on the individualspectrum gr i HH(
A/R ; Z p ) .The starting point of the extension of the above strategy to invariants defined from THH is thefollowing generalization of Bökstedt’s theorem.
Theorem 6.3.
Let R be a perfectoid ring. Then one has an isomorphism THH ∗ ( R ; Z p ) ≃ R [ σ ] ,for | σ | = 2 . Moreover, one has π ∗ (THH( R ; Z p ) tC p ) = R [ u ± ] for | u | = 2 , and the Frobenius ϕ : THH( R ; Z p ) → THH( R ; Z p ) tC p exhibits the source as the connective cover of the target. Theorem 6.3 reduces to Bökstedt’s theorem for R = F p , and is extended to an arbitrary perfectoidring in [BMS19, Sec. 6]. The case of R = O C p had been previously proved in [Hes06]. See also[HN19, Sec. 1.3] for an account of this result. The basic strategy is to bootstrap from the case R = F p , using the Hochschild–Kostant–Rosenberg theorem and that for any map of perfectoidrings R → R ′ , the p -completed relative cotangent complex L R ′ /R vanishes.In loc. cit. , the constructions TC −∗ ( R ; Z p ) , TP ∗ ( R ; Z p ) are also identified. Let A inf = A inf ( R ) bethe Witt vectors of R ♭ , so one has a canonical surjection θ : A inf → R with kernel generated by anonzerodivisor ξ ∈ A inf . Then one has isomorphisms: TC −∗ ( R ; Z p ) = A inf [ x, σ ] / ( xσ = ξ ) , | x | = − , | σ | = 2 (8) TP ∗ ( R ; Z p ) = A inf ( R )[ u ± ] , | u | = 2 . (9)Here σ ∈ π TC − ( R ; Z p ) is a lift of the generator in π THH( R ; Z p ) . With respect to these isomor-phisms, the canonical map TC −∗ ( R ; Z p ) → TP( R ; Z p ) carries x to u − and σ to ξu . The cyclotomicFrobenius ϕ : TC −∗ ( R ; Z p ) → TP ∗ ( R ; Z p ) carries σ u and x ϕ ( ξ ) u − , and is the Witt vectorFrobenius on π .Identifying TC −∗ ( R ; Z p ) , TP ∗ ( R ; Z p ) for quasiregular semiperfectoids is significantly more difficult(and the description in purely algebraic terms is a major result of [BMS19, BS19]). To begin with,we make the simple observation that these are concentrated in even degrees. Corollary 6.4 (Evenness for quasiregular semiperfectoids) . Let A be a quasiregular semiperfectoid R -algebra. Then THH ∗ ( A ; Z p ) is concentrated in even degrees. Consequently, TC −∗ ( A ; Z p ) , TP ∗ ( A ; Z p ) are concentrated in even degrees.Proof. This follows from the equivalence(10)
THH( A ; Z p ) ⊗ THH( R ; Z p ) R ≃ HH(
A/R ; Z p ) , the Hochschild–Kostant–Rosenberg filtration (which shows that the latter is concentrated in evendegrees). Then the T -homotopy fixed point and Tate spectral sequences prove the remaining claims. (cid:3) Similarly from (10) one obtains:
Corollary 6.5.
Let A be a smooth algebra over the perfectoid ring R . Then THH ∗ ( A ; Z p ) ≃ R [ σ ] ⊗ R Ω ∗ R/ Z p with | σ | = 2 . With respect to the above equivalence, the motivic filtration on
THH( A ; Z p ) is such that σ belongs to filtration and Ω A/R belongs to filtration . This is not a Postnikov filtration, soit seems difficult to construct the filtration on THH( A ; Z p ) purely within the setting of smooth R -algebras. Thus, one needs to use instead the quasisyntomic site.In particular, it follows from Corollary 6.4 that the constructions THH( − ; Z p ) , TC − ( − ; Z p ) , TP( − ; Z p ) ,when considered as objects of Shv(QSyn , Sp) , have homotopy groups concentrated in even degrees.In fact, the same holds for
TC( − ; Z p ) . Theorem 6.6 (The odd vanishing conjecture, [BS19, Sec. 14]) . The quasisyntomic sheaf
TC( − ; Z p ) has homotopy groups concentrated in even degrees. Theorem 6.6 (which was conjectured in [BMS19]) is much more difficult than Corollary 6.4. Inparticular, the evenness of
TC( R ; Z p ) does not hold for an arbitrary quasiregular semiperfectoidring, and the proof relies on André’s lemma and the theory of prismatic cohomology. Definition 6.7 (The motivic filtrations) . The motivic filtration on
THH( − ; Z p ) (resp. TC − ( − ; Z p ) , TP( − ; Z p ) , TC( − ; Z p ) )is given as the double speed Postnikov filtration in Shv(QSyn , Sp) , in other words
Fil ≥ i THH( − ; Z p ) is the i -th connective cover of the quasisyntomic sheaf THH( − ; Z p ) .We define the objects for A ∈ QSyn , b ∆ A { i } = gr i TP( A ; Z p )[ − i ] , (11) N ≥ i b ∆ A { i } = gr i TC − ( A ; Z p )[ − i ] , (12) Z p ( i )( A ) = gr i TC( A ; Z p )[ − i ] . (13)All these define sheaves of p -complete, coconnective spectra on QSyn . Remark 6.8 (The motivic filtrations on
QRSPerfd ) . A priori, the motivic filtrations are defined us-ing the abstract theory of sheaves of spectra, and the t -structure there. However, if A ∈ QRSPerfd ,the motivic filtrations on
THH( A ; Z p ) , TC − ( A ; Z p ) , TP( A ; Z p ) are very explicit: they are simplythe double-speed Postnikov filtrations on these individual spectra. In other words, when restricted toquasiregular semiperfectoid rings, the individual homotopy groups of THH( − ; Z p ) , TC − ( − ; Z p ) , TP( − ; Z p ) form sheaves of spectra, cf. [BMS19, Sec. 7]. In particular, for a quasiregular semiperfectoid ring A , we have b ∆ A { i } = π i TP( A ; Z p ) ; this is an invertible module over b ∆ A = π TP( A ; Z p ) .Indeed, the object b ∆ A = gr TP( A ; Z p ) (for R quasisyntomic) is perhaps the most fundamentalof all the above structures and is closely related to prismatic cohomology [BS19]. Let us discusssome of the structure that it carries, which follows directly from its definition.Let A ∈ QRSPerfd . The
Nygaard filtration on b ∆ A = π TP( A ; Z p ) = π TC − ( A ; Z p ) is thefiltration that comes from the homotopy fixed point spectral sequence. In particular, we define N ≥ i b ∆ A = π ( τ ≥ i THH( A ; Z p )) h T ⊂ b ∆ A . This defines a descending, multiplicative, and completefiltration on b ∆ A such that N ≥ i b ∆ A / N ≥ i +1 b ∆ A = π i THH( A ; Z p ) . By descent, we obtain the Nygaardfiltration on b ∆ A for all quasisyntomic A . For A quasiregular semiperfectoid, b ∆ A { i } = π i TP( A ; Z p ) is an invertible b ∆ A -module (which can be trivialized, but not canonically in general) and the notationabove N ≥ i b ∆ A { i } = π i TC − ( A ; Z p ) is consistent. The Frobenius gives an endomorphism ϕ : b ∆ A → b ∆ A which for A ∈ QRSPerfd comes from the cyclotomic Frobenius. The filtration and the Frobenius
OME RECENT ADVANCES IN TOPOLOGICAL HOCHSCHILD HOMOLOGY 17 interact: we also have “divided” Frobenii ϕ i : N ≥ i b ∆ A { i } → b ∆ A { i } for i ≥ , which arise from thecyclotomic Frobenius on π i .If A is a quasiregular semiperfectoid algebra over the perfectoid ring R , then π i THH( A ; Z p ) has a finite filtration whose associated graded terms are (the p -completions of) V j L A/R [ − j ] for ≤ j ≤ i . This gives strong control over the associated graded terms of the Nygaard filtration N ≥∗ b ∆ A in terms of the cotangent complex of A/R . Moreover, as A ranges over quasiregularsemiperfectoid R -algebras, we can trivialize the Breuil–Kisin twists b ∆ A { i } for i ∈ Z using thedescription of TC −∗ ( R ; Z p ) , TP ∗ ( R ; Z p ) . In particular, we have that ϕ becomes divisible by ϕ ( ξ ) i (one typically writes e ξ = ϕ ( ξ ) ) on N ≥ i b ∆ A and we have a divided Frobenius ϕ/ e ξ i : N ≥ i b ∆ A → b ∆ A .By descent, we obtain this structure for any quasisyntomic R -algebra. Example 6.9.
In the base case of the perfectoid ring R , we have b ∆ R = A inf and N ≥ i b ∆ R = ξ i A inf .Given a quasiregular semiperfectoid R -algebra A , the ideal ( ξ ) is b ∆ A is well-defined (and is containedin N ≥ b ∆ A ); however, it depends on the choice of perfectoid ring R . On the other hand, the ideal ( e ξ ) = ( ϕ ( ξ )) is well-defined purely in terms of A without reference to R . In fact, it is the kernel ofthe map b ∆ A = π (TP( A ; Z p )) → π (THH( A ; Z p ) tC p ) .In particular, by analyzing topological Hochschild homology and its homotopical structure, oneobtains the above quasisyntomic sheaf of rings, equipped with the Frobenius and filtration. Thisis a structure of great interest to p -adic arithmetic geometry in mixed characteristic. For formallysmooth algebras over a perfectoid ring, this agrees with the construction of A inf -cohomology of[BMS18] (and later [BS19]).In the next couple of sections, we will discuss the situation in more detail in characteristic p ,where one recovers the theory of crystalline cohomology.7. Derived de Rham cohomology
In this section, we discuss some of the properties of p -adic derived de Rham cohomology, after[Bha12b]; see also [SZ18]. Fix a base ring k . Definition 7.1 (Derived de Rham cohomology [Ill72, Sec. VIII.2]) . Let R be a k -algebra. The derived de Rham cohomology L Ω R/k ∈ D ( k ) is the left derived functor of the functor P Ω • P/k sending a polynomial k -algebra to its de Rham complex considered as an E ∞ -algebra over k . More-over, L Ω R/k is equipped with the descending, multiplicative derived Hodge filtration n L Ω ≥∗ R/k o obtained as the left Kan extension of the naive filtration on the de Rham complex of a polynomial k -algebra (i.e., the i th filtration piece consists of j -forms for j ≥ i ). Remark 7.2 (The Hodge completion) . The Hodge completion of derived de Rham cohomology isoften more tractable. For example, for a smooth k -algebra R , the Hodge completion of L Ω R/k agreeswith the usual de Rham complex; this follows by considering the map from derived to underivedde Rham cohomology (with respective Hodge filtrations), and using that V i L R/k = Ω iR/k for
R/k smooth.
Example 7.3 (Derived de Rham cohomology in characteristic zero) . Let k = C , and let R be afinitely generated C -algebra. On the one hand, derived de Rham cohomology of animated C -algebrasis easily seen to be the constant functor with value C in this case; indeed, this follows because thede Rham complex of a polynomial C -algebra is acyclic in positive degrees. On the other hand,the Hodge completion of derived de Rham cohomology agrees with the singular cohomology (with C -coefficients) of the associated complex points. This is a classical result of Grothendieck [Gro66]for R smooth; compare [Bha12a] for a discussion in general.In the sequel, we will only consider the p -adic version of derived de Rham cohomology, and wewill simply drop the p -completion from the notation. We will also often drop the p -completionnotation on the cotangent complex and its wedge powers. Construction 7.4 (The derived conjugate filtration) . Let A → B be a map of animated F p -algebras. By left Kan extension of the Postnikov filtration (and using the Cartier isomorphism)we see that L Ω B/A admits a natural B (1) := B ⊗ A,ϕ A -structure and an increasing, multiplicative,and exhaustive filtration Fil ∗ conj L Ω B/A in D ( B (1) ) ; the associated graded pieces are given by gr i = V i L (1) B/A [ − i ] .A key consequence of the derived conjugate filtration, the fact that differential forms and thecotangent complex agree for smooth algebras, and the Cartier isomorphism for smooth algebras,is the following result. Note that it shows that derived de Rham cohomology behaves entirelydifferently in characteristic p than in characteristic zero. Theorem 7.5 (Bhatt [Bha12b, Cor. 3.10]) . Given a smooth map A → B of rings, the p -completederived de Rham cohomology L Ω B/A agrees with the p -complete de Rham cohomology Ω • B/A (withderived and classical Hodge filtrations matching).
Remark 7.6.
Let R be a Cartier smooth F p -algebra. Then the natural map L Ω R/ F p → Ω • R/ F p isan equivalence respecting Hodge filtrations. This also follows from the conjugate filtration. In fact,for each i , the map L ( τ ≤ i Ω R/ F p ) → τ ≤ i Ω • R/ F p is an equivalence; one sees this on associated gradedterms, whence it follows from the assumptions.A further aspect of the p -adic theory is the appearance of certain p -adic period rings whenone applies p -adic derived de Rham cohomology to certain large rings, shown in [Bei12] in theHodge-completed case and explored further in [Bha12b]. This phenomenon arises from the naturalappearance of divided powers, cf. [SZ18, Prop. 3.16] for a detailed account. Example 7.7 (Divided powers via derived de Rham cohomology) . Consider the map Z p [ x ] → Z p .Then the p -adic derived de Rham cohomology is given by the p -complete divided power algebra (cid:16) Z p h x i i ! i(cid:17) ˆ p : more precisely, the natural map Z p [ x ] → L Ω Z / Z p [ x ] exhibits the target as the p -adicdivided power completion of ( x ) in the source.To see this, we observe that everything involved has a grading. Formally, we work in the ∞ -category of nonnegatively graded animated rings R ⋆ with R = Z p . The construction L Ω B/A carries through in this ∞ -category, and it is not difficult to see that the Hodge filtration con-verges for grading reasons (that is, L Ω ≥ iB/A is concentrated in degrees ≥ i ). In the graded ∞ -category, the isomorphism L Ω Z p / Z p [ x ] = (cid:16) Z p h x i i ! i(cid:17) ˆ p follows by passage to the associated gradedof the Hodge filtration gr ∗ ( L Ω B/A ) = V ∗ L B/A [ −∗ ] , using L Z p / Z p [ x ] = Z p [1] and the décalage iso-morphism V i ( M [1]) = Γ i ( M )[ i ] . Passing from the graded ∞ -category to the ungraded one, weconclude the desired isomorphism. One also uses here that if A → B is a map of animated nonnegatively graded rings with A = B = Z p , then A → L Ω B/A is an isomorphism in degree ; this follows easily from the case of a polynomial algebra. OME RECENT ADVANCES IN TOPOLOGICAL HOCHSCHILD HOMOLOGY 19
In particular, if A is a p -complete ring and x ∈ A is a nonzerodivisor, then the p -adic derived deRham cohomology of A → A/x is simply the p -complete divided power envelope of ( x ) ; this followsfrom the above by base-change. Construction 7.8 (Derived de Rham cohomology as a quasisyntomic sheaf) . Let R be a qua-sisyntomic ring; for simplicity we assume R is p -torsionfree or an F p -algebra. On the category ofquasisyntomic R -algebras, the construction A L Ω A/R defines a sheaf of spectra, which belongs tothe heart of the t -structure (in fact, it takes discrete values on quasiregular semiperfectoid algebras).This follows from reducing modulo p and the derived conjugate filtration. Similarly, A L Ω ≥ iA/R defines a sheaf of spectra (also in the heart). Construction 7.9 (The Hodge-completed variant) . Let R be a quasisyntomic ring. On the cate-gory of quasisyntomic R -algebras, the constructions A \ L Ω A/R , \ L Ω ≥ iA/R defines a sheaf of cocon-nective spectra, which belongs to the heart of the t -structure (in fact, it takes discrete values onquasiregular semiperfectoid algebras). This follows from the Hodge filtration.The cohomology theories of [BMS19], for algebras over a perfectoid base, can be described as“deformations” of (Hodge-completed) de Rham cohomology, which therefore plays a central rolein the theory. This arises as the combination of the following two results. The first (cf. [BMS19,Sec. 5]) gives a close relationship between de Rham and periodic cyclic homology. Other proofs(which work outside the p -complete context) have been given by Antieau [Ant19], Moulinos–Robalo–Toen [MRT19], and Raksit [Rak20]. For the result, we write HC − = HH h T , HP = HH t T . Theorem 7.10.
Let R be a quasisyntomic ring, and let A be a quasisyntomic R -algebra such that L R/A is the suspension of a p -completely flat module (e.g., A could be quasiregular semiperfectoid).Then we have natural isomorphisms π i HP(
A/R ; Z p ) = \ L Ω A/R , π i HC − ( A/R ; Z p ) = \ L Ω ≥ iA/R . In particular, by quasisyntomic descent, we obtain multiplicative, convergent exhaustive Z -indexeddescending filtrations for any quasisyntomic R -algebra A , on HC − ( A/R ; Z p ) , HP(
A/R ; Z p ) with gr i HP(
A/R ; Z p ) = \ L Ω A/R [2 i ] , gr i HC − ( A/R ; Z p ) = \ L Ω ≥ iA/R [2 i ] . Remark 7.11.
In characteristic zero and for
A/R smooth, the analogs of these filtrations arecanonically split (e.g., by Adams operations), and the connection between periodic cyclic and deRham cohomology is classical, cf. [Lod98, Sec. 5.1.12]. However, these filtrations are not canonicallysplit in positive characteristic, and the induced spectral sequences from de Rham cohomology toperiodic cyclic homology need not degenerate for smooth projective varieties [ABM19].The second result, which comes from analyzing the structure of
THH( R ; Z p ) , states that TP gives a one-parameter deformation of HP , for algebras over a perfectoid base. Theorem 7.12.
Let R be a perfectoid ring, and let A be any R -algebra. Then there is a naturalequivalence (14) TP( A ; Z p ) /ξ = HP( A/R ; Z p ) . More precisely, we have an equivalence of E ∞ -algebras TP( A ; Z p ) ⊗ TP( R ; Z p ) HP(
R/R ; Z p ) =HP( A/R ; Z p ) . Using Theorem 6.3 and the surrounding discussion, we have that TP ∗ ( R ; Z p ) = A inf [ u ± ] and HP(
R/R ; Z p ) = R [ u ± ] ; the map TP ∗ ( R ; Z p ) → HP ∗ ( R/R ; Z p ) has kernel generatedby the element ξ ∈ A inf . Compare also [AMN18] for a discussion of related results. By considering (14) for A a quasiregular semiperfectoid R -algebra, combining with Theorem 7.10,and using the definitions of the motivic filtrations, we find that(15) b ∆ A /ξ = \ L Ω A/R . By quasisyntomic descent, we obtain (15) for all quasisyntomic R -algebras A . In particular, b ∆ A gives a one-parameter deformation of (Hodge-completed) derived de Rham cohomology. Remark 7.13 (Non-Nygaard complete prismatic cohomology) . Given a perfectoid ring R , one candefine a “Nygaard decompleted” version ∆ − of b ∆ − , which deforms derived de Rham cohomologyrather than its Hodge completion. Namely, one considers the quasisyntomic sheaf b ∆ − and restrictsto formally smooth R -algebras, and then left Kan extends from formally smooth (or p -completepolynomial) R -algebras to all p -complete R -algebras, as functors to ( p, ξ ) -complete E ∞ -algebrasover A inf . This yields a construction A ∆ A which provides a deformation along the parameter ξ of derived de Rham cohomology, i.e., one has functorial equivalences ∆ A /ξ ≃ L Ω A/R , whichtherefore also restricts to a sheaf on quasisyntomic R -algebras (and belongs to the heart). At leasta priori, this construction depends on the choice of the perfectoid ring R mapping to A . However,in [BS19, Sec. 7] a purely algebraic construction of ∆ − is given (on quasiregular semiperfectoidrings, from which one can descend) that makes clear that ∆ − can genuinely be defined on the wholequasisyntomic site, without the choice of a perfectoid base. Similarly, ∆ − is still equipped with aNygaard filtration (cid:8) N ≥∗ ∆ − (cid:9) such that the completion with respect to this filtration is c ∆ − ; thisfollows because the associated graded terms of the Nygaard filtration are left Kan extended from p -complete polynomial rings as proved in [AMMN20, Cor. 5.21].We have seen that p -adic derived de Rham cohomology coincides with the “underived” versionfor smooth algebras. More generally, there is a similar description in the case of a locally completeintersection singularity (or a quasisyntomic ring) in terms of the divided power de Rham complexof a polynomial algebra surjecting onto it. This fact is essentially the comparison between crys-talline cohomology and derived de Rham cohomology [Bha12b] and the classical description (dueto Berthelot [Ber74, Sec. V.2.3]) of crystalline cohomology in terms of the divided power de Rhamcomplex, cf. also [BdJ11] for another approach. We do not review the general theory of dividedpower structures in detail and give an ad hoc construction. Construction 7.14 (Divided power envelopes of free algebras) . Let ( A, I ) be a pair consisting ofa p -torsionfree Z ( p ) -algebra and an ideal I ⊂ A . Suppose that A is a polynomial Z ( p ) -algebra and I ⊂ A is the ideal generated by a collection of the polynomial generators, i.e., ( A, I ) is a free object(in the evident sense) in the category of such pairs.We define the divided power algebra D I ( A ) to be the subalgebra of A ⊗ Q generated by A and the elements y i i ! , y ∈ I ; this is also the divided power envelope (cf. for instance [Sta20, Tag07H7]). We have a descending multiplicative filtration n Fil ≥∗ D I ( A ) o defined by the divided powers: Fil ≥ r D I ( A ) is the ideal generated by all elements y j ...y jmm j ! ...j m ! for j + · · · + j m ≥ r for the y k ∈ I . Onthe A -algebra D I ( A ) , we have a flat connection d : D I ( A ) → D I ( A ) ⊗ A Ω A/ Z ( p ) (extended from d : A → Ω A ), and this connection satisfies the Griffiths transversality property: d (Fil ≥ r D I ( A )) ⊂ Fil ≥ r − D I ( A ) ⊗ A Ω A . In particular, we can form the divided power de Rham complex Ω • D I ( A ) = D I ( A ) → D I ( A ) ⊗ A Ω A → D I ( A ) ⊗ A Ω A → . . . , OME RECENT ADVANCES IN TOPOLOGICAL HOCHSCHILD HOMOLOGY 21 and this is in turn equipped with a multiplicative filtration such that
Fil ≥ r Ω • D I ( A ) = Fil ≥ r D I ( A ) → Fil ≥ r − D I ( A ) ⊗ A Ω A → Fil ≥ r − D I ( A ) ⊗ A Ω A → . . . . Let ( A, I ) be a free pair as above, and let A , I denote the reductions modulo p . We denoteby ( − ) (1) the Frobenius twist along A , so A /ϕ ( I ) = ( A /I ) (1) , for instance. Then one checks(cf. [Bha12b, Lem. 3.42] and [BMS19, Prop. 8.11]) that D I ( A ) /p admits an ascending, exhaustive,multiplicative filtration (an analog of the conjugate filtration) such that gr D I ( A ) /p = A /ϕ ( I ) = ( A /I ) (1) and in general(16) gr i D I ( A ) /p = (Γ iA /I ( I /I )) (1) . Explicitly, the i th stage of the filtration on D I ( A ) /p is the A /ϕ ( I ) -module generated by y j ...y jmm j ! ...j m ! for j + . . . j m ≤ pi . Construction 7.15 (Derived divided power envelopes) . Given any pair ( A, I ) consisting of a Z ( p ) -algebra A and an ideal I ⊂ A , we can simplicially resolve the pair in terms of pairs ( B, J ) which are free in the sense above. Taking simplicial resolutions, we obtain the derived dividedpower envelope LD I ( A ) (a priori, an animated ring) equipped with its filtration Fil ≥∗ LD I ( A ) anda connection satisfying Griffiths transversality. By left Kan extension of (16), there exists ananalogous increasing, multiplicative, and exhaustive filtration on LD I ( A ) /p .Suppose that ( A, I ) is a pair such that A, A/I are p -torsionfree, the Frobenius on A = A/p is flat(e.g.,
A/p is smooth over a perfect ring), and such that the p -completion of L A/I/A is p -completelyflat. In particular, it follows from the conjugate filtration (and reducing modulo p ) that LD I ( A ) is a p -torsionfree, discrete ring. In particular, it is simply the subring of A ⊗ Q generated by the y i i ! , y ∈ I , and (by taking resolutions) one sees that it is actually the divided power envelope in theusual sense [Sta20, Tag 07H7]. We will only be interested in this case.We now record the main result. Again, we emphasize that this result is essentially the comparisonbetween crystalline and derived de Rham cohomology as in [Bha12b]. Theorem 7.16 ( L Ω via the divided power de Rham complex) . Suppose ( A, I ) is a pair suchthat A, A/I are p -torsionfree, A/p is Cartier smooth and has flat Frobenius, and such that the p -completion of L A/I/A is p -completely flat. Then there is a natural multiplicative, filtered isomor-phism between the p -adic derived de Rham cohomology L Ω A/I/ Z and the p -completed divided powerde Rham complex Ω • D I ( A ) .Proof. We will use a similar argument as in Theorem 7.5. In fact, it suffices to show that for apair ( A, I ) satisfying the above conditions, Ω • D I ( A ) is quasi-isomorphic modulo p to its left Kanextension from free pairs; for a free pair the natural map induces a filtered quasi-isomorphism Ω • D I ( A ) → Ω • ( A/I ) / Z ( p ) by the Poincaré lemma. For this, we will produce an appropriate filtrationon Ω • D I ( A ) /p .By our assumptions, D I ( A ) is simply the subring of A ⊗ Q generated by the divided powers of I .Thus, we can use the conjugate filtration Fil ≤∗ conj D I ( A ) /p , as in (16), which is defined by left Kanextension from the case of a free algebra. From its definition (and left Kan extension), we see that A = A/p -connection on D I ( A ) /p is compatible with the conjugate filtration. In particular, wehave an ascending, exhaustive, multiplicative filtration on Ω • D I ( A ) /p such that gr i is the de Rham complex over A of the A -module-with-connection Γ iA /I ( I /I ) (1) . It thus suffices to show thatthis de Rham complex,(17) Γ iA /I ( I /I ) (1) → Γ iA /I ( I /I ) (1) ⊗ A Ω A → Γ iA /I ( I /I ) (1) ⊗ A Ω A → . . . as a functor from such pairs ( A, I ) to D ( F p ) , is left Kan extended from the free objects. Now the A -connection on Γ iA /I ( I /I ) (1) is the canonical (Frobenius descent) connections, cf. [Bha12b,Lem. 3.44]: explicitly, this follows because any ip -th divided power γ ip ( y ) , y ∈ I is a flat section forthis connection. Since this is a Frobenius descent connection, the Cartier isomorphism (valid since A is Cartier smooth) goes into effect: the j th cohomology of (17) is given by Γ iA /I ( I /I ) (1) ⊗ A Ω iA . Thus, it follows that (17) is left Kan extended from free pairs, whence the result. (cid:3) The ring A crys In this section, we will construct the functor A b ∆ A together with the Nygaard filtration anddivided Frobenii in characteristic p . We describe the construction purely algebraically here, andin the next section will outline the proof that it is compatible with the construction arising fromtopological Hochschild homology.We first need the divided power construction for ideals containing p (and where the dividedpowers are compatible with the canonical divided powers on ( p ) ). The construction is analogous tothat of Construction 7.14; however, we will not have the analog of the Hodge filtration. Construction 8.1 (Derived divided powers for ideals containing p ) . Let ( A, I ) be a pair consistingof a Z ( p ) -algebra and an ideal I ⊂ A containing p .Suppose first ( A, I ) is free: in other words, that A is a polynomial ring and I ⊂ A is the idealgenerated by a subset of the polynomial generators together with p . In this case, we define D I ( A ) to be the subring of A ⊗ Q generated by A and n y i i ! o y ∈ I . In general, we define the derived dividedpowers LD I ( A ) by simplicially resolving the pair ( A, I ) by free objects, and taking the inducedsimplicial resolution of divided power envelopes.As before, LD I ( A ) defines an animated ring, and its rationalization is simply A . To control it ingeneral, we again use the conjugate filtration. This gives that if ( A, I ) is a free pair, then D I ( A ) /p has an ascending filtration as in (16). In particular, we find that if the pair ( A, I ) is such that A is p -torsionfree, A/p is Cartier smooth with flat Frobenius, and L A/I/ ( A/p ) is the suspension ofa flat A/I -module, then LD I ( A ) is discrete and p -torsionfree; we will thus simply write D I ( A ) .In particular, again by taking resolutions and comparing, one verifies the universal property that D I ( A ) is actually the divided power envelope of ( A, I ) compatible with the canonical divided powerson ( p ) . Definition 8.2 (The rings A inf , A crys ) . Let R ∈ QRSPerf F p .(1) The ring R ♭ (the tilt of R ) is defined as the inverse limit perfection of R , i.e., R ♭ = lim ←− ϕ R .This comes with a natural map R ♭ → R , and our assumption implies that this map issurjective.(2) The ring A inf ( R ) is defined to be W ( R ♭ ) ; we have a natural surjective map(18) θ : W ( R ♭ ) → R. In particular, A inf ( R ) is the universal p -complete pro-nilpotent thickening of R . OME RECENT ADVANCES IN TOPOLOGICAL HOCHSCHILD HOMOLOGY 23 (3) The ring A crys ( R ) is defined as the p -complete (derived) divided power envelope of thesurjection θ (whose kernel includes ( p ) ). Remark 8.3 (Properties of A crys ) . Our assumptions imply that LD ker θ W ( R ♭ ) is p -torsionfree.Therefore, A crys ( R ) can equivalently be obtained by taking the subring of A inf ( R )[1 /p ] generatedby the divided powers of ker( θ ) , and then p -adically completing again. In particular, it is actually adiscrete p -torsionfree, p -complete ring (and not an animated ring). Moreover, it is the p -completionof the (classical) divided power envelope of θ compatible with divided powers on ( p ) , since theclassical and derived divided power envelopes coincide. Remark 8.4.
The choice of the map θ is in some sense arbitrary. Given any perfect F p -algebra P with a surjection P ։ R , we could instead construct A crys ( R ) as the p -complete divided powerenvelope of W ( P ) ։ R ; this does not change the outcome. Example 8.5.
Let R be the ring F p [ x /p ∞ ] / ( x ) . Then A crys ( R ) is the p -adic completion of thesubring Z p [ x /p ∞ , x i i ! ] i ≥ ⊂ Q p [ x /p ∞ ] .The construction R A crys ( R ) defines a sheaf of spectra (which actually has image in discretespectra) on QRSPerf F p . Indeed, since A crys takes values in p -complete, p -torsionfree abelian groups,it suffices to observe that A crys ( R ) /p defines a sheaf; but this in turn follows from the conjugatefiltration as in (16) and descent for the cotangent complex and its wedge powers. In fact, one canexplicitly identify its reduction modulo p as a sheaf on QRSPerf F p . One can give a proof of thisusing derived divided powers for F p -algebras.We will be interested in the case of certain lci singularities, and first we will need the followingresult. Theorem 8.6 (Cf. [BMS19, Prop. 8.12]) . For R ∈ QRSPerf F p , we have a natural isomorphism A crys ( R ) /p = L Ω R/ F p . One can also prove the following closely related result, identifying A crys ( R ) with the derived deRham cohomology of any p -adic lift. Theorem 8.7.
Let S be a quasisyntomic ring which is p -torsionfree and such that R = S/p isquasiregular semiperfect. Then we have a natural isomorphism A crys ( R ) = L Ω S/ Z p .Proof. Let S ♭ denote the inverse limit perfection of S/p ; we have a map W ( S ♭ ) → S which is sur-jective modulo p by our assumptions, hence surjective. By Theorem 7.16, it follows that L Ω S/ Z p = L Ω S/W ( S ♭ ) is the p -complete derived divided power envelope of the surjection W ( S ♭ ) → S . Simi-larly, by construction A crys ( R ) is the p -complete derived divided power envelope of the surjection W ( S ♭ ) → S/p compatible with the divided powers on ( p ) . But it is easy to see that for any pair ( A, I ) with A, A/I p -torsionfree, the derived divided power envelope of ( A, I ) and the derived di-vided power envelope of ( A, ( I, p )) (where the latter is taken compatible with divided powers on p ) agree: indeed, this follows by left Kan extension from the polynomial case, when the result isclear. (cid:3) By descent, we obtain from A crys ( − ) a sheaf of spectra on QSyn , which is a p -adic lift of the sheaf L Ω − / F p . One can show that this is precisely derived crystalline cohomology , i.e., the functor on F p -algebras obtained by left Kan extending (absolute) crystalline cohomology. In particular, the basiccomparison theorems in crystalline (or de Rham–Witt) theory yields that de Rham cohomology ofsmooth F p -algebras admits a p -adic lift given by crystalline cohomology. Left Kan extending to quasisyntomic F p -algebras, one obtains a p -adic lift of L Ω − / F p , and one shows that this is preciselythe above one. In other words: Theorem 8.8 (Cf. [BMS19, Th. 8.14]) . For a quasiregular semiperfect F p -algebra R , there is anatural isomorphism between the derived crystalline cohomology of R (or the derived de Rham–Witt cohomology of R ), obtained by left Kan extending crystalline cohomology from polynomial F p -algebras, and the ring A crys ( R ) . In particular, by descent from quasiregular semiperfect F p -algebras, the construction of thering A crys ( R ) provides another approach to crystalline cohomology; of course, the approach is notessentially different from the classical one, since the definition of divided powers is fundamental tothe crystalline site and the basic construction of crystalline cohomology. A key insight of [BMS19] isthat topological Hochschild homology provides a new (and fundamentally different) approach to theconstruction of crystalline cohomology, where divided powers arise very naturally (instead of beingintroduced by fiat). Most importantly, this approach has the advantage of working equally well inmixed characteristic, where it reproduces the prismatic cohomology [BS19]. Before formulating theresults, we need one more ingredient. Definition 8.9 (The Nygaard filtration on A crys ) . Let R be a quasiregular semiperfect F p -algebra.We define the Nygaard filtration {N ≥∗ A crys ( R ) } such that N ≥ i A crys ( R ) ⊂ A crys ( R ) consists ofthose elements x ∈ A crys ( R ) such that p i | ϕ ( x ) , where ϕ : A crys ( R ) → A crys ( R ) denotes theendomorphism induced by Frobenius. By construction, since everything involved is p -torsionfree,we have an induced divided Frobenius map ϕ/p i : N ≥ i A crys ( R ) → A crys ( R ) . Example 8.10.
Consider the case where R = F p [ x /p ∞ ] / ( x ) as in Example 8.5, so that A crys ( R ) is the p -completion of the ring Z p [ x /p ∞ , x j j ! ] . Then N ≥ i A crys ( R ) is the p -completion of the subringgenerated by p i − j ( x j j ! ) for all j ≥ . Example 8.11 (The case of a δ -lift) . More generally, let S be a p -torsionfree, p -complete δ -ring such that R = S/p is quasiregular semiperfect. Then we have the canonical identification A crys ( R ) = L Ω S , and the Nygaard filtration on A crys ( R ) is the tensor product of the Hodge filtrationon L Ω S and the p -adic filtration on Z p . This can be seen by left Kan extension from the case of an(ind-)smooth algebra, using [BMS19, Sec. 8.1.2].Although the definition of the Nygaard filtration in this manner does not make the claim imme-diately evident, each N ≥ i A crys ( R ) turns out to define a sheaf of p -complete spectra on QRSPerf F p .One way to see this is to use the following proposition, which gives very strong control over theNygaard filtration: Proposition 8.12 (Cf. [BMS19, Th. 8.14(2)]) . Let R ∈ QRSPerf F p . For each i ≥ , the map ϕ/p i induces an isomorphism N ≥ i A crys ( R ) / N ≥ i +1 A crys ( R ) ∼ −→ Fil conj , ≤ i ( A crys ( R ) /p ) . More precisely, ϕ/p i (by construction) gives a well-defined map N ≥ i A crys ( R ) / N ≥ i +1 A crys ( R ) ∼ −→ A crys ( R ) /p ; the claim is that this map has image in the i th stage of the conjugate filtration (whichis also the i th stage of the de Rham conjugate filtration under Theorem 8.6), and induces anisomorphism onto its image. In fact, one can also use the actual crystalline cohomology of R , as one sees by the universal property of A crys . OME RECENT ADVANCES IN TOPOLOGICAL HOCHSCHILD HOMOLOGY 25
Since the Nygaard filtration gives a sheaf of p -complete spectra on QRSPerf F p (thanks to Propo-sition 8.12 and the conjugate filtration), by quasisyntomic descent we obtain a filtration on thederived crystalline cohomology of quasisyntomic F p -algebra (in particular any smooth algebra overa perfect field). This is the classical construction of the Nygaard filtration. To describe this, wereview some fundamentals of de Rham–Witt theory, after [Ill79].Suppose R is a smooth algebra over a perfect field k . In this case, one has the de Rham–Wittcomplex W Ω • R of [Ill79], which gives an explicit p -torsionfree, p -adically complete commutativedifferential graded algebra representing the crystalline cohomology of R , W R = W Ω R d → W Ω R d → W Ω R → . . . , and equipped with a map of commutative dg-algebras W Ω • R → Ω • R/k which induces a quasi-isomorphism W Ω • R /p → Ω • R/k . The object W Ω • R is equipped with operators of graded abeliangroups F, V : W Ω • R → W Ω • R recovering the Witt vector Frobenius and Verschiebung in degree zeroand satisfying the identities F V = V F = p, dF = pF d, V d = pdV, (and F dV = d , which is actually a consequence by p -torsionfreeness). Another construction of W Ω • R as a “strict Dieudonné algebra” (and universal property/construction in that category) isgiven in [BLM18].The endomorphism ϕ of the commutative differential graded algebra W Ω • R induced by the Frobe-nius on R is given in degree i by p i F . Then one can define a descending, multiplicative Nygaardfiltration on the differential graded algebra W Ω • R such that(19) N ≥ i W Ω • R = p i − V W Ω R → p i − V W Ω R → · · · → V W Ω i − R → W Ω iR → W Ω i +1 R → . . . . By construction, the Frobenius ϕ (i.e., p d F in degree d ) is divisible by p i on N ≥ i W Ω • R ⊂ W Ω • R and we can define a divided Frobenius (of cochain complexes) ϕ/p i : N ≥ i W Ω • R → W Ω • R . Moreover, one checks by an explicit homological calculation that the map of cochain complexes(20) ϕ/p i : N ≥ i W Ω • R / N ≥ i +1 W Ω • R → W Ω • R /p → Ω • R/k induces a quasi-isomorphism from the source to the i -truncation of the target (the source livesin degrees ≤ i ). All this can also be developed in the generality of strict Dieudonné complexes;compare [BLM18, Sec. 8].The Nygaard filtration is somewhat subtle in general. Indeed, a purely crystalline approach tothe Nygaard filtration is not expected to exist (in filtration degrees ≥ p ).The basic comparison result is that the Nygaard filtration on A crys ( R ) recovers by descent theabove Nygaard filtration on crystalline cohomology. Proposition 8.13 (Cf. [BMS19, Th.8.14(3)]) . The Nygaard filtration (cid:8) N ≥∗ A crys ( R ) (cid:9) descends tothe above Nygaard filtration (in the derived category) on crystalline cohomology for smooth algebrasover a perfect field. To summarize, from the above discussion, we have an equivalence between the following twoconstructions of sheaves on
QSyn F p . For convenience, we will use the first notation LW Ω − .(1) The derived functor R LW Ω R of the derived de Rham–Witt cohomology on polynomial(or smooth) F p -algebras, together with its Nygaard filtration N ≥∗ LW Ω R obtained by leftKan extending the Nygaard filtration (19) on polynomial rings. (2) The quasisyntomic sheaf R R Γ QSyn (Spec
R, A crys ( − )) given on the basis QRSPerf F p bythe construction A crys ( − ) , equipped with the Nygaard filtration { R Γ QSyn (Spec R, N ≥∗ A crys ( − )) } obtained by descending the Nygaard filtration on A crys of quasiregular semiperfect F p -algebras (Definition 8.9).In fact, either construction of the Nygaard filtration gives a map in the filtered derived categoryfor any R ∈ QSyn F p ϕ/p ∗ : N ≥∗ LW Ω R → p ∗ LW Ω R where the target denotes the p -adic filtration on the derived de Rham–Witt cohomlogy.We next need to discuss the completion of this sheaf with respect to the Nygaard completion.Again, there are two equivalent ways to proceed. The first is simply to take the completion inthe filtered derived category of (cid:8) N ≥∗ LW Ω R (cid:9) for any R ∈ QSyn F p , e.g., as constructed by leftKan extension from polynomial algebras; we denote this by n N ≥∗ \ LW Ω R o . The second (whichwe describe below) is to take the Nygaard completion of A crys ( − ) for quasiregular semiperfect F p -algebras, and then descend. Construction 8.14 (The Nygaard completion of A crys ( − ) ) . Let R ∈ QRSPerf F p . Then we definethe Nygaard completion [ A crys ( R ) to be the completion of the ring A crys ( R ) with respect to theNygaard filtration, [ A crys ( R ) = lim ←− i A crys ( R ) / N ≥ i A crys ( R ) . We also obtain the Nygaard filtration N ≥∗ [ A crys ( R ) obtained by completion, and the completed divided Frobenius ϕ/p i : N ≥ i [ A crys ( R ) → [ A crys ( R ) for i ≥ . Proposition 8.15.
For R ∈ QRSPerfd F p , the ring [ A crys ( R ) is p -complete and p -torsionfree. More-over, we have a natural isomorphism [ A crys ( R ) /p ≃ \ L Ω R/ F p for R ∈ QRSPerf F p .Proof. First, A crys ( R ) is clearly derived p -complete as an inverse limit of modules of boundedtorsion. The p -torsionfreeness follows because for x ∈ A crys ( R ) , if px ∈ N ≥ i A crys ( R ) , then x ∈N ≥ i − A crys ( R ) ; this is evident from the definition of the Nygaard filtration. Next, one verifies thatthe map A crys ( R ) /p → L Ω R/ F p (as in Theorem 8.6) carries N ≥ i A crys ( R ) to the i th stage of theHodge filtration L Ω ≥ iR/ F p , e.g., by calculating explicitly in the case of F p [ x /p ∞ ] / ( x ) . The result thenfollows, cf. [BMS19, Th. 8.14]. (cid:3) When we descend from the quasiregular semiperfect F p -algebras, we find that \ LW Ω R /p ≃ \ L Ω R/ F p for R ∈ QSyn F p , i.e., Nygaard completed LW Ω − gives a p -adic lift of Hodge-completedderived de Rham cohomology. For smooth algebras over a perfect ring, it follows that the Hodgeor Nygaard completion does nothing. More generally, by Remark 7.6, this also follows for Cartiersmooth algebras: Corollary 8.16.
Suppose R is a Cartier smooth F p -algebra (e.g., a smooth algebra over a perfectfield). Then the map LW Ω R → \ LW Ω R is an equivalence, i.e., the Nygaard filtration is automati-cally complete. Proposition 8.17.
If the F p -algebra R is Cartier smooth, then the map in the filtered derivedcategory ϕ/p ∗ : (cid:8) N ≥∗ LW Ω R (cid:9) → { p ∗ LW Ω R } has the property that on gr i the map is the truncation τ ≤ i . (Equivalently, the map exhibits thesource as the connective cover in the Beilinson t -structure, cf. [BMS19, Sec. 5] , of the target.) OME RECENT ADVANCES IN TOPOLOGICAL HOCHSCHILD HOMOLOGY 27
Proof.
Indeed, this follows because on associated graded terms, the divided Frobenius is identifiedwith the map L ( τ ≤ i Ω • ) R → L Ω • R/ F p , thanks to Proposition 8.12 by descent or (8.12) (and thefollowing discussion) by left Kan extension. The hypothesis of Cartier smoothness implies that thismap implements τ ≤ i -truncation, i.e., L ( τ ≤ i Ω • R ) = τ ≤ i Ω • R . (cid:3) The motivic filtrations for quasiregular semiperfect F p -algebras In this section, we describe the identification between the associated graded pieces N ≥ i ∆ R { i } of the motivic filtration on TC − ( R ; Z p ) and the Nygaard-completed Nygaard pieces of crystallinecohomology, for R a quasisyntomic F p -algebra. By the usual descent we may work with QRSPerf F p .Let R ∈ QRSPerf F p be a quasiregular semiperfect F p -algebra. From the cyclotomic spectrum THH( R ) , we obtain the following data:(1) The spectra TC − ( R ) = THH( R ) h T , TP( R ) = THH( R ) t T , which have homotopy groupsconcentrated in even degrees, calculated via the T -homotopy fixed point and Tate spectralsequences.(2) An identification TP( R ) /p = HP( R/ F p ) (a special case of (14)).(3) The cyclotomic Frobenius ϕ : TC − ( R ) → TP( R ) .Consequently, given a quasiregular semiperfect F p -algebra R ∈ QRSPerf F p , we obtain the fol-lowing data:(1) A p -adically complete, p -torsionfree ring c ∆ R def = π (TC − ( R )) and an endomorphism ϕ cyc : c ∆ R → c ∆ R . induced by the cyclotomic Frobenius.(2) A descending, multiplicative complete filtration N ≥∗ c ∆ R arising from the homotopy fixedpoint spectral sequence; in particular, N = i ∆ R = π i THH( R ) . Moreover, we have N ≥ i c ∆ R = x i π i TC − ( R ) ⊂ π TC − ( R ) .(3) The property that ϕ cyc ( N ≥ i b ∆ R ) ⊂ p i b ∆ R .(4) A canonical, multiplicative isomorphism b ∆ R /p ≃ \ L Ω R/ F p .(5) When R is perfect, c ∆ R = W ( R ) , the endomorphism ϕ cyc identifies with the (Witt vector)Frobenius, the filtration N ≥∗ c ∆ R is the p -adic filtration. Theorem 9.1 (Cf. [BMS19, Th. 8.17]) . For R ∈ QRSPerf F p , there is a functorial isomorphismof rings [ A crys ( R ) ≃ c ∆ R , carrying the Nygaard filtration on [ A crys ( R ) to {N ≥∗ b ∆ R } . The cyclotomicFrobenius ϕ cyc on c ∆ R agrees with the endomorphism induced by the Frobenius ϕ : R → R . Corollary 9.2 (The motivic filtration) . For R ∈ QSyn F p , there is a convergent and exhaustivedescending Z -indexed multiplicative filtration Fil ≥∗ TC − ( R ) , Fil ≥∗ TP( R ) such that gr i TC − ( R ) = N ≥ i \ LW Ω R [2 i ] (21) gr i TP( R ) = \ LW Ω R [2 i ] . (22) With respect to these filtrations, the cyclotomic Frobenius is the divided Frobenius ϕ/p i on the i thgraded piece. We will give a proof of these results below, in a slightly different manner than [BMS19]. Whilethere is an explicit topological argument in [BMS19] in the case of certain algebras, we argue insteadusing the following rigidity property of crystalline cohomology as a p -adic deformation of de Rhamcohomology. Theorem 9.3 (Cf. [BLM18, Th. 10.1.2]) . Let R F ( R ) , QRSPerf F p → Rings be a functoron quasiregular semiperfect F p -algebras taking values in p -adically complete, p -torsionfree rings.Suppose given an isomorphism of ring-valued functors F ( − ) /p ≃ L Ω − / F p . Then there is a uniqueisomorphism F ( − ) ≃ A crys ( − ) lifting the specified isomorphism modulo p .Proof of Theorem 9.1. To begin with, we cannot directly apply Theorem 9.1 since for a quasiregularsemiperfect R ∈ QRSPerf F p , we have c ∆ R /p = \ L Ω R/ F p , i.e., we obtain the Hodge completion of thederived de Rham cohomology. We thus need to first “decomplete;” this will follow Remark 7.13 (inthe case where the perfectoid base is F p ). To do this, we define the construction R
7→ {N ≥∗ ∆ R } on QSyn F p by restricting R
7→ {N ≥∗ c ∆ R } to finitely generated polynomial F p -algebras and thenleft Kan extending to all quasisyntomic F p -algebras. By construction (and Theorem 7.5), it followsthat R ∆ R is a p -adic deformation of R L Ω R/ F p ; in particular, ∆ defines a sheaf of spectra on QSyn F p . It is easy to see that the completion of the filtered sheaf (cid:8) N ≥∗ ∆ R (cid:9) is indeed n N ≥∗ c ∆ R o (since the associated graded terms of n N ≥∗ c ∆ R o , i.e., π ∗ THH( R ) , are already left Kan extendedfrom their unfolding to polynomial algebras).It follows from Theorem 9.3 that there is a unique functorial isomorphism (for R ∈ QRSPerf F p ) ∆ R ≃ A crys ( R ) for R ∈ QRSPerf F p compatible with the isomorphism mod p to L Ω R/ F p ; in particular,this isomorphism is compatible the projection maps to R . Moreover, again by left Kan extensionthe cyclotomic Frobenius defines an endomorphism ϕ cyc : ∆ R → ∆ R carrying N ≥ i ∆ R into p i ∆ R . We observe that ϕ cyc : ∆ R → ∆ R (or A crys ( R ) → A crys ( R ) ) is necessarilythe endomorphism induced by functoriality from the Frobenius ϕ : R → R . Note first that this isindeed the case when R is perfect, by Theorem 6.3, and ∆ R = A crys ( R ) = W ( R ) . It follows thatwhen R is semiperfect, we have by naturality (along R ♭ → R ) a commutative diagram W ( R ♭ ) = A crys ( R ♭ ) (cid:15) (cid:15) ϕ cyc / / W ( R ♭ ) (cid:15) (cid:15) A crys ( R ) ϕ cyc / / A crys ( R ) . It follows that ϕ cyc and A crys ( ϕ ) are both endomorphisms of A crys ( R ) which agree when restrictedto W ( R ♭ ) ; now taking divided power envelopes and p -completing again show that they agree.We have now shown that there is an isomorphism ∆ R ≃ A crys ( R ) , compatible with Frobenii(the cyclotomic Frobenius and the Frobenius induced by functoriality), so we simply write ϕ . Now ϕ ( N ≥ i ∆ R ) ⊂ p i ∆ R . It follows that under the above comparison, we have N ≥ i ∆ R ⊂ N ≥ i A crys ( R ) assubmodules of ∆ R = A crys ( R ) : that is, the filtration coming from THH is contained in the Nygaardfiltration. It remains to show that both filtrations are actually equal. Given this, it will follow that c ∆ R = [ A crys ( R ) , compatible with filtrations and Frobenii (by p -adic continuity).It suffices to show that the inclusion N ≥ i ∆ R ⊂ N ≥ i A crys ( R ) is an equality for each i and for thering R = F p [ x /p ∞ ] / ( x ) . Indeed, the inclusion will then be an equality for any tensor product ofsuch rings. Now any quasiregular semiperfect F p -algebra admits a surjection from a tensor product R ′ of such rings which also induces a surjection on cotangent complexes, from which we see that ∆ R ′ → ∆ R and A crys ( R ′ ) → A crys ( R ) induce surjections on filtered pieces (cf. [BMS19, Prop. 8.12]for this argument). Thus we can reduce to the case R = F p [ x /p ∞ ] / ( x ) . OME RECENT ADVANCES IN TOPOLOGICAL HOCHSCHILD HOMOLOGY 29
Suppose R = F p [ x /p ∞ ] / ( x ) . we know that ∆ R = A crys ( R ) is the p -completion of Z p [ x /p ∞ , x j j ! ] .Indeed, it suffices (thanks to the explicit description of the Nygaard filtration in this case, and since p ∈ N ≥ ∆ R ) to see that x i i ! belongs to the image of N ≥ i ∆ R → ∆ R or equivalently maps to zero inthe quotient ∆ R / N ≥ i ∆ R . For this, we use a grading argument, also used in [BLM18, Sec. 10.2]. Tothis end, we can replace R by k [ x /p ∞ ] / ( x ) for k a perfect field containing a transcendental element t ; this admits an automorphism given by sending x i t i x i , i ∈ Z [1 /p ] ≥ . Now x i i ! has weight i with respect to the induced grading (from this automorphism), while the weights of ∆ R / N ≥ i ∆ R are less than i as one sees from comparing N j ∆ R = π j THH( R ) , which admits a finite filtration by V j ′ L R/ F p [ − j ′ ] for j ′ ≤ j . (cid:3) We now unwind the construction explicitly for Cartier smooth algebras and in particular verifythe Segal conjecture. In the context of topological Hochschild homology, the
Segal conjecture refersto the assertion that the cyclotomic Frobenius ϕ : THH( R ) → THH( R ) tC p should be an equivalencein sufficiently high degrees after p -completion. The reason for the name comes from the case R = S ;in this case, THH( S ) = S and the Frobenius (or the unit map) S → S tC p is actually a p -adicequivalence [Gun80, Lin80], which is the special case of the Segal Burnside ring conjecture forthe group C p (in general a theorem of Carlsson [Car84]). In the classical approach to topologicalcyclic homology, this implies that the genuine C p n -fixed points of THH agree p -adically with the C p n -homotopy fixed points for all n ≥ , an insight due to [Tsa98, BBLNR14] and formalized in[NS18, Cor. II.4.9]. Many cases in which topological cyclic homology has been effectively computedfor ring spectra such as [AR02, Aus10] are cases where the Segal conjecture holds, and the Segalconjecture for THH seems to play a central role in the theory.Given this, it would be of interest to better understand the class of quasisyntomic rings for whichsome version of the Segal conjecture holds; this seems closely related to some type of regularity ofthe ring. Since everything in sight is endowed with a motivic filtration, one expects to see the Segalconjecture at the level of filtered pieces.We will describe this in characteristic p . First, if R is any F p -algebra, we have an equiva-lence of spectra THH( R ) tC p ≃ HP( R/ F p ) ≃ TP( R ) /p , cf. [BMS19, Prop. 6.4]. Consequently,for R ∈ QSyn F p we can define a complete, Z -indexed descending multiplicative motivic filtra-tion on THH( R ) tC p such that gr i = \ L Ω R/ F p [2 i ] . With respect to this, the cyclotomic Frobe-nius ϕ : THH( R ) → THH( R ) tC p is a filtered map, and on gr i it is given by the map ϕ/p i : N ≥ i ∆ R / N ≥ i +1 ∆ R → b ∆ R /p = \ L Ω R/ F p ; this follows from the commutative diagram TC − ( R ) (cid:15) (cid:15) ϕ / / TP( R ) (cid:15) (cid:15) THH( R ) ϕ / / THH( R ) tC p and descent from QRSPerf F p , since the cyclotomic Frobenius realizes the divided Frobenius. Notethat TC − ( R ) /x = THH( R ) for x ∈ π − (TC − ( F p )) as in Theorem 6.3 and the following discussion.The next result (for smooth algebras over a perfect field) appears in [BMS19, Cor. 8.18]. Thelast part had previously appeared in [Hes18]. See also [HW19] for a recent proof at p = 2 using topological Hochschild homology. Corollary 9.4.
For R/ F p Cartier smooth, the cyclotomic Frobenius
THH( R ) → THH( R ) tC p hasthe property that on gr i [ − i ] , it identifies with the ( − i ) -connective cover τ ≤ i (Ω ∗ R/ F p ) → Ω ∗ R/ F p . If Ω iR/ F p = 0 for i > d (e.g., R could be smooth of dimension d over a perfect ring), then THH( R ) → THH( R ) tC p has ( d − -truncated homotopy fiber.Proof. This follows from Proposition 8.17, given the above discussion and description of the map
THH( R ) → THH( R ) tC p on associated graded pieces. (cid:3) Question 9.5.
Note that the results of [BMS19, Sec. 9] (as well as [BS19]) prove the Segal con-jecture for smooth algebras over a perfectoid ring. The Segal conjecture does hold for p -completeregular noetherian rings (with mild finiteness hypotheses), cf. [Mat20, Sec. 5] for an argument thatrelies on the Beilinson–Lichtenbaum conjecture for the generic fiber. For smooth algebras over thering of integers in a p -adic field (for p > ), this was proved (in a purely p -adic fashion, using TR )in [HM03, HM04]. Can one prove a filtered version of the Segal conjecture in such cases? Remark 9.6.
Another approach to the motivic filtration on TP in characteristic p has been givenin [AN18], using the expression TP( − ; Z p ) = (TR( − ; Z p )) t T proved in loc. cit, using that TR ∗ fora regular F p -algebra recovers the de Rham–Witt complex [Hes96]. This approach does not seem torecover the filtration in mixed characteristic, though.10. The Z p ( i ) : an example and some questions In this section, we revisit the calculation of the p -adic K -theory (or equivalently the topolog-ical cyclic homology) of the dual numbers over a perfect field k of characteristic p . This cal-culation is due to Hesselholt–Madsen [HM97b], using the methods of equivariant stable homo-topy theory, and has since been extended and generalized in various directions (see for instance[HM97a, Hes05, AGH09, AGHL14]). The calculation (more generally for truncated polynomialalgebras) was recently revisited by Speirs [Spe20], who gave another approach using the Nikolaus–Scholze formula (6) for TC .To begin with, let us discuss some aspects of the motivic filtration on TC in particular. Construction 10.1 (The motivic filtration on
TC( − ; Z p ) ) . Given an animated Z p -algebra R , thereis a natural complete, descending, multiplicative Z ≥ -indexed filtration Fil ≥∗ TC( R ; Z p ) with(23) gr i TC( R ; Z p ) = Z p ( i )( R )[2 i ] , i ≥ . For quasisyntomic rings, the motivic filtration is defined using descent as before: it is the doublespeed Postnikov filtration (in
Shv(QSyn , Sp) ). In particular, for a quasisyntomic ring R , we havethe expression(24) Z p ( i )( R ) = fib(id − ϕ i : N ≥ i b ∆ R { i } → b ∆ R { i } ) . This is a consequence of (6), using that the two terms above are as the graded quotients of TC − ( R ; Z p ) , TP( R ; Z p ) . Now a key feature of TC( − ; Z p ) (not shared by TC − ( − ; Z p ) , TP( − ; Z p ) ) isthat it commutes with sifted colimits, [CMM18, Theorem G]. Moreover, one can check that the mo-tivic filtration on TC( − ; Z p ) when defined in the above manner on quasisyntomic rings is actuallyleft Kan extended from p -complete polynomial algebras, and has the property that Fil ≥ i TC( R ; Z p ) is ( i − -connective, cf. [AMMN20, Theorem 5.1]. Using these facts, one can left Kan extend themotivic filtration on TC( − ; Z p ) from p -complete polynomial algebras to all p -complete animatedrings. This will agree with the previous definition on quasisyntomic rings and, because of theconnectivity statement, will converge. OME RECENT ADVANCES IN TOPOLOGICAL HOCHSCHILD HOMOLOGY 31
Let us describe the Z p ( i ) in characteristic p . Construction 10.2 (The Z p ( i ) in characteristic p ) . For R an animated F p -algebra, we recallthat we have the derived de Rham–Witt cohomology LW Ω R , the Nygaard filtration N ≥ i , and thedivided Frobenius ϕ/p i : N ≥ i LW Ω R → LW Ω R . There is a natural isomorphism(25) Z p ( i )( R ) = fib(id − ϕ/p i ) : N ≥ i LW Ω R → LW Ω R . In fact, for R ∈ QSyn F p , this expression after Nygaard completion is a consequence of (24) and theidentification of b ∆ − and \ LW Ω − on QSyn F p . Then the main observation is that the Nygaard comple-tion is actually superfluous in the expression for the Z p ( i ) , by a p -adic continuity argument, cf. theproof of [BMS19, Prop. 8.20]. Left Kan extending from finitely generated polynomial algebras, weobtain (25) in general.Using the above, one can identify the Z p ( i ) on regular F p -algebras as the pro-étale cohomologyof the logarithmic de Rham–Witt sheaves W Ω i log [ − i ] , cf. [BMS19, Cor. 8.21]; as in [KM18] this canbe generalized to Cartier smooth algebras. By the results of [GL00], this shows that the Z p ( i ) arein fact p -adic étale motivic cohomology for regular F p -schemes. Remark 10.3 (The Frobenius action) . In general, if R is any F p -algebra, a key feature of the Z p ( i ) is that the Frobenius ϕ : R → R acts as multiplication by p i on Z p ( i )( R ) ; this is evident from itsexpression (25). In particular, the motivic filtration on TC( R ; Z p ) becomes, after rationalization,the eigenspace decomposition based on the Frobenius action, and consequently splits canonically.This is of course analogous to the motivic filtration on K -theory, which rationally diagonalizes theAdams operations.We will give here a description of the Z p ( i ) of k [ x ] /x (which by the motivic filtration easily givesthe description of TC( k [ x ] /x ) ); the calculation is very close to that of [Spe20]. The key observationis that k [ x ] /x admits a natural lift to a quasisyntomic δ -ring, so we can use the divided powerde Rham complex to compute everything. Our strategy is to use the fact (cf. Example 8.11) thatif R is a p -torsionfree, p -complete δ -ring in QSyn , then there is a functorial divided Frobenius ϕ/p i : L Ω ≥ iR/ Z p → L Ω R/ Z p , and the Nygaard filtration on L Ω R/ Z p = LW Ω R/p is the tensor productof the p -adic filtration and the Hodge filtration. This is a consequence of the case of p -completepolynomial δ -rings (by a left Kan extension argument), where it follows by a direct comparisonbetween the de Rham complex of R and the de Rham–Witt complex of R/p . Compare [BMS19,Sec. 8.1.2].Now let ( A, I ) be a pair consisting such that A, A/I are equipped with the compatible structureof δ -rings, and suppose both A, A/I are p -complete, p -torsionfree, and quasisyntomic. Suppose A/p is Cartier smooth and the Frobenius on
A/p is flat. As we have seen, the Nygaard filtration on LW Ω A/ ( I,p ) = L Ω A/I/ Z p identifies with the tensor product of the Hodge filtration and the p -adicfiltration. Using Theorem 7.16, we can identify the de Rham complex (with its Hodge filtration,and Frobenius) of A/I as the divided power de Rham complex of A , with divided powers along I , and with the divided power filtration. In light of this, we obtain an explicit cochain complexrepresenting Z p ( i )( A/ ( I, p )) .Indeed, we construct the divided power de Rham complex Ω • D I ( A ) with the (cochain-level) Ny-gaard filtration N ≥∗ Ω • D I ( A ) (defined as the tensor product filtration as above). The Frobenius lift ϕ : A → A induces a Frobenius lift ϕ on Ω • D I ( A ) which becomes divisible on the subcomplex This crucially uses that I is preserved by δ . N ≥ i Ω • D I ( A ) , and we can take(26) Z p ( i )( A/I ) = fib (cid:18) N ≥ i Ω • D I ( A ) ϕ/p i − −−−−−→ Ω • D I ( A ) (cid:19) . To see this, we may reduce by descent to the case where
A/I is quasiregular semiperfect and A is a perfect δ -ring (where the Frobenius ϕ is an isomorphism), and then the above is effectivelythe definition. Thus, we get an expression for Z p ( i )( A/I ) as the mapping fiber of a cochain mapbetween explicit cochain complexes.In this section, we illustrate the above method by proving the following result. By the motivicfiltration on TC , this reproves the result of [HM97b, Th. 8.2] (and by [McC97] yields the calculationof K ∗ ( k [ x ] /x ; Z p ) since K ∗ ( k ) = Z p in degree zero, [Hil81, Kra80]). Theorem 10.4.
Let k be a perfect F p -algebra for p > . Then for i > , we have that Z p ( i )( k [ x ] /x ) has no cohomology in degree outside . Moreover, H ( Z p ( i )( k [ x ] /x )) is isomorphic to a direct sum L ≤ d ≤ i − , ( d, p )=1 W n ( i,d ) ( k ) where n = n ( i, d ) is chosen such that p n − d ≤ i − < p n d .Proof. In the above strategy, we consider the example ( A, I ) = ( W ( k )[ x ] , ( x )) where the δ -structureis such that δ ( x ) = 0 , so the Frobenius lift carries x x p . It follows that LW Ω k [ x ] /x is given bythe p -completion of the divided power de Rham complex(27) W ( k ) (cid:20) x, x j j ! (cid:21) j ≥ → W ( k ) (cid:20) x, x j j ! (cid:21) j ≥ dx. The Hodge filtration is as in Example 8.11, given by the divided power filtration and the naivefiltration. That is, L Ω ≥ iW ( k )[ x ] /x corresponds to the p -completion of the subcomplex(28) M j ≥ i W ( k ) (cid:26) x j j ! , x j +1 j ! (cid:27) j ≥ i d −→ M j ≥ i − W ( k ) (cid:26) x j j ! , x j +1 j ! (cid:27) dx Also, the Frobenius is defined on the complex by sending x x p .It follows now that N ≥ i LW Ω k [ x ] /x is given by the p -completion of the complex(29) M j ≥ p max( i − j, W ( k ) (cid:26) x j j ! , x j +1 j ! (cid:27) d −→ M j ≥ p max( i − j − , W ( k ) (cid:26) x j j ! , x j +1 j ! (cid:27) dx. In particular, thanks to (26), we obtain that Z p ( i )( k [ x ] /x ) is the mapping fiber of ϕ/p i − fromthe complex (29) to (27).Let us evaluate the Z p ( i )( k [ x ] /x ) . In this case, we can evaluate the cohomology of LW Ω k [ x ] /x .Note that only H is nonzero, and it has a natural grading (where | x | = 1 ); with respect to thisgrading, we have easily from (27) (cid:0) H ( LW Ω k [ x ] /x ) (cid:1) d = ( W ( k ) /d d odd d even . Explicitly, the generator in degree d = 2 j + 1 is x j j ! dx . OME RECENT ADVANCES IN TOPOLOGICAL HOCHSCHILD HOMOLOGY 33
Similarly, from (29) we see that N ≥ i LW Ω k [ x ] /x has cohomology concentrated in (cohomological)degree , and with respect to the internal grading we have H ( N ≥ i LW Ω k [ x ] /x ) d = W ( k ) /pd d = 2 j + 1 , j < iW ( k ) /d d = 2 j + 1 , j ≥ i d even . Explicitly, the generator in degree j + 1 is p max( i − j − x j j ! dx .Now we need to understand the canonical and divided Frobenius maps.(1) Consider ϕ/p i : N ≥ i LW Ω k [ x ] /x → LW Ω k [ x ] /x . This map multiplies the internal gradingby p . Suppose j < i . Then it carries the class of p i − j − x j j ! dx into p − i p i − j − x p j d ( x p ) j ! = p − j x (2 j +1) p − j ! dx. Now p j j ! , ( jp )! , and ( (2 j +1) p − )! agree up to p -adic units. It follows from this that for d =2 j +1 for j < i , ϕ/p i carries the degree d summand of H ( N ≥ i LW Ω k [ x ] /x ) d isomorphicallyto the degree d summand of H ( LW Ω k [ x ] /x ) pd .(2) For j ≥ i , the class ϕ/p i carries the generator in degree d = 2 j + 1 to a nonunit multiple ofthe generator in degree pd .(3) For d = 2 j + 1 for j ≥ i − , the canonical map induces an isomorphism on degree d summands.Fix an odd integer d ≥ such that d is not divisible by p . In this case, we consider the map ofabelian groups(30) M a ≥ H ( N ≥ i LW Ω k [ x ] /x ) p a d ϕ/p i − −−−−−→ M a ≥ H ( LW Ω k [ x ] /x ) p a d This suffices for the calculation of Z p ( i )( k [ x ] /x ) , since we can decompose the map ϕ/p i − oversuch d .Let n = n ( i, d ) be such that p n − d ≤ i − < p n d . We claim that (30) is surjective, and thekernel is W n ( i,d ) ( k ) if d ≤ i − (and zero if d > i − ). The map(31) M a ≥ n H ( N ≥ i LW Ω k [ x ] /x ) p a d ϕ/p i − −−−−−→ M a ≥ n H ( LW Ω k [ x ] /x ) p a d is seen to be an isomorphism after p -completion since the canonical map is an isomorphism whilethe divided Frobenius is divisible by p . Thus, to prove the claim about (30), it suffices to quotientby the summands for a ≥ n , and to consider the map(32) M a Can one calculate the Z p ( i ) of F p -algebras with worse than lci singularities?A basic example would be the case of a square-zero extension k ⊕ V , for k a perfect field and V a k -vector space, where the K -theory is calculated in [LM08]. This calculation has been extendedto perfectoid rings by Riggenbach [Rig20], using the approach to TC of [NS18].In mixed characteristic, one knows [BMS19, Sec. 10] that for formally smooth algebras over O C ,the Z p ( i ) are given by the truncated p -adic nearby cycles of the usual Tate twists on the genericfiber; they thus are closely related to integral p -adic Hodge theory. For i ≤ p − , or when oneworks up to bounded denominators, it is shown in [AMMN20] that that the Z p ( i ) recover “syntomiccohomology” in a form essentially due to [FM87, Kat87]. It would be interesting to carry out morecalculations of the Z p ( i ) and TC in mixed characteristic.Finally, the K -theory of Z /p is only known in a limited range [Bru01, Ang11]. Question 10.6. Can one compute Z p ( i )( Z /p n ) (and thus the K -theory of Z /p n ) for n > ?The work [BCM20] uses prismatic cohomology to show that L K (1) K ( Z /p n ) = 0 for n ≥ ; thisfact (and some generalizations) are also proved by different methods in [LMMT20, Mat20]. However,accessing the p -adic K -groups (or the Z p ( i ) ) themselves seems to be substantially more difficult. Astacky approach to prismatic cohomology has been proposed by Drinfeld [Dri20] and Bhatt–Lurie,and in particular one expects that the coherent cohomology of the object Σ ′′ introduced in loc. cit. should be related to the Z p ( i ) . We hope that an increased understanding of the structure of Σ ′′ andof prismatic cohomology in general will also shed some light on these K -theoretic questions. Thevery recent work of Liu–Wang [LW20] on calculating TC( O K ; F p ) via descent-theoretic methods isan important step in this direction. References [AB19] Johannes Anschütz and Arthur-César Le Bras, Prismatic Dieudonné theory , arXiv preprintarXiv:1907.10525 (2019).[ABM19] Benjamin Antieau, Bhargav Bhatt, and Akhil Mathew, Counterexamples to hochschild–kostant–rosenberg in characteristic p , arXiv preprint arXiv:1909.11437 (2019).[AGH09] Vigleik Angeltveit, Teena Gerhardt, and Lars Hesselholt, On the K -theory of truncated polynomialalgebras over the integers , J. Topol. (2009), no. 2, 277–294. MR 2529297[AGHL14] Vigleik Angeltveit, Teena Gerhardt, Michael A. Hill, and Ayelet Lindenstrauss, On the algebraic K -theory of truncated polynomial algebras in several variables , J. K-Theory (2014), no. 1, 57–81.MR 3177818[AGV72] M. Artin, A. Grothendieck, and J. L. Verdier, Théorie des topos et cohomologie étale des schémas (SGA4) , Lecture Notes in Mathematics, Springer-Verlag, 1972.[AMGR17] David Ayala, Aaron Mazel-Gee, and Nick Rozenblyum, Factorization homology of enriched ∞ -categories , arXiv preprint arXiv:1710.06414 (2017).[AMMN20] Benjamin Antieau, Akhil Mathew, Matthew Morrow, and Thomas Nikolaus, On the Beilinson fibersquare , arXiv preprint arXiv:2003.12541 (2020).[AMN18] Benjamin Antieau, Akhil Mathew, and Thomas Nikolaus, On the Blumberg-Mandell Künneth theoremfor TP , Selecta Math. (N.S.) (2018), no. 5, 4555–4576. MR 3874698 OME RECENT ADVANCES IN TOPOLOGICAL HOCHSCHILD HOMOLOGY 35 [AN18] Benjamin Antieau and Thomas Nikolaus, Cartier modules and cyclotomic spectra , to appear in J. Amer.Math. Soc., arXiv preprint arXiv:1809.01714 (2018).[And18] Yves André, La conjecture du facteur direct , Publ. Math. Inst. Hautes Études Sci. (2018), 71–93.MR 3814651[Ang11] Vigleik Angeltveit, On the algebraic K -theory of Witt vectors of finite length , arXiv preprintarXiv:1101.1866 (2011).[Ant19] Benjamin Antieau, Periodic cyclic homology and derived de Rham cohomology , Ann. K-Theory (2019),no. 3, 505–519. MR 4043467[Aok20] Ko Aoki, Tensor triangular geometry of filtered objects and sheaves , arXiv preprint arXiv:2001.00319(2020).[AR02] Christian Ausoni and John Rognes, Algebraic K -theory of topological K -theory , Acta Math. (2002),no. 1, 1–39. MR 1947457[Aus10] Christian Ausoni, On the algebraic K -theory of the complex K -theory spectrum , Invent. Math. (2010), no. 3, 611–668. MR 2609252[Avr99] Luchezar L. Avramov, Locally complete intersection homomorphisms and a conjecture of Quillen on thevanishing of cotangent homology , Ann. of Math. (2) (1999), no. 2, 455–487. MR 1726700[BBLNR14] Marcel Bökstedt, Robert R. Bruner, Sverre Lunøe-Nielsen, and John Rognes, On cyclic fixed points ofspectra , Math. Z. (2014), no. 1-2, 81–91. MR 3150193[BCM20] Bhargav Bhatt, Dustin Clausen, and Akhil Mathew, Remarks on K (1) -local K -theory , Selecta Math.(N.S.) (2020), no. 3, Paper No. 39, 16. MR 4110725[BdJ11] Bhargav Bhatt and Aise Johan de Jong, Crystalline cohomology and de rham cohomology , arXiv preprintarXiv:1110.5001 (2011).[Bei12] A. Beilinson, p -adic periods and derived de Rham cohomology , J. Amer. Math. Soc. (2012), no. 3,715–738. MR 2904571[Ber74] Pierre Berthelot, Cohomologie cristalline des schémas de caractéristique p > , Lecture Notes in Math-ematics, Vol. 407, Springer-Verlag, Berlin-New York, 1974. MR 0384804[BG16] Clark Barwick and Saul Glasman, Cyclonic spectra, cyclotomic spectra, and a conjecture of Kaledin ,arXiv preprint arXiv:1602.02163 (2016).[BGH18] Clark Barwick, Saul Glasman, and Peter Haine, Exodromy , arXiv preprint arXiv:1807.03281 (2018).[BGT13] Andrew J. Blumberg, David Gepner, and Gonçalo Tabuada, A universal characterization of higheralgebraic K -theory , Geom. Topol. (2013), no. 2, 733–838. MR 3070515[Bha] Bhargav Bhatt, An imperfect ring with a trivial cotangent complex , Available at .[Bha12a] , Completions and derived de Rham cohomology , arXiv preprint arXiv:1207.6193 (2012).[Bha12b] , p -adic derived de Rham cohomology , arXiv preprint arXiv:1204.6560 (2012).[BHM93] M. Bökstedt, W. C. Hsiang, and I. Madsen, The cyclotomic trace and algebraic K -theory of spaces ,Invent. Math. (1993), no. 3, 465–539. MR 1202133[BIM19] Bhargav Bhatt, Srikanth B. Iyengar, and Linquan Ma, Regular rings and perfect(oid) algebras , Comm.Algebra (2019), no. 6, 2367–2383. MR 3957103[BK86] Spencer Bloch and Kazuya Kato, p -adic étale cohomology , Inst. Hautes Études Sci. Publ. Math. (1986),no. 63, 107–152. MR 849653[BLM18] Bhargav Bhatt, Jacob Lurie, and Akhil Mathew, Revisiting the de Rham–Witt complex , to appear inAstérisque, preprint available at arXiv:1805.05501 (2018).[Blo86] Spencer Bloch, Algebraic cycles and higher K -theory , Adv. in Math. (1986), no. 3, 267–304.MR 852815[BM15] Andrew J. Blumberg and Michael A. Mandell, The homotopy theory of cyclotomic spectra , Geom. Topol. (2015), no. 6, 3105–3147. MR 3447100[BMS18] Bhargav Bhatt, Matthew Morrow, and Peter Scholze, Integral p -adic Hodge theory , Publ. Math. Inst.Hautes Études Sci. (2018), 219–397. MR 3905467[BMS19] , Topological Hochschild homology and integral p -adic Hodge theory , Publ. Math. Inst. HautesÉtudes Sci. (2019), 199–310. MR 3949030[Bru01] Morten Brun, Filtered topological cyclic homology and relative K -theory of nilpotent ideals , Algebr.Geom. Topol. (2001), 201–230. MR 1823499[BS15] Bhargav Bhatt and Peter Scholze, The pro-étale topology for schemes , Astérisque (2015), no. 369, 99–201. MR 3379634 [BS19] Bhargav Bhatt and Peter Scholze, Prisms and prismatic cohomology , arXiv preprint arXiv:1905.08229(2019).[Car84] Gunnar Carlsson, Equivariant stable homotopy and Segal’s Burnside ring conjecture , Ann. of Math. (2) (1984), no. 2, 189–224. MR 763905[CM19] Dustin Clausen and Akhil Mathew, Hyperdescent and étale K-theory , arXiv preprint arXiv:1905.06611(2019).[CMM18] Dustin Clausen, Akhil Mathew, and Matthew Morrow, K -theory and topological cyclic homology ofhenselian pairs , to appear in J. Amer. Math. Soc., arXiv preprint arXiv:1803.10897 (2018).[Cn06] Guillermo Cortiñas, The obstruction to excision in K -theory and in cyclic homology , Invent. Math. (2006), no. 1, 143–173. MR 2207785[ČS19] Kęstutis Česnavičius and Peter Scholze, Purity for flat cohomology , arXiv preprint arXiv:1912.10932(2019).[DGM13] Bjørn Ian Dundas, Thomas G. Goodwillie, and Randy McCarthy, The local structure of algebraic K-theory , Algebra and Applications, vol. 18, Springer-Verlag London, Ltd., London, 2013. MR 3013261[DHI04] Daniel Dugger, Sharon Hollander, and Daniel C. Isaksen, Hypercovers and simplicial presheaves , Math.Proc. Cambridge Philos. Soc. (2004), no. 1, 9–51. MR 2034012[Dri20] Vladimir Drinfeld, Prismatization , arXiv preprint arXiv:2005.04746 (2020).[FM87] Jean-Marc Fontaine and William Messing, p -adic periods and p -adic étale cohomology , Current trendsin arithmetical algebraic geometry (Arcata, Calif., 1985), Contemp. Math., vol. 67, Amer. Math. Soc.,Providence, RI, 1987, pp. 179–207. MR 902593[Fon13] Jean-Marc Fontaine, Perfectoïdes, presque pureté et monodromie-poids (d’après Peter Scholze) , no.352, 2013, Séminaire Bourbaki. Vol. 2011/2012. Exposés 1043–1058, pp. Exp. No. 1057, x, 509–534.MR 3087355[FS02] Eric M. Friedlander and Andrei Suslin, The spectral sequence relating algebraic K -theory to motiviccohomology , Ann. Sci. École Norm. Sup. (4) (2002), no. 6, 773–875. MR 1949356[Gab92] Ofer Gabber, K -theory of Henselian local rings and Henselian pairs , Algebraic K -theory, commutativealgebra, and algebraic geometry (Santa Margherita Ligure, 1989), Contemp. Math., vol. 126, Amer.Math. Soc., Providence, RI, 1992, pp. 59–70. MR 1156502[Gep19] David Gepner, An introduction to higher categorical algebra , Handbook of homotopy theory (HaynesMiller, ed.), CRC Press/Chapman and Hall, 2019.[GH99] Thomas Geisser and Lars Hesselholt, Topological cyclic homology of schemes , Algebraic K -theory (Seat-tle, WA, 1997), Proc. Sympos. Pure Math., vol. 67, Amer. Math. Soc., Providence, RI, 1999, pp. 41–87.MR 1743237[GH06a] , Bi-relative algebraic K -theory and topological cyclic homology , Invent. Math. (2006), no. 2,359–395. MR 2249803[GH06b] , The de Rham-Witt complex and p -adic vanishing cycles , J. Amer. Math. Soc. (2006), no. 1,1–36. MR 2169041[GL00] Thomas Geisser and Marc Levine, The K -theory of fields in characteristic p , Invent. Math. (2000),no. 3, 459–493. MR 1738056[Goo86] Thomas G. Goodwillie, Relative algebraic K -theory and cyclic homology , Ann. of Math. (2) (1986),no. 2, 347–402. MR 855300[GR03] Ofer Gabber and Lorenzo Ramero, Almost ring theory , Lecture Notes in Mathematics, vol. 1800,Springer-Verlag, Berlin, 2003. MR 2004652[GR04] Ofer Gabber and Lorenzo Ramero, Foundations for almost ring theory – release 7.5 , 2004.[Gro66] A. Grothendieck, On the de Rham cohomology of algebraic varieties , Inst. Hautes Études Sci. Publ.Math. (1966), no. 29, 95–103. MR 199194[GRW89] S. Geller, L. Reid, and C. Weibel, The cyclic homology and K -theory of curves , J. Reine Angew. Math. (1989), 39–90. MR 972360[Gun80] Jeremy Gunawardena, Segal’s Burnside ring conjecture for cyclic groups of odd prime order , JT Knightprize essay, Cambridge (1980).[Hes96] Lars Hesselholt, On the p -typical curves in Quillen’s K -theory , Acta Math. (1996), no. 1, 1–53.MR 1417085[Hes05] , K -theory of truncated polynomial algebras , Handbook of K -theory. Vol. 1, 2, Springer, Berlin,2005, pp. 71–110. MR 2181821 OME RECENT ADVANCES IN TOPOLOGICAL HOCHSCHILD HOMOLOGY 37 [Hes06] , On the topological cyclic homology of the algebraic closure of a local field , An alpine anthologyof homotopy theory, Contemp. Math., vol. 399, Amer. Math. Soc., Providence, RI, 2006, pp. 133–162.MR 2222509[Hes18] , Topological Hochschild homology and the Hasse-Weil zeta function , An alpine bouquet of al-gebraic topology, Contemp. Math., vol. 708, Amer. Math. Soc., Providence, RI, 2018, pp. 157–180.MR 3807755[Hil81] Howard L. Hiller, λ -rings and algebraic K -theory , J. Pure Appl. Algebra (1981), no. 3, 241–266.MR 604319[HM97a] Lars Hesselholt and Ib Madsen, Cyclic polytopes and the K -theory of truncated polynomial algebras ,Invent. Math. (1997), no. 1, 73–97. MR 1471886[HM97b] , On the K -theory of finite algebras over Witt vectors of perfect fields , Topology (1997), no. 1,29–101. MR 1410465[HM03] , On the K -theory of local fields , Ann. of Math. (2) (2003), no. 1, 1–113. MR 1998478[HM04] , On the De Rham-Witt complex in mixed characteristic , Ann. Sci. École Norm. Sup. (4) (2004), no. 1, 1–43. MR 2050204[HN19] Lars Hesselholt and Thomas Nikolaus, Topological cyclic homology , Handbook of homotopy theory(Haynes Miller, ed.), CRC Press/Chapman and Hall, 2019.[Hoy14] Marc Hoyois, A quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula , Algebr.Geom. Topol. (2014), no. 6, 3603–3658. MR 3302973[HW19] Jeremy Hahn and Dylan Wilson, Real topological hochschild homology and the Segal conjecture , arXivpreprint arXiv:1911.05687 (2019).[Ill71] Luc Illusie, Complexe cotangent et déformations. I , Lecture Notes in Mathematics, Vol. 239, Springer-Verlag, Berlin-New York, 1971. MR 0491680[Ill72] , Complexe cotangent et déformations. II , Lecture Notes in Mathematics, Vol. 283, Springer-Verlag, Berlin-New York, 1972. MR 0491681[Ill79] , Complexe de de Rham-Witt et cohomologie cristalline , Ann. Sci. École Norm. Sup. (4) (1979), no. 4, 501–661. MR 565469[Jar87] J. F. Jardine, Simplicial presheaves , J. Pure Appl. Algebra (1987), no. 1, 35–87. MR 906403[Kat70] Nicholas M. Katz, Nilpotent connections and the monodromy theorem: Applications of a result ofTurrittin , Inst. Hautes Études Sci. Publ. Math. (1970), no. 39, 175–232. MR 291177[Kat87] Kazuya Kato, On p -adic vanishing cycles (application of ideas of Fontaine-Messing) , Algebraic geom-etry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 207–251.MR 946241[KL15] Kiran S. Kedlaya and Ruochuan Liu, Relative p -adic Hodge theory: foundations , Astérisque (2015),no. 371, 239. MR 3379653[KM18] Shane Kelly and Matthew Morrow, K -theory of valuation rings , arXiv preprint arXiv:1810.12203 (2018).[Kra80] Ch. Kratzer, λ -structure en K -théorie algébrique , Comment. Math. Helv. (1980), no. 2, 233–254.MR 576604[KST18] Moritz Kerz, Florian Strunk, and Georg Tamme, Towards Vorst’s conjecture in positive characteristic .[Kun69] Ernst Kunz, Characterizations of regular local rings of characteristic p , Amer. J. Math. (1969),772–784. MR 252389[Lev06] Marc Levine, Chow’s moving lemma and the homotopy coniveau tower , K -Theory (2006), no. 1-2,129–209. MR 2274672[Lev08] , The homotopy coniveau tower , J. Topol. (2008), no. 1, 217–267. MR 2365658[Lin80] Wen Hsiung Lin, On conjectures of Mahowald, Segal and Sullivan , Math. Proc. Cambridge Philos. Soc. (1980), no. 3, 449–458. MR 556925[LM08] Ayelet Lindenstrauss and Randy McCarthy, The algebraic K -theory of extensions of a ring by directsums of itself , Indiana Univ. Math. J. (2008), no. 2, 577–625. MR 2414329[LMMT20] Markus Land, Akhil Mathew, Lennart Meier, and Georg Tamme, Purity in chromatically localizedalgebraic K -theory , arXiv preprint arXiv:2001.10425 (2020).[Lod98] Jean-Louis Loday, Cyclic homology , second ed., Grundlehren der Mathematischen Wissenschaften [Fun-damental Principles of Mathematical Sciences], vol. 301, Springer-Verlag, Berlin, 1998, Appendix E byMaría O. Ronco, Chapter 13 by the author in collaboration with Teimuraz Pirashvili. MR 1600246[LT19] Markus Land and Georg Tamme, On the K -theory of pullbacks , Ann. of Math. (2) (2019), no. 3,877–930. MR 4024564 [Lur09] Jacob Lurie, Higher topos theory , Annals of Mathematics Studies, vol. 170, Princeton University Press,Princeton, NJ, 2009. MR 2522659[Lur18] Jacob Lurie, Spectral algebraic geometry , 2018.[LW20] Ruochuan Liu and Guozhen Wang, Topological cyclic homology of local fields , arXiv preprintarXiv:2012.15014 (2020).[Mad94] Ib Madsen, Algebraic K -theory and traces , Current developments in mathematics, 1995 (Cambridge,MA), Int. Press, Cambridge, MA, 1994, pp. 191–321. MR 1474979[Mat20] Akhil Mathew, On K (1) -local TR , arXiv preprint arXiv:2005.08744 (2020).[McC97] Randy McCarthy, Relative algebraic K -theory and topological cyclic homology , Acta Math. (1997),no. 2, 197–222. MR 1607555[MRT19] Tasos Moulinos, Marco Robalo, and Bertrand Toën, A universal HKR theorem , arXiv preprintarXiv:1906.00118 (2019).[MSV97] J. McClure, R. Schwänzl, and R. Vogt, T HH ( R ) ∼ = R ⊗ S for E ∞ ring spectra , J. Pure Appl. Algebra (1997), no. 2, 137–159. MR 1473888[NS18] Thomas Nikolaus and Peter Scholze, On topological cyclic homology , Acta Math. (2018), no. 2,203–409. MR 3904731[Qui73] Daniel Quillen, Higher algebraic K -theory. I , Algebraic K -theory, I: Higher K -theories (Proc. Conf.,Battelle Memorial Inst., Seattle, Wash., 1972), 1973, pp. 85–147. Lecture Notes in Math., Vol. 341.MR 0338129[Rak20] Arpon Raksit, Hochschild homology and the derived de Rham complex revisited , arXiv preprintarXiv:2007.02576 (2020).[Ras18] Sam Raskin, On the dundas-goodwillie-mccarthy theorem , arXiv preprint arXiv:1807.06709 (2018).[Rig20] Noah Riggenbach, On the algebraic K -theory of double points , arXiv preprint arXiv:2007.01227 (2020).[RS10] Andreas Rosenschon and V. Srinivas, The Griffiths group of the generic abelian 3-fold , Cycles, motivesand Shimura varieties, Tata Inst. Fund. Res. Stud. Math., vol. 21, Tata Inst. Fund. Res., Mumbai, 2010,pp. 449–467. MR 2906032[Sch02] Chad Schoen, Complex varieties for which the Chow group mod n is not finite , J. Algebraic Geom. (2002), no. 1, 41–100. MR 1865914[Sch12] Peter Scholze, Perfectoid spaces , Publ. Math. Inst. Hautes Études Sci. (2012), 245–313. MR 3090258[Spe20] Martin Speirs, On the K -theory of truncated polynomial algebras, revisited , Adv. Math. (2020),107083, 18. MR 4070307[Sta20] The Stacks project authors, The Stacks Project , https://stacks.math.columbia.edu , 2020.[SZ18] Tamás Szamuely and Gergely Zábrádi, The p -adic Hodge decomposition according to Beilinson , Alge-braic geometry: Salt Lake City 2015, Proc. Sympos. Pure Math., vol. 97, Amer. Math. Soc., Providence,RI, 2018, pp. 495–572. MR 3821183[Tot16] Burt Totaro, Complex varieties with infinite Chow groups modulo 2 , Ann. of Math. (2) (2016),no. 1, 363–375. MR 3432586[Tsa98] Stavros Tsalidis, Topological Hochschild homology and the homotopy descent problem , Topology (1998), no. 4, 913–934. MR 1607764[TT90] R. W. Thomason and Thomas Trobaugh, Higher algebraic K -theory of schemes and of derived cate-gories , The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA,1990, pp. 247–435. MR 1106918[Voe03] Vladimir Voevodsky, Motivic cohomology with Z / -coefficients , Publ. Math. Inst. Hautes Études Sci.(2003), no. 98, 59–104. MR 2031199[Voe11] , On motivic cohomology with Z /l -coefficients , Ann. of Math. (2) (2011), no. 1, 401–438.MR 2811603[Yek18] Amnon Yekutieli, Flatness and completion revisited , Algebr. Represent. Theory21