aa r X i v : . [ m a t h . K T ] M a y ON K (1) -LOCAL TR AKHIL MATHEW
Abstract.
We discuss some general properties of TR and its K (1) -localization. We prove thatafter K (1) -localization, TR of H Z -algebras is a truncating invariant in the sense of Land–Tamme,and deduce h -descent results. We show that for regular rings in mixed characteristic, TR isasymptotically K (1) -local, extending results of Hesselholt–Madsen. As an application of thesemethods and recent advances in the theory of cyclotomic spectra, we construct an analog ofThomason’s spectral sequence relating K (1) -local K -theory and étale cohomology for K (1) -local TR . Introduction
The topological cyclic homology,
TC( R ) , of a ring (or ring spectrum) R is a basic invariant in-troduced by Bökstedt–Hsiang–Madsen [BHM93] (see also Dundas–Goodwillie–McCarthy [DGM13]and Nikolaus–Scholze [NS18]) with many applications in algebraic K -theory. Its p -adic completion TC( R ; Z p ) arises as the fixed points of an operator called Frobenius on another invariant TR( R ; Z p ) ,which plays a central role in the approach to TC via equivariant stable homotopy theory. The con-struction TR( − ; Z p ) is often of arithmetic significance; for instance, the foundational calculations[HM03, HM04, GH06] of the p -adic K -theory of local fields F are based on the relationship between TR( O F ; Z p ) and the de Rham–Witt complex with log poles of O F .In this paper, we prove some structural results about TR and how it relates to its K (1) -localization, throughout at an implicit prime number p . We will only consider p -typical TR andafter p -adic completion. The operation of K (1) -localization when applied to algebraic K -theory isdramatically simplifying, as shown by Thomason [Tho85, TT90]; in particular, K (1) -local K -theorysatisfies étale descent and admits a descent spectral sequence from étale cohomology under mildhypotheses, cf. [CM19] for a modern account. Here we study analogs of some of these propertiesfor L K (1) TR( − ) . Our starting point is the following. Theorem 1.1.
As a functor on connective H Z -algebras, L K (1) TR( − ) is a truncating invariantin the sense of Land–Tamme [LT19] . In other words, if R is a connective H Z -algebra, then L K (1) TR( R ) ∼ −→ L K (1) TR( π R ) . Theorem 1.1 refines results of [LMT20, BCM20]. In loc. cit., it is shown that L K (1) K ( − ) and(equivalently) L K (1) TC( − ) are truncating invariants of connective H Z -algebras, which ultimatelyfollows from the claim(1) L K (1) K ( Z /p n ) = 0 , n ≥ , and more generally (and consequently) for any H Z -algebra R ,(2) L K (1) K ( R ) ∼ −→ L K (1) K ( R [1 /p ]) . Date : May 28, 2020.
In [BCM20], (1) is proved via a calculation in prismatic cohomology; in [LMT20], (1) is provedusing some unstable chromatic homotopy theory. Our proof of Theorem 1.1 (which also gives a newproof of (1)) is based on a direct TC -theoretic argument via estimation of exponents of nilpotenceof the Bott element; in fact, it yields a slightly stronger result (Theorem 3.1 below).The property of being truncating yields many pleasant features of the construction L K (1) TR( − ) :by [LT19], one obtains cdh -descent and excision. Since we are working K (1) -locally, we can combinethis with results of [CMNN20] to obtain h -descent: Theorem 1.2.
Any K (1) -local localizing invariant which is truncating, such as L K (1) TR( − ) , sat-isfies h -descent on qcqs schemes. In particular, L K (1) TR( − ) satisfies étale descent. This is not so surprising, since TR( − ; Z p ) itself(like all Hochschild-theoretic invariants) actually satisfies flat descent, cf. [BMS19, Sec. 3]. However,Theorem 1.2 (together with (1)) leads to étale descent in the generic fiber. Since TR( R ; Z p ) of aring R depends only on the (derived) p -adic completion of R , we can informally view L K (1) TR( R ) as an invariant of the “rigid space” associated to R [1 /p ] . Example 1.3 (Galois descent in the generic fiber) . Let R → S be a finite, finitely presented mapof rings. Suppose we have a finite group G acting on S such that R [1 /p ] → S [1 /p ] is G -Galois.Then for any K (1) -local localizing invariant E which is truncating, we have E ( R ) ∼ −→ E ( S ) hG . Recall that the Lichtenbaum–Quillen conjecture, refined by the Beilinson–Lichtenbaum conjec-ture proved by Voevodsky–Rost (cf. [HW19] for an account), predicts that for Z [1 /p ] -algebras A satisfying mild finiteness conditions, the p -adic K -theory K ( A ; Z p ) is “asymptotically K (1) -local:”that is, the map K ( A ; Z p ) → L K (1) K ( A ; Z p ) is an equivalence in high enough degrees. We nextdiscuss analogs of such statements for TR( R ; Z p ) for p -adic rings R . Indeed, in [HM03, HM04], itis shown that if R is smooth of relative dimension d over a discrete valuation ring O K of mixedcharacteristic with perfect residue field of characteristic p > , then TR( R ; F p ) → L K (1) TR( R ; F p ) is d -truncated; more precisely, this is a consequence of the relationship shown in loc. cit. with theabsolute de Rham–Witt complex. We prove this asymptotic K (1) -locality more generally for regu-lar rings satisfying F -finiteness hypotheses from the Beilinson–Lichtenbaum conjecture applied tothe generic fiber as well as the connection between TR and p -typical curves [Hes96]. We expectthat there should be a purely p -adic proof of this result (as in [HM03, HM04] in the smooth case). Theorem 1.4.
Let R be a p -torsionfree excellent regular noetherian ring. Suppose that R/p isfinitely generated as a module over its subring of p th powers. Suppose furthermore that for all p ∈ Spec( R ) containing ( p ) , we have dim R p + log p [ κ ( p ) : κ ( p ) p ] ≤ d for some d ≥ . Then the map TR( R ; F p ) → L K (1) TR( R ; F p ) is ( d − -truncated. Using the theory of topological Cartier modules of Antieau–Nikolaus [AN18], we relate the prop-erty of
TR( R ; F p ) being “asymptotically K (1) -local” to the extensively studied Segal conjecturefor THH( R ) , i.e., the condition that the cyclotomic Frobenius ϕ : THH( R ; F p ) → THH( R ; F p ) tC p should be an equivalence in high degrees. In particular, we obtain a version of the Segal conjecturefor THH( R ) when R is regular. We expect that there should be a filtered version of this statement,using the motivic filtrations of Bhatt–Morrow–Scholze [BMS19]. Corollary 1.5.
Let R be as in Theorem 1.4. Then the cyclotomic Frobenius ϕ : THH( R ; F p ) → THH( R ; F p ) tC p is ( d − -truncated. Finally, we study the analog of Thomason’s spectral sequence [Tho85, TT90] from étale coho-mology to K (1) -local algebraic K -theory. For a scheme X over Z [1 /p ] satisfying mild finiteness N K (1) -LOCAL TR conditions (to wit: X should be qcqs of finite Krull dimension, with a uniform bound on the mod p virtual cohomological dimensions of the residue fields, [RØ06, CM19]), one has a convergent spectralsequence E s,t = H s ( X, Z p ( t )) = ⇒ π t − s L K (1) K ( X ) . We can construct a similar spectral sequence for L K (1) TR under significantly stronger finitenessand regularity conditions, arising from a natural filtration. To formulate the E -term (or the gradedpieces of this filtration), we use the arc p -topology of [BM18]. Definition 1.6 (The arc p -topology and arc p -cohomology) . We say that a map of derived p -completerings R → R ′ is an arc p -cover if every map R → V for V a rank valuation ring which is p -completeand such that p = 0 , there exists an extension of rank valuation rings V → W and a commutativediagram R (cid:15) (cid:15) / / R ′ (cid:15) (cid:15) V / / W .
The arc p -topology is the finitary Grothendieck topology on the opposite of the category of derived p -complete rings defined such that a family { R → R ′ α } α ∈ A is a covering family if and only if thereexists a finite subset A ′ ⊂ A such that R → Q α ∈ A ′ R ′ α is an arc p -cover.Given any functor F from derived p -complete rings to abelian groups, we let R Γ arc p (Spec( R ) , F ( − )) denote the arc p -cohomology of F ( − ) on a derived p -complete ring R . Example 1.7. (1) We can consider the arc p -cohomology of the structure presheaf O , R Γ arc p (Spec( R ) , O ) .This is closely related to the perfectoidizations considered in [BS19, Sec. 7–8], which workwith the p -complete arc -topology rather than the arc -topology. For R = Z p , it is not diffi-cult to see that this is the continuous Gal( Q p ) -homotopy invariants of the derived saturation ( O C ) ∗ for C = c Q p (in the sense of almost ring theory [GR03]).(2) We consider the arc p -cohomology of the Witt vector presheaf W ( O ) , denoted R R Γ arc p (Spec( R ) , W ( O )) ,as well as its p -adic Tate twists W ( O )( i ) for i ∈ Z . Theorem 1.8.
Let K be a complete nonarchimedean field of mixed characteristic (0 , p ) with ringof integers O K and residue field k with [ k : k p ] < ∞ . Suppose either K is discretely valued or K is perfectoid. Let R be a formally smooth O K -algebra. Then there exists a natural convergent,exhaustive Z -indexed descending filtration Fil ≥∗ L K (1) TR( R ) on L K (1) TR( R ) such that (3) gr i L K (1) TR( R ) ≃ R Γ arc p (Spec( R ) , W ( O )( i ))[2 i ] . Theorem 1.8 is effectively an étale hyperdescent (in the generic fiber) result together with thecalculation for perfectoids. For illustration, we specialize to the case where R = O K , for K discretelyvalued with perfect residue field. Then the filtration (3) arises via a type of pro-Galois descent inthe generic fiber. If L/K is G -Galois, Example 1.3 and Theorem 1.4 imply that TR( O K ; F p ) → TR( O L ; F p ) hG is a -truncated map. However, this does not help with passage to K since TR doesnot commute with filtered colimits; note that TR has a simple form for O K , cf. [Hes06]. Usingagain the theory of topological Cartier modules as a “decompletion” of the theory of cyclotomicspectra, cf. [AN18], we prove the following pro-Galois result (cf. Example 6.9): For set-theoretic reasons, to define arc p -cohomology we should fix a cutoff cardinal. We will only considersituations where the choice of cutoff cardinal does not affect the result. AKHIL MATHEW
Theorem 1.9.
Let K be a complete, discretely valued field of characteristic zero with perfect residuefield k of odd characteristic p . Let TR( O K | K ) denote the cofiber of the transfer map TR( k ) → TR( O K ) . Then the natural map induces an equivalence TR( O K | K ; F p ) → τ ≥ Tot (cid:16)
TR( O K ; F p ) ⇒ TR( O K ⊗ K K ; F p ) →→→ . . . (cid:17) . The idea that TR should satisfy this type of pro-Galois descent in the generic fiber is expressedin [Hes02]; in particular Conjecture 5.1 of loc. cit. predicts a related (but stronger) conclusion atthe level of homotopy groups (in particular, the vanishing of higher Galois cohomology groups inthe associated descent spectral sequence with Q p / Z p coefficients). Conventions.
We write Sp for the ∞ -category of spectra and S for the sphere spectrum. We usethe theory of cyclotomic spectra in the form developed in [NS18], as well as the theory of topologicalCartier modules developed in [AN18]; we write CycSp for the ∞ -category of cyclotomic spectra.Given an E ∞ -ring B , a homogeneous element x ∈ π ∗ ( B ) , and a B -module M , we often write M/x for the cofiber of multiplication by x on M . In the case x = p , we will often write this by ; F p ,e.g., THH( B ; F p ) refers to the cofiber of p on THH( B ) . A B -algebra always refers, unless otherwisespecified, to an E -algebra in B -modules. Acknowledgments.
I would like to thank Benjamin Antieau, Lars Hesselholt, Matthew Morrow,Thomas Nikolaus, Wiesława Nizioł, and Peter Scholze for many helpful conversations related to thesubject of this paper. I would especially like to thank Bhargav Bhatt and Dustin Clausen for theircollaboration in [BCM20] and many helpful discussions. Finally, I would like to thank BenjaminAntieau, Lennart Meier, and Georg Tamme for several comments and corrections on a draft. Thiswork was done while the author was a Clay Research Fellow.2.
Generalities on K (1) -local truncating invariants Let B be a base connective E ∞ -ring. In this section, we work with localizing invariants onsmall B -linear idempotent-complete stable ∞ -categories. Unlike in [BGT13], we do not assumecompatibility with filtered colimits, so for us a localizing invariant is simply a functor from (small,idempotent-complete) B -linear stable ∞ -categories to spectra which carries Verdier quotient se-quences to cofiber sequences. Following [LT19], we say that such a localizing invariant E is trun-cating if for every connective B -algebra A , we have E ( A ) ∼ −→ E ( Hπ A ) . This implies [LT19, Th. B]that E satisfies excision. Example 2.1.
The constructions L K (1) K ( − ) , L K (1) TC( − ) are truncating on connective H Z -algebras, as verified in [LMT20]. For commutative p -complete rings, the two invariants actuallyagree (we do not know if this is true for noncommutative p -complete rings, cf. [BCM20, Question2.20]). Below we will show that L K (1) TR( − ) is truncating.In the rest of the section, we will assume for simplicity of notation that B is discrete; by theassumption of truncatedness, this does not affect any of the results. Proposition 2.2.
Let E be a K (1) -local localizing invariant of B -linear ∞ -categories which istruncating. Then, on the category of discrete B -algebras:(1) E is nilinvariant.(2) E annihilates any B -algebra C which is annihilated by a power of p .(3) Let A → A ′ be a map of B -algebras which is a p -isogeny. Then E ( A ) → E ( A ′ ) is anequivalence. N K (1) -LOCAL TR Proof.
For (1), the fact that E is nilinvariant follows from [LT19, Th. B]. For (2), since E isnilinvariant, we may assume C is an F p -algebra, so that E ( C ) is a K ( F p ; Z p ) = H Z p -module (thelast identification by [Qui72]); since E is K (1) -local we get E ( C ) = 0 .For (3), the kernel of A → A ′ is annihilated by a power of p , so by (2) (and excision) we canassume that A ⊂ A ′ . Let n ≫ , so p n A ′ ⊂ A . Then the diagram A (cid:15) (cid:15) / / A ′ (cid:15) (cid:15) A/ ( p n A ′ ∩ A ) / / A ′ /p n A ′ is a Milnor square of rings. Applying the localizing invariant E and using excision and (2) again,we conclude (3). (cid:3) In the next result, we use Voevodsky’s h -topology for possibly non-noetherian schemes; in otherwords, the topology generated by finitely presented v -covers (also called universally subtrusivemorphisms), cf. [Ryd10] or [BS17, Sec. 2]. Theorem 2.3 ( h -descent for truncating K (1) -local invariants) . Let E be a K (1) -local localizing in-variant on B -linear ∞ -categories which is truncating. Then E satisfies h -descent on quasi-compactand quasi-separated (qcqs) B -schemes.Proof. By the results of [LT19, App. A], E satisfies cdh -descent, and in particular satisfies excisionfor abstract blow-up squares. The results of [CMNN20] imply that E satisfies finite locally freedescent. Since E also satisfies Nisnevich descent (as does any localizing invariant [TT90]), weobtain that E satisfies fppf descent thanks to [Sta20, Tag 05WN]. By [BS17, Th. 2.9], h -descent(for any sheaf) is implied by fppf descent and excision for abstract blow-up squares. Combiningthese facts, we conclude. (cid:3) Example 2.4.
Let E be as above. Let π : X ′ → X be a finitely presented proper morphism(e.g., a finitely presented closed immersion) of qcqs B -schemes such that X ′ [1 /p ] ∼ −→ X [1 /p ] . Then E ( X ) ∼ −→ E ( X ′ ) . In fact, by cdh descent, we have a pullback square E ( X ) (cid:15) (cid:15) / / E ( X ′ ) (cid:15) (cid:15) E ( X ⊗ F p ) / / E ( X ′ ⊗ F p ) , and the terms on the bottom vanish by Proposition 2.2.In the next result, we use the notion of nilpotence of a group action, cf. [Mat18, Sec. 4.1] or[CM19, Def. 2.17] for accounts, or [MNN17] for the general setup in equivariant stable homotopytheory. Let G be a finite group. The collection of nilpotent objects of the ∞ -category Fun(
BG,
Sp) is the thick subcategory generated by the objects which are induced from the trivial subgroup. Foran algebra object of
Fun(
BG,
Sp) , nilpotence holds if and only if the Tate construction vanishes. Amodule over a nilpotent algebra object in
Fun(
BG,
Sp) is itself nilpotent.
Corollary 2.5 (Galois descent in the generic fiber) . Let E be a K (1) -local localizing invariant of B -linear ∞ -categories which is truncating. Let R → S be a finite and finitely presented map of B -algebras. Let G be a finite group acting on S via R -algebra maps. Suppose that R [1 /p ] → S [1 /p ] is AKHIL MATHEW G -Galois. Then the natural map induces an equivalence E ( R ) ∼ −→ E ( S ) hG . Moreover, the G -actionon E ( S ) is nilpotent.Proof. Replacing S with S × R/p , we may assume without loss of generality that R → S is an h -cover. Then, by Theorem 2.3, we have(4) E ( R ) ≃ Tot( E ( C ( R → S ) • )) = Tot( E ( S ) ⇒ E ( S ⊗ R S ) →→→ . . . ) . Since the group G acts on S , we have a natural map of cosimplicial rings from the Čech nerve C ( R → S ) • to the standard resolution S ⇒ Q G S →→→ for G acting on S (which calculates S hG ). Thismap of cosimplicial rings is an isogeny in each degree: for example, in degree , S ⊗ R S → Q G S isan isogeny because it is a map of finitely presented R -modules (cf. [Sta20, Tag 0564]) which inducesan isomorphism after inverting p thanks to the Galois hypothesis. Therefore, by Proposition 2.2,we find that the map induces an equivalence after applying E , and we find from (4), E ( R ) ∼ −→ Tot( E ( S ) ⇒ E ( Y G S ) →→→ . . . ) = E ( S ) hG , which is the desired claim. Finally, to see that the G -action on E ( S ) is nilpotent, we use that E ( S ) is a module G -equivariantly over L K (1) K ( S ) ∼ −→ L K (1) K ( S [1 /p ]) (via (2)), and the G -action on L K (1) K ( S [1 /p ]) is nilpotent by [CMNN20, Th. 5.6] (cf. also [CM19, Lem. 4.20]). (cid:3) The truncating property of L K (1) TR( − ) In this section, we prove the following basic result. Throughout, we fix a connective, K (1) -acyclic E ∞ -ring B , e.g., H Z . Theorem 3.1.
The construction L K (1) TR( − ) is truncating on connective E - B -algebras. Moregenerally, for any set Q , the construction L K (1) ( Q Q TR( − )) is truncating on connective E - B -algebras. The proof of Theorem 3.1 will rely on a K (1) -acyclicity criterion for cyclotomic spectra (Propo-sition 3.6), which will use some elementary estimates for exponents of nilpotence with respect tothe Bott element β .In the following, we let ku denote the connective topological K -theory spectrum, so π ∗ ( ku ) = Z [ β ] with | β | = 2 . Since B is K (1) -acyclic, the associative ring spectrum B ⊗ ku/p is annihilated by apower of β . Our strategy is roughly based on bounding the exponents of nilpotence for β in thefixed points THH( B ) C pn ⊗ ku/p and in particular showing that they are O ( p n ) .Recall that ku is complex-oriented, leading to the following result. Proposition 3.2.
Let M be a ku -module equipped with an S -action. Then for each n ≥ , thenatural map induces an equivalence (5) M hS ⊗ ku BS ku BC pn ∼ −→ M hC pn . Proof.
Compare [MNN17, Sec. 7.4]. In fact, via the projection formula, M hC pn ≃ ( M ⊗ ku ku S /C pn ) hS .Here ku S /C pn denotes the ku -valued function spectrum of S /C p n with the corresponding S -action, and the tensor product is taken in Fun( BS , Mod( ku )) . Let V n denote the one-dimensionalcomplex representation of S where z ∈ S acts by multiplication by z p n , and let S ( V n ) denote This states informally that if A is a connective B -algebra, then the fiber of the map TR( A ; F p ) → TR( Hπ ( A ); F p ) has the property that each degree is annihilated by a power of v (depending on the degree; note that this conditionis slightly stronger than the fiber simply being K (1) -acyclic). N K (1) -LOCAL TR the unit circle in V n as an S -space, so S ( V n ) ≃ S /C p n . The Spanier–Whitehead dual of theEuler sequence in Fun( BS , Sp) , S ( V n ) + → S → S V n and the complex-orientability of ku togethershow that ku S /C pn ∈ Fun( BS , Mod( ku )) belongs to the thick subcategory generated by the unit.Therefore, applying the right adjoint ( − ) hS : Fun( BS , Mod( ku )) → Mod( ku BS ) , we find thatthe natural map M hS ⊗ ku BS ( ku S /C pn ) hS → ( M ⊗ ku ku S /C pn ) hS = M hC pn is an equivalence, whence the result. (cid:3) By complex orientability, we have π ∗ ( ku BS ) = Z [ β ][[ x ]] , | β | = 2 , | x | = − . Consider the formal group law over Z [ β ] given by f ( u, v ) = u + v + βuv ; this is the formal grouplaw associated to the complex oriented ring spectrum ku , and is homogeneous of degree − if | u | , | v | = − . We have an equivalence ku hS /x ≃ ku . More generally, let [ p n ]( x ) ∈ π ∗ ( ku BS ) denote the p -series of the formal group law f ; modulo p , we have [ p n ]( x ) ≡ β p n − x p n since [ p n ]( x ) ishomogeneous of degree − and recovers the multiplicative formal group law under the specialization β . By the Eilenberg–Moore spectral sequence or Gysin sequence (of which (5) is a form), wehave(6) ku BC pn = ku BS / ([ p n ]( x )) . Proposition 3.3.
Let A be a connective E ∞ -ring spectrum with S -action. Suppose that β r = 0 in π ∗ ( A ⊗ ku/p ) . Then in π ∗ (( A ⊗ ku ) hC pn /p ) , we have β p n − rp n = 0 .Proof. Let R = A ⊗ ku/p , so R is an associative ku -algebra spectrum with S -action. Considerthe S -homotopy fixed points R hS , which is a ku BS -algebra. Since p = 0 in π R , we have byProposition 3.2 and (6), R hC pn = R hS ⊗ ku BS ku BC pn = R hS / [ p n ]( x ) = R hS / ( β p n − x p n ) . Our assumption is that β r = 0 in π ∗ ( R ) , which means that we can write β r = xv ∈ π ∗ ( R hS ) forsome v since R = R hS /x . It follows that in π ∗ ( R hC pn ) , we have β p n − rp n = β p n − ( xv ) p n = 0 . (cid:3) Now we apply the above to the E ∞ -ring THH( B ) equipped with its S -action; recall that B isassumed connective and K (1) -acyclic. Proposition 3.4.
There exists a constant κ > such that for each i > :(1) We have that β κi = 0 in ( ku/p ⊗ THH( B )) hC i .(2) Let N be a THH( B ) ⊗ ku -module in Fun( BS , Sp) . Suppose that N (as a THH( B ) ⊗ ku -module in Fun( BS , Sp) ) is induced from the cyclic group C i ⊂ S . Then for any t , N hC t /p is annihilated by β κi .Proof. To prove (1), using the transfer and restricting to a p -Sylow subgroup, we may reduce tothe case where i is a power of p , say i = p n . Then the claim follows from Proposition 3.3, since β is nilpotent in ku/p ⊗ THH( B ) since B is K (1) -acyclic. It follows that we can find a κ such that β κ i = 0 in ( ku/p ⊗ THH( B )) hC i for all i > .For (2), consider a THH( B ) ⊗ ku -module N in Fun( BS , Sp) which is the induction of a
THH( B ) ⊗ ku -module N ′ in Fun( BC i , Sp) . It follows that N hS /p = ( N ′ /p ) hC i is a module over ( ku/p ⊗ THH( B )) hC i , writing homotopy orbits as a module over homotopy fixed points; this is therefore AKHIL MATHEW annihilated by β κ i by the previous part of the result. For any t , let W t be the one-dimensionalcomplex representation of S where z acts by multiplication by z t , so the unit circle S ( W t ) + =( S /C t ) + as S -spaces. We have N hC t = ( N ⊗ S ( W t ) + ) hS by the projection formula for inductionand restriction along C t ⊂ S . Then the Euler sequence S ( W t ) + → S → S W t (as in the proofof Proposition 3.2) and the complex-orientability of ku yields a fiber sequence N hC t → N hS → Σ N hS . Therefore, N hC t /p is annihilated by β κ i ; taking κ = 2 κ we conclude. (cid:3) For the next result, we use the notion of a (nonnegatively) graded cyclotomic spectrum , cf. [AMMN20,Sec. 3 and App. A] or [Bru01]. A graded cyclotomic spectrum consists of a graded spectrum X = L i ≥ X i equipped with an S -action together with a graded S -equivariant cyclotomic Frobe-nius ϕ i : X i → X tC p pi for each i . Given a nonnegatively graded E -ring R , the topological Hochschildhomology THH( R ) acquires the structure of a graded cyclotomic spectrum. Given a graded cyclo-tomic spectrum X , we can consider a graded cyclotomic spectrum X ≥ i where we only consider thegraded summands in degrees ≥ i ; this gives any graded cyclotomic spectrum a natural descendingfiltration. The filtration quotients X ≥ i /X ≥ pi have trivialized Frobenius because of the grading, andtheir TR can be thus described explicitly: Construction 3.5 ( TR of cyclotomic spectra with zero Frobenius) . Suppose X ∈ CycSp ≥ withtrivialized Frobenius. Then, as in [AN18, Rem. 2.5] and [NS18, Cor. II.4.7], we obtain a naturalequivalence(7) TR( X ) ≃ Y i ≥ X hC pi . Proposition 3.6 ( K (1) -acyclicity criterion) . Let X be a positively graded cyclotomic spectrum withthe structure of THH( B ) -module. Suppose for each i > :(1) X i is (2 p + 1) κi -connected (where κ is as in Proposition 3.4).(2) As a THH( B ) -module in Fun( BS , Sp) , X i is induced from the cyclic group C i ⊂ S .Then for any set Q , Q Q TR( X ) is K (1) -acyclic.Proof. For simplicity of notation, we write TR Q ( − ) = Q Q TR( − ) . The construction TR Q ( − ) isexact and commutes with geometric realizations on CycSp ≥ ; therefore, it commutes with tensoringwith ku . Without loss of generality, we can therefore assume that X is a THH( B ) ⊗ ku -module ingraded cyclotomic spectra. Here we regard ku as a trivial cyclotomic spectrum, i.e., via the imageof the unique symmetric monoidal functor Sp → CycSp .We consider the descending filtration { Fil ≥ n X = X ≥ p n } n ≥ on the cyclotomic spectrum X ; notethat the associated graded terms have trivialized Frobenii for degree reasons. This yields a filtrationon the spectrum TR Q ( X ) , with Fil ≥ n TR Q ( X ) = TR Q ( X ≥ p n ) . Since TR( − ) preserves connectivity,our assumptions imply that Fil ≥ n TR Q ( X ) is (2 p + 1) κp n -connective.We have by (7), gr n TR Q ( X ) = L p n ≤ i
Let R be a connective E -ring spectrum, and let x ∈ π t ( R ) be an element. Let n Fil ≥ n Y o n ≥ be a filtered R -module spectrum. Suppose that there exist functions f, g : N → N suchthat:(1) gr n ( Y ) is annihilated by x f ( n ) .(2) Fil ≥ n ( Y ) is g ( n ) -connective.(3) g ( n ) − t P n − i =0 f ( i ) → ∞ for n → ∞ .Then Y [1 /x ] = 0 .Proof. Let y ∈ π s ( Y ) . For each n > , then the class x f (0)+ f (1)+ ··· + f ( n − y ∈ π ∗ ( Y ) naturally liftsto π s + t ( f (0)+ ··· + f ( n − (Fil ≥ n Y ) . But for n ≫ , the connectivity of Fil ≥ n Y forces this last groupto vanish. Therefore, the image of y in π s ( Y [1 /x ]) vanishes as desired. (cid:3) We now use the following basic calculation of
THH of a free associative algebra, as a spectrumequipped with S -action. Versions of this are classical in ordinary Hochschild homology, cf. [Lod98,Sec. 3.1]. In the language of factorization homology, this result is a special case of the calculationof the factorization homology of a free algebra, [AFT17, Prop. 4.13] and [AF15, Prop. 5.5]. Theorem 3.8 ( THH of a free associative algebra) . Let M be a spectrum, and let T ( M ) = L n ≥ M ⊗ n be the free E -algebra spectrum generated by M . Then there is a natural equivalencein Fun( BS , Sp) , (8) THH( T ( M )) ≃ M n ≥ Ind S C n ( M ⊗ n ) , where we use the natural C n -action on M ⊗ n by permuting the factors.Proof. The results of loc. cit. (applied to the framed manifold S ) imply that THH( T ( M )) ≃ R S T ( M ) = L n ≥ (Conf n ( S ) + ⊗ M ⊗ n ) h Σ n , for Conf n ( S ) the configuration space of n orderedpoints on the circle. One checks now (cf. [CJ98, Ex. II.14.4]) that Conf n ( S ) , as a space with S × Σ n -action, is homotopy equivalent to ( S × Σ n ) /C n (with C n embedded diagonally), whencethe claim. (cid:3) Proposition 3.9.
Let
M, N be connective spectra. Then the map of cyclotomic spectra
THH( T ( M ⊕ N )) ⊗ THH( B ) → THH( M ) ⊗ THH( B ) induces an equivalence on L K (1) ( Q Q TR( − )) for any set Q if N is ≥ κ (2 p + 1) -connective.Proof. We can consider the tensor algebra T ( M ⊕ N ) as a graded E -ring spectrum where M isplaced in degree zero and N is placed in degree . In this case, if we collect the terms in (8), we findthat the i th graded piece of THH( T ( M ⊕ N )) is the component which is i -homogeneous. Explicitly,for any subset I ⊂ h n i = { , , . . . , n } , we write ( M, N ) ⊗ ( h n i\ I,I ) for the ordered tensor product of n factors, where the j th factor is M if j / ∈ I and N if j ∈ I . Expanding (8) gives(9) THH( T ( M ⊕ N )) i = M n ≥ Ind S C n M I ⊂h n i , | I | = i ( M, N ) ⊗ ( h n i\ I,I ) . Here the C n -action on the parenthesized term in (9) permutes the various summands. Note inparticular that the stabilizer of I ⊂ h n i in the n th summand is a subgroup of a cyclic group C i ⊂ C n since | I | = i . In particular, it follows that the i th graded piece of THH( T ( M ⊕ N )) is induced from C i ⊂ S . Furthermore, since N is ≥ κ (2 p + 1) -connective, it follows that the i thgraded piece of THH( T ( M ⊕ N )) is ≥ κ (2 p + 1) i -connective. By Proposition 3.6, it follows thatthe positively graded part of THH( T ( M ⊕ N )) ⊗ THH( B ) has vanishing K (1) -local Q Q TR( − ) forany set Q , whence the result. (cid:3) For the next result, if R is an E -ring spectrum and N an ( R, R ) -bimodule, we let T R ( N ) = L n ≥ N ⊗ R N ⊗ R · · · ⊗ R N be the free E -algebra under R generated by N . Proposition 3.10.
Let R be a connective B -algebra. Let N be a ≥ κ (2 p + 1) -connective ( R, R ) -bimodule. Then the map THH( T R ( N )) → THH( R ) induces an equivalence on L K (1) ( Q Q TR( − )) for any set Q .Proof. Simplicially resolving N by free ( R, R ) -bimodules (since Q Q TR( − ) commutes with geomet-ric realizations), we may assume that N is free on generators (possibly infinitely many) in degrees ≥ κ (2 p + 1) . Simplicially resolving R by free B -algebras, we may assume that R is free as well, onsome classes in degrees ≥ . In this case, the result follows from Proposition 3.9. (cid:3) Lemma 3.11.
Let E be a localizing invariant of B -linear ∞ -categories. Suppose that there exists k ≥ such that for every connective B -algebra R , we have E ( R ) ∼ −→ E ( τ ≤ k R ) . Then E is truncating.Proof. We show by descending induction that for any i ≥ and for any connective B -algebra R ,the map R → τ ≤ i R induces an equivalence on E ; taking i = 0 gives the theorem. For i ≥ k , wealready know the claim by assumption. Suppose we know the claim for i + 1 ; to prove the claimfor i , we need to see that τ ≤ i +1 R → τ ≤ i R induces an equivalence on E . Now τ ≤ i +1 R → τ ≤ i R is asquare-zero extension, i.e., we have a pullback diagram of B -algebras τ ≤ i +1 R (cid:15) (cid:15) / / H ( π R ) (cid:15) (cid:15) τ ≤ i R / / H ( π R ) ⊕ H ( π i +2 R )[ i + 2] . Since E is a localizing invariant, the main result of Land–Tamme [LT19] yields an B -algebra e R with underlying spectrum H ( π R ) ⊗ τ ≤ i +1 R τ ≤ i R fitting into a commutative diagram of B -algebras τ ≤ i +1 R (cid:15) (cid:15) / / H ( π R ) (cid:15) (cid:15) τ ≤ i R / / e R which is carried to a pullback by E . But the map H ( π R ) → e R is an equivalence in degrees ≤ i + 1 and therefore induces an equivalence on E by the inductive hypothesis. Therefore, E ( τ ≤ i +1 R ) → E ( τ ≤ i R ) is an equivalence, whence the inductive step and the result. (cid:3) Proof of Theorem 3.1.
As before, we write TR Q ( − ) = Q Q TR( − ) for a set Q . For any connective B -algebra R , we claim that R → τ ≤ κ (2 p +1) R induces an equivalence on L K (1) TR Q ( − ) . Indeed, thisfollows from Proposition 3.10 because we can simplicially resolve τ ≤ κ (2 p +1) R using free R -algebrasover free ( R, R ) -bimodules on classes in degrees ≥ κ (2 p + 1) and since TR Q ( − ) commutes withgeometric realizations. The result now follows from Lemma 3.11. (cid:3) This type of argument is also used in [LMT20, Sec. 2.5]. N K (1) -LOCAL TR Corollary 3.12 (Cf. Land–Meier–Tamme [LMT20]) . The invariants L K (1) K ( − ) , L K (1) TC( − ) aretruncating on connective B -algebras.Proof. The result for L K (1) TC( − ) follows from Theorem 3.1 by taking Frobenius fixed points.The result for L K (1) K ( − ) is a formal consequence since the fiber of the trace K ( − ) → TC( − ) istruncating by the Dundas–Goodwillie–McCarthy theorem [DGM13]. (cid:3) Question 3.13.
For i ≥ , let T ( i ) denote a height i telescope. In [LMT20], it is shown that T ( n ) -local K -theory (and hence T ( n ) -local TC( − ) ) is truncating on connective T (1) ⊕ · · · ⊕ T ( n ) -acyclicring spectra. Is there a TR analog of this result? Note also that the result proved in loc. cit. (inthe height one case) is stronger than Corollary 3.12 since there it is not assumed that one worksover a base B , as we do. Corollary 3.14 (Land–Meier–Tamme [LMT20], Bhatt–Clausen–Mathew [BCM20]) . For any n ,we have L K (1) K ( Z /p n ) = 0 .Proof. Take B = H Z in Theorem 3.1, so L K (1) TC( − ) is truncating and therefore nilinvarianton connective H Z -algebras. Then the result follows because the Dundas–Goodwillie–McCarthytheorem and comparison with F p yields L K (1) K ( Z /p n ) = L K (1) TC( Z /p n ) , but the above showsthat this equals L K (1) TC( F p ) = 0 . (cid:3) Asymptotic K (1) -locality In this section, we show (Theorem 4.8) that TR is asymptotically K (1) -local for a regular ringsatisfying mild hypotheses, using the Beilinson–Lichtenbaum conjecture. This result is due toHesselholt–Madsen in the case of smooth algebras over a DVR with perfect residue field of charac-teristic > , which we begin by reviewing. Theorem 4.1 (Hesselholt–Madsen [HM03, HM04]) . Let K be a complete, discretely valued field ofcharacteristic zero with ring of integers O K ⊂ K and perfect residue field k of characteristic p > .Let R be a smooth O K -algebra of relative dimension d . Then the map TR( R ; F p ) → L K (1) TR( R ; F p ) is d -truncated.Proof. Without loss of generality, we can assume that µ p ⊂ K , since otherwise TR( R ; F p ) is aretract of TR( R [ ζ p ]; F p ) via the transfer. In this case, the result follows from [HM04, Th. E]. Indeed, loc. cit. gives a calculation of the cofiber TR( R | R K ; F p ) of the transfer map TR( R ⊗ O K k ; F p ) → TR( R ; F p ) . Since TR( R ⊗ O K k ; F p ) is K (1) -acyclic and d -truncated in view of the identification[Hes96] with the de Rham–Witt complex of R ⊗ O K k , it suffices to verify the (stronger) claim that TR( R | R K ; F p ) → L K (1) TR( R | R K ; F p ) is ≤ d − -truncated. Equivalently (cf. Lemma 5.8 below), itsuffices to show that the cofiber of the Bott element on TR( R | R K ; F p ) is ≤ d + 1 -truncated. In fact,this follows from the calculation in loc. cit. , once we know that the absolute de Rham–Witt complex W Ω ∗ ( R,M R ) is p -divisible in degrees ≥ d + 2 . This in turn follows from the case where R = O K ,cf. [HM03, Cor. 3.2.7]; the case of polynomial rings via the functor P , cf. [HM04, Lem. 7.1.4] andits proof; and finally étale base-change [HM04, Lem 7.1.1]. (cid:3) Construction 4.2 ( TR as p -typical curves) . Let R be an animated ring. We let(10) C ( R ) = lim ←− n Ω K ( R [ x ] /x n , ( x )) Also called a simplicial commutative ring; cf. [ČS19] for a discussion of this terminology. denote the spectrum of curves on the K -theory of R , defining a functor from animated rings tospectra. By [Hes96, Th. 3.1.9], if R is a discrete Z /p j -algebra for some j , we have a naturalexpression of TR( R ; Z p ) as a summand of C ( R ; Z p ) . Left Kan extending both sides to animatedrings (since both TR( − ) and C ( − ) commute with geometric realizations), we obtain that TR( R ; Z p ) is naturally a summand of C ( R ) for R an animated Z /p j -algebra. Passing to the limit over j and using the p -adic continuity of K -theory [CMM18, Th. 5.21], we find that for any p -completeanimated ring R , TR( R ; Z p ) is a summand of C ( R ; Z p ) .Next, we discuss the comparison between the p -adic K -theory of R and R [1 /p ] , cf. [Niz08,Lem. 3.5] for this argument. Proposition 4.3.
Let R be a regular noetherian ring of finite Krull dimension. Suppose for every x ∈ Spec(
R/p ) , we have [ κ ( x ) : κ ( x ) p ] ≤ p d . Then the map K ( R ; F p ) → K ( R [1 /p ]; F p ) is d -truncated.Proof. The homotopy fiber of the map in question is, by Quillen’s dévissage theorem, the mod pK -theory of the abelian category of finitely generated R/p -modules, i.e., the mod p G -theory of
R/p , G ( R/p ; F p ) . So it suffices to show that G ( R/p ; F p ) is concentrated in degrees ≤ d . Using thefiltration by codimension of support and dévissage (cf. [Qui73, Th. 5.4]), we find that G ( R/p ; F p ) has a filtration whose associated graded terms are direct sums of the K ( κ ( x ); F p ) for x ∈ Spec(
R/p ) ;it therefore suffices to show that these terms are d -truncated. But now the Geisser–Levine theorem[GL00] implies that for any field E of characteristic p , there is an embedding K ∗ ( E ; F p ) ֒ → Ω ∗ E/ F p .This implies that K ( κ ( x ); F p ) is dim κ ( x ) Ω κ ( x ) / F p -truncated. But dim κ ( x ) Ω κ ( x ) / F p = log p [ κ ( x ) : κ ( x ) p ] ≤ d by the theory of p -bases [Mat86, Th. 26.5]. Combining these facts, the result follows. (cid:3) Proposition 4.4 (Rosenschon–Østvær [RØ06]) . Let R be a regular noetherian Z [1 /p ] -algebra offinite Krull dimension. Suppose that vcd p ( κ ( x )) ≤ d for all x ∈ Spec( R [1 /p ]) . Then the map K ( R ; F p ) → L K (1) K ( R ; F p ) has ≤ d − -truncated homotopy fiber.Proof sketch. Using Nisnevich descent [TT90], we reduce to the case where R is henselian local,and even a field of characteristic = p by Gabber–Suslin rigidity [Gab92]; note that we do nothave to worry about the distinction between connective and nonconnective K -theory by regularity.Then, the result follows from the Beilinson–Lichtenbaum conjecture (proved by Voevodsky–Rost,cf. [HW19]) describing the associated graded terms of the motivic filtration on K ( R ; F p ) , see e.g.,[CM19, Sec. 6.2] for an account. (cid:3) Proposition 4.5.
Let R be an excellent normal domain which is henselian along an ideal I ⊂ R containing ( p ) . Suppose that for all p ∈ Spec( R ) containing I , we have dim R p +log p [ κ ( p ) : κ ( p ) p ] ≤ p d . Then for all q ∈ Spec( R [1 /p ]) , the residue field κ ( q ) has p -cohomological dimension at most d + 1 .Proof. By standard continuity arguments, it suffices to show that for any affine open U = Spec( R [1 /f ]) ⊂ Spec( R [1 /p ]) and any constructible p -torsion sheaf F on U , we have H n ( U, F ) = 0 for n > d + 1 .Denote by j : U ֒ → Spec( R ) the open inclusion, and denote by i : Spec( R/I ) ֒ → Spec( R ) be theclosed embedding. Using that Spec(
R/I ) has p -cohomological dimension ≤ and the affine analogof proper base change [Gab94, Hub93] applied to the henselian ideal I ⊂ R , we see that it sufficesto show that i ∗ R n j ∗ F = 0 for n > d .Working stalkwise on R and using the compatibility of étale cohomology with filtered colimits,we can now reduce to the case where R is an excellent, strictly henselian normal local domain N K (1) -LOCAL TR with residue field k (and I ⊂ R is contained in the maximal ideal), since excellence and normalityare preserved by strict henselization (cf. [Gre76] for the former). The statement then becomes that H n ( U, F ) = 0 for n > d , which follows from the bound of Gabber–Orgogozo [GO08, Th. 6.1] (notingthat the p -dimension of the residue field k is log p [ k : k p ] since k is separably closed). (cid:3) Remark 4.6.
Suppose that R is a ring such that R /p is finite type over a perfect field k ofcharacteristic p . Then for every prime ideal p ∈ Spec( R ) containing ( p ) , we have dim( R ) p +log p [ κ ( p ) : κ ( p ) p ] ≤ dim( R ) . Indeed, we have dim( R ) p + dim( R / p ) ≤ dim( R ) . Therefore, itsuffices to prove log p [ κ ( p ) : κ ( p ) p ] = dim( R / p ) . But in this case, both are the transcendence degreeof the field κ ( p ) over k , cf. [Mat80, Th. 27.B]. Lemma 4.7.
Let R be an F p -algebra. Suppose that for all residue fields κ of R , we have [ κ : κ p ] ≤ p d for some d ≥ . Then for all residue fields κ ′ of R [[ x ]] , we have [ κ ′ : κ ′ p ] ≤ p d +1 .Proof. We have that R [[ x ]] is generated by p elements over the subring generated by its p th powersand by R . For every residue field κ of R , it follows that R [[ x ]] ⊗ R κ is generated by p elementsover the subring generated by p th powers together with κ . In particular, it is generated by p d +1 elements as a module over its p th powers. It follows that the same holds for any residue field of R [[ x ]] ⊗ R κ and, varying κ , we obtain the conclusion for every residue field of R . (cid:3) Theorem 4.8.
Let R be an excellent, p -torsionfree regular noetherian ring. Suppose R/p is finitelygenerated as a module over its subring of p th powers. Suppose furthermore that for all p ∈ Spec( R ) containing ( p ) , we have dim R p +log p [ κ ( p ) : κ ( p ) p ] ≤ d for some d ≥ . Then the map TR( R ; F p ) → L K (1) TR( R ; F p ) is ( d − -truncated.Proof. Without loss of generality, we can assume that R is p -henselian (since henselization pre-serves excellence, cf. [Gre76]), so dim R ≤ d . By Construction 4.2, it suffices to show that C ( R ) /p → L K (1) C ( R ) /p is ( d − -truncated. Now C ( R ) is the desuspension of the fiber of themap lim ←− n K ( R [ x ] /x n ) → K ( R ) . With p -adic coefficients, the fact that R/p is finitely generated asa module over its p th powers allows us to pass to the limit [CMM18, Th. F] and we obtain C ( R ; F p ) = Ω K ( R [[ x ]] , ( x ); F p ) . Since K ( R ) is a retract of K ( R [[ x ]]) , it suffices to show that the fiber of K ( R [[ x ]]; F p ) → L K (1) K ( R [[ x ]]; F p ) is d -truncated. We will verify this by comparing both sides with the intermediate term K ( R [[ x ]][1 /p ]; F p ) .First, for every characteristic p residue field κ of R , corresponding to a prime ideal p ⊂ R containing ( p ) , we have [ κ : κ p ] ≤ p d − by our assumption, since dim R p ≥ . Now R [[ x ]] is alsoa p -torsionfree regular ring of dimension dim( R ) + 1 . For every characteristic p residue field κ ′ of R [[ x ]] , we have [ κ ′ : κ ′ p ] ≤ p d by Lemma 4.7. By Proposition 4.3, it follows that K ( R [[ x ]]; F p ) → K ( R [[ x ]][1 /p ]; F p ) is d -truncated.Second, we apply Proposition 4.5 to the ring R [[ x ]] and the ideal I = ( p, x ) . The power seriesring R [[ x ]] remains excellent (and regular) since R is excellent, thanks to [KS16]. For any primeideal p ∈ Spec( R [[ x ]]) containing I , we let p = p ∩ R ⊂ R , so that R [[ x ]] p / ( x ) = R p . Then dim( R [[ x ]] p ) = dim( R p ) + 1 and κ ( p ) = κ ( p ) . Thus, Proposition 4.5 applies to R [[ x ]] (with d replaced by d +1 ) and we find that the characteristic zero residue fields of R [[ x ]] have p -cohomologicaldimension at most d + 2 . Therefore, by Proposition 4.4, K ( R [[ x ]][1 /p ]; F p ) → L K (1) K ( R [[ x ]][1 /p ]) is ( d − -truncated.Combining the above, we find that the composite map K ( R [[ x ]]; F p ) → L K (1) K ( R [[ x ]]; F p ) ≃ L K (1) K ( R [[ x ]][1 /p ]; F p ) (the last identification by (2)) is d -truncated, whence the result. (cid:3) This recovers in particular Theorem 4.1. More generally, we have:
Example 4.9.
Let R be a d -dimensional regular, excellent p -torsionfree noetherian ring with R/p finitely generated over its p th powers. Suppose that ( R/p ) red is finite type over a perfect field (andnecessarily of dimension d − ). Then Theorem 4.8 (in view of Remark 4.6) applies to show that TR( R ; F p ) → L K (1) TR( R ; F p ) is ( d − -truncated.5. The Segal conjecture
In this section, we discuss the relationship between the following two properties of a cyclotomicspectrum X :(1) TR( X ) agrees with its K (1) -localization in high enough degrees.(2) The cyclotomic Frobenius ϕ X : X → X tC p is an equivalence in high enough degrees.The first property is the Lichtenbaum–Quillen style statement discussed in the previous section,and verified for THH( R ) under regularity and finiteness hypotheses. The second property is oftenreferred to as the “Segal conjecture” since for X = THH( S ) , the Frobenius S → S tC p is an equivalenceby the Segal conjecture for C p , proved in [Lin80, Gun80]. The Segal conjecture has been studiedextensively for THH( R ) for R a ring (or ring spectrum).We first show the implication (1) = ⇒ (2) . We use the theory of topological Cartier modules ofAntieau–Nikolaus [AN18], which we begin by briefly reviewing. Definition 5.1. A topological Cartier module M is an object of Fun( BS , Sp) together with maps V : M hC p → M and F : M → M hC p in Fun( BS , Sp) together with a homotopy between the com-posite and the norm map M hC p → M hC p (considered S ≃ S /C p -equivariantly). The collectionof topological Cartier modules is naturally organized into a presentable stable ∞ -category.Given a bounded-below, p -typical cyclotomic spectrum X , we can consider TR( X ) as a topolog-ical Cartier module, and we have an identification X ≃ cofib( V ) . Under these identifications, thecyclotomic Frobenius X → X tC p is obtained from F : TR( X ) → TR( X ) hC p by taking cofibers by V on both domain and codomain and identifying TR( X ) tC p ≃ X tC p as (TR( X ) hC p ) tC p = 0 [NS18,Lem. I.2.1]. On bounded-below objects, this construction establishes a fully faithful embeddingfrom cyclotomic spectra into topological Cartier modules, with image given by the V -completeobjects [AN18, Th. 3.21]. Proposition 5.2.
Let X be a connective, p -complete cyclotomic spectrum whose underlying spec-trum is K (1) -acyclic. Suppose the map TR( X ) → L K (1) TR( X ) is d -truncated. Then the Frobenius ϕ : X → X tC p is d -truncated. In the case L K (1) TR( X ) = 0 , the result is [AN18, Prop. 2.25]. Proof.
Since X is K (1) -acyclic, it follows that V : TR( X ) hC p → TR( X ) is K (1) -locally an equiv-alence. The K (1) -localization L K (1) TR( X ) acquires the structure of a topological Cartier module aswell by K (1) -localizing F, V and using the comparison map L K (1) (TR( X ) hC p ) → ( L K (1) TR( X )) hC p .The composite map ( L K (1) TR( X )) hC p → ( L K (1) TR( X )) hC p is an equivalence after p -completionsince Tate constructions vanish in the K (1) -local category. Since we saw that V is an equiva-lence on L K (1) TR( X ) after p -completion, it follows that the Frobenius on L K (1) TR( X ) induces anequivalence L K (1) TR( X ) ∼ −→ ( L K (1) TR( X )) hC p . N K (1) -LOCAL TR For a topological Cartier module Y , we consider the fiber of F = F Y : Y → Y hC p , whichwe denote fib( F ) . As we saw above, fib( F ) is contractible for the topological Cartier module L K (1) TR( X ) . Moreover, fib( F ) is d -truncated for the topological Cartier module fib(TR( X ) → L K (1) TR( X )) because this topological Cartier module is itself d -truncated. In particular, we findthat fib( F : TR( X ) → TR( X ) hC p ) is d -truncated. Taking the cofiber on both the domain andcodomain of the Verschiebung, we find that fib( F : TR( X ) → TR( X ) hC p ) = fib( ϕ : X → X tC p ) which is therefore d -truncated as desired. (cid:3) Combining Theorem 4.1, Proposition 5.2, and Theorem 4.8, we obtain the following result.Versions of the Segal conjecture have been studied by many authors. For instance, it is known that
THH( Z p ; F p ) → THH( Z p ; F p ) tC p is an equivalence on connective covers [BM94, Lem. 6.5]. Compare[HM03] for the more general case of a DVR O K of mixed characteristic and perfect residue fieldof characteristic p > . The Segal conjecture for smooth algebras in characteristic p appears as[Hes18, Prop. 6.6] and (at the filtered level) [BMS19, Cor. 8.18]. The Segal conjecture has also beenverified for certain ring spectra as well, cf. [AR02, LNR11, AKQ17]. Corollary 5.3 (The Segal conjecture for regular rings) . Let R be a p -torsionfree excellent reg-ular noetherian ring with R/p finitely generated over its p th powers. Suppose that for all p ∈ Spec( R ) containing ( p ) , we have dim R p + log p [ κ ( p ) : κ ( p ) p ] ≤ d . Then the map THH( R ; F p ) → THH( R ; F p ) tC p is ( d − -truncated. (cid:3) Question 5.4.
Is it possible to prove a filtered version of this result, with respect to the motivicfiltrations [BMS19] on both sides?We next discuss the converse direction. Here we only prove the result under a more restrictivehypothesis, namely for cyclotomic spectra which are ku -modules. Example 5.5.
Let C = c Q p . If R is a O C -algebra, then the cyclotomic trace makes THH( R ; Z p ) intoa ku -module in view of the equivalence ku ˆ p ≃ K ( O C ; Z p ) , cf. [Niz98, Lem. 3.1], [Sus83], and [Hes06].One can improve this slightly: K ( \Z p [ ζ p ∞ ]; Z p ) has the structure of an E ∞ -algebra under ku , so thesame applies to any \Z p [ ζ p ∞ ] -algebra R . This follows from three facts. First, K ( \Z p [ ζ p ∞ ]; Z p ) ≃ K ( \Q p ( ζ p ∞ ); Z p ) , cf. [HN19, Lem. 1.3.7] for this argument (which only uses that the field \Q p ( ζ p ∞ ) is perfectoid). Second, since \Q p ( ζ p ∞ ) is perfectoid, it has p -cohomological dimension ≤ by thetilting equivalence, whence K ( \Q p ( ζ p ∞ ); Z p ) → L K (1) K ( \Q p ( ζ p ∞ ); Z p ) is the connective cover map,cf. Proposition 4.4. Third, L K (1) K ( Z [ ζ p ∞ ]) has the structure of an E ∞ -algebra under KU andhence ku , cf. [BCM20, Construction 3.7]. Lemma 5.6.
Let M ∈ Fun( BS , Mod( H F p )) be an r -connected object. Suppose that there exists amap f : M → M ′ in Fun( BS , Mod( H F p )) such that M ′ is induced (as an object with S -action)and such that f is an equivalence on τ ≥ r . Then there exists a map f M → M of r -connected objectsin Fun( BS , Mod( H F p )) which induces an equivalence on τ ≥ r +1 and such that f M is induced.Proof. The homotopy groups π ∗ ( M ) , π ∗ ( M ′ ) form graded modules over the ring π ∗ ( F p [ S ]) = F p [ ǫ ] /ǫ , | ǫ | = 1 . By assumption, π ∗ ( M ′ ) is a free graded F p [ ǫ ] /ǫ -module and π ∗ ( M ) is thesubmodule of those elements in degree ≥ r . Choosing an F p -subspace V of π r ( M ) = π r ( M ′ ) whichis complementary to ǫπ r − ( M ′ ) ⊂ π r ( M ′ ) , we can modify M to form f M with π r ( f M ) = V ; it is noweasy to see that π ∗ ( f M ) is a free graded F p [ ǫ ] /ǫ -module, so we can conclude. (cid:3) Proposition 5.7.
Let X be a connective, p -complete ku -module in CycSp . Suppose that ϕ : X → X tC p is d -truncated. Then the fiber of TR( X ) /p → L K (1) TR( X ) /p is d + 3 -truncated.Proof. It follows that the cyclotomic spectrum Y = X/ ( p, β ) = X ⊗ ku H F p ∈ CycSp ≥ has theproperty that ϕ : Y → Y tC p is d + 4 -truncated. It follows that the comparison map Y C p → Y hC p is d + 4 -truncated as well, via the fiber square Y C p (cid:15) (cid:15) / / Y hC p (cid:15) (cid:15) Y ϕ / / Y tC p . Now for each n ≥ , we have a pullback diagram [NS18, Lem. II.4.5],(11) Y C pn R (cid:15) (cid:15) / / Y hC pn (cid:15) (cid:15) Y C pn − / / Y tC pn . The bottom horizontal map is Y C pn − → Y hC pn − ϕ hCpn − −−−−−−→ ( Y tC p ) hC pn − , and the last term isidentified with Y tC pn using the Tate orbit lemma, cf. [NS18, Lem. II.4.1].Now Y C p → Y hC p is an equivalence in degrees ≥ d +6 , hence Y C pn − → Y hC pn − is an equivalencein degrees ≥ d +6 by [NS18, Cor. II.4.9] (a generalization of results of Tsalidis [Tsa98] and Bökstedt–Bruner–Lunøe-Nielsen–Rognes [BBLNR14]). Since ϕ : Y → Y tC p is an equivalence in degrees ≥ d + 6 , it follows that the bottom horizontal map in (11) is an equivalence in degrees ≥ d + 6 .Note also that Y tC p is a module over F tC p p in Fun( BS , Mod( H F p )) ; since the latter has induced S -action, it follows that the former does too.By Lemma 5.6, we can find a ≥ d + 6 -connected Y ′ with an S -equivariant map Y ′ → Y which induces an equivalence in degrees ≥ d + 7 and such that Y ′ is induced as an object of Fun( BS , Mod( H F p )) . It follows that the map Y ′ hC pn → Y hC pn is an equivalence in degrees ≥ d + 7 . But the commutative square Y ′ hC pn (cid:15) (cid:15) / / Y hC pn (cid:15) (cid:15) Y ′ tC pn = 0 / / Y tC pn now shows that any α ∈ π r ( Y hC pn ) for r ≥ d + 7 has vanishing image in π r ( Y tC pn ) . Using thecommutative square (11) again (since the horizontal maps are equivalences in degrees ≥ d + 6 ),it follows that the restriction map Y C pn → Y C pn − induces the zero map in degrees ≥ d + 7 .Consequently, taking the inverse limit over R , we find that TR( Y ) ∈ Sp ≤ d +6 , whence the fiber of TR( X ) /p → L K (1) TR( X ) /p belongs to Sp ≤ d +3 by Lemma 5.8. (cid:3) Lemma 5.8.
Let M be a connective, p -complete ku -module spectrum. Then the following areequivalent:(1) M/ ( p, β ) is concentrated in degrees ≤ d + 3 .(2) The fiber of M/p → L K (1) M/p is concentrated in degrees ≤ d . N K (1) -LOCAL TR Proof.
Suppose (1). Let N = M/p . It suffices to show that the fiber of N → L K (1) N = N [1 /β ] isconcentrated in degrees ≤ d . Let F = fib( N → N [1 /β ]) . If π i ( F ) = 0 , then the long exact sequenceshows either that there exists a β -power torsion element in π i ( N ) or π i +1 ( N ) → π i +1 ( N [1 /β ]) isnot surjective. In the former case, we obtain a nonzero element in π i +3+2 j ( N/β ) for some j ≥ ,which by our assumptions forces i ≤ d . In the latter case, there exists x ∈ π i +1+2 j ( N ) for some j > which cannot be divided by β , whence we obtain a nontrivial class in π i +1+2 j ( N/β ) , againforcing i ≤ d . This proves that F is d -truncated, as desired. The fact that (2) implies (1) followsbecause L K (1) M/p is acted upon invertibly by β . (cid:3) Remark 5.9.
Let X = THH( F p ) ; in this case, we have that THH( F p ) → THH( F p ) tC p is ( − -truncated. This computation is due to Hesselholt–Madsen [HM97, Sec. 5], and is refined in [NS18,App. IV-4]. Meanwhile TR( F p ) = H Z p → L K (1) TR( F p ) = 0 is -truncated modulo p . Thus, thebound of Proposition 5.7 is the best possible. Proposition 5.10.
Let R be a p -torsionfree perfectoid ring. Let R be a formally smooth R -algebra(with respect to the p -adic topology) of relative dimension d . Then the map ϕ : THH( R ; F p ) → THH( R ; F p ) tC p is ( d − -truncated, and the map L K (1) TR( R ; F p ) → L K (1) TR( R ; F p ) is ( d − -truncated.Proof. Let X i , i ∈ C be a diagram of spectra. Suppose that for each i ∈ C , the map X i → L K (1) X i is m -truncated for some m . Then the map lim ←− i X i → L K (1) (lim ←− i X i ) is m -truncated,and L K (1) (lim ←− i X i ) → lim ←− i L K (1) X i is an equivalence. This follows because L K (1) ( − ) annihilatesbounded-above spectra. Applying the above observation, and using flat descent of TR( − ; Z p ) (whichfollows from [BMS19, Sec. 3]) and THH( − ; Z p ) , THH( − ; Z p ) tC p , we reduce to the case where R is a \Z p [ ζ p ∞ ] -algebra (e.g., using André’s lemma, cf. [BS19, Th. 7.12]), so THH( R ; Z p ) is a ku ˆ p -modulein CycSp ≥ (Example 5.5). With this reduction in mind, the first claim implies the second in viewof Proposition 5.7 (applied to the cyclotomic spectrum THH( R ; F p ) ).Thus, we prove the first claim (i.e., the Segal conjecture for THH( R ; F p ) ); this result appears as[BMS19, Cor. 9.12] in the case where R is the ring of integers in a complete, algebraically closednonarchimedean field of mixed characteristic, and in [BMS19, Sec. 6] when R = R .Let ( A, I ) be the perfect prism associated to the perfectoid ring R , and let e ξ ∈ I be a generator.We use the prismatic cohomology ∆ R/A of [BS19] and its Nygaard completion b ∆ R/A . For a qua-sisyntomic R -algebra S , there are [BMS19] complete, exhaustive descending Z -indexed filtrationson TC − ( S ; Z p ) , TP( S ; Z p ) with gr i TC − ( S ; Z p ) ≃ N ≥ i b ∆ S/A [2 i ] , gr i TP( S ; Z p ) ≃ b ∆ S/A [2 i ] , and the map ϕ : TC − ( S ; Z p ) → TP( S ; Z p ) on graded pieces is given by the prismatic dividedFrobenius(12) ϕ/ e ξ i : N ≥ i b ∆ S/A [2 i ] → b ∆ S/A [2 i ] , cf. [BS19, Sec. 13] for the comparison between prismatic cohomology and the objects of [BMS19].Here we have trivialized the Breuil–Kisin twists involved since we are over the base perfectoid ring R , and we can compute the Nygaard-completed absolute prismatic cohomology as the Nygaardcompleted relative prismatic cohomology over R .Now the map ϕ : TC − ( S ; Z p ) → TP( S ; Z p ) arises by taking S -invariants on the cyclotomicFrobenius ϕ : THH( S ; Z p ) → THH( S ; Z p ) tC p . Both sides of this map are filtered (again, as in [BMS19]) and the associated graded pieces are gr i THH( S ; Z p ) ≃ N i b ∆ S/A [2 i ] , gr i THH( S ; Z p ) = ( b ∆ S/A ) / e ξ [2 i ] , and the cyclotomic Frobenius is induced from (12). But since R is formally smooth over R , theNygaard filtration is complete and ϕ/ e ξ i induces an equivalence N i ∆ R/A ∼ −→ τ ≤ i ( ∆ R/A / e ξ ) , cf. [BS19,Sec. 12.4], where the right-hand-side is the i th stage of the conjugate filtration on ∆ R/A / e ξ . Usingthe Hodge–Tate comparison, [BS19, Th. 4.10], it follows that the cohomology groups of ∆ R/A / e ξ are p -torsion-free. It follows that gr i ϕ : gr i THH( R ; F p ) → gr i THH( R ; F p ) tC p exhibits the domain as the i -connective cover of the codomain, and therefore has ( i − -connectedhomotopy fiber for all i . The codomain is d -truncated by the Hodge–Tate comparison, since R/R is formally smooth of relative dimension d ; therefore, gr i ϕ is an equivalence for i ≥ d . Therefore,the fiber of ϕ : THH( R ; F p ) → THH( R ; F p ) tC p has a complete filtration such that gr i is ( i − -truncated for all i and contractible for i ≥ d . This proves that ϕ : THH( R ; F p ) → THH( R ; F p ) tC p is ( d − -truncated, whence the result. (cid:3) Pro-Galois descent
In this section, we prove a type of pro-Galois descent for TR in the generic fiber, which is relatedto the conjecture in [Hes02]. The basic example is when K is a characteristic zero local field, andone tries to relate TR( O K ; Z p ) (computed by [HM03]) with the “continuous” homotopy fixed pointsfor the Galois group Gal(
K/K ) on TR( O K ; Z p ) , before or after K (1) -localization. The advantageis that the latter is much more tractable, cf. [Hes06] for the calculation of TR( O K ; Z p ) . For finiteGalois extensions in the generic fiber, these claims follow from section 2. However, there are someadditional subtleties to extend to pro-Galois descent because TR fails to commute with filteredcolimits.6.1. An auxiliary construction.
Let B be a base ring. Let E be a K (1) -local, localizing invarianton B -linear ∞ -categories which is truncating. Let R be a p -adically complete B -algebra. Given a K (1) -local truncating invariant E , we now describe a construction of a sheaf on the finite étale siteof R [1 /p ] . For every finite étale R [1 /p ] -algebra S , we can choose a “ring of integers” S which isfinite and finitely presented with S [1 /p ] = S and consider E ( S ) . It is not difficult to see that thisonly depends on S and that it defines the desired sheaf; to make the functoriality precise, we useleft Kan extension. Construction 6.1 ( E as a sheaf on the finite étale site of the generic fiber) . Let R be a p -adicallycomplete B -algebra and let R = R [1 /p ] . Using the K (1) -local, localizing invariant E which isassumed to be truncating, we define a sheaf F E of spectra on the finite étale site of Spec( R ) asfollows. Let C denote the category of finite, finitely presented R -algebras S with S [1 /p ] étaleover R , and let C denote the category of finite étale R -algebras. Consider the functor F : C → C given by inverting p . Note that:(1) F is essentially surjective. That is, given any finite étale R -algebra S , there exists a finite,finitely presented R -algebra S with S = S [1 /p ] . In fact, consider any finite R -subalgebra S ′ ⊂ S with S ′ [1 /p ] = S . The algebra S ′ is not necessarily finitely presented, but it is adirected colimit of finite, finitely presented R -algebras S ( α )0 under surjective maps; one ofthem will have S ( α )0 [1 /p ] = S , and can be taken for S . N K (1) -LOCAL TR (2) If S ∈ C , then C × C C /S is a filtered category. In fact, the subcategory of C × C C /S spannedby those S such that the structure map S [1 /p ] → S is an isomorphism is itself filteredand cofinal. This follows similarly by comparing S with its image in S .We consider the functor S E ( S ) on the category C of finite, finitely presented R -algebras S with S [1 /p ] finite étale over R . Then, to define F E on finite étale R -algebras, we left Kanextend E ( − ) along the functor C → C . Explicitly, it follows from Example 2.4 that if S is a finiteétale R -algebra and S is a finite, finitely presented R -algebra with S [1 /p ] ≃ S , then we havea canonical equivalence F E ( S ) ≃ E ( S ) . It also follows from h -descent that F E is indeed a sheafof K (1) -local spectra on the finite étale site of Spec( R ) . Note that it is a sheaf of modules overthe sheaf S L K (1) K ( S ) , which is also the sheaf F L K (1) K because L K (1) K ( − ) is insensitive toinverting p on H Z -algebras as in (2). Proposition 6.2 (Hypercompleteness of F E ) . Notation as above, suppose there is a uniform boundon the mod p cohomological dimensions of the residue fields of R and R has finite Krull dimension.Then F E defines a hypercomplete sheaf on the finite étale site of Spec( R ) .Proof. Our hypotheses imply that
Spec( R ) has finite mod p étale cohomological dimension. By the K ( π, property (cf. [Sch13, Th. 4.9] which assumes noetherian hypotheses, but one can pass tothe limit using the Fujiwara–Gabber theorem, cf. e.g. [BM18, Th. 6.11]), it follows that the étalefundamental group π et1 (Spec( R )) has finite mod p cohomological dimension, which implies that thenotion of hypercompleteness for sheaves of p -complete spectra on the finite étale site of Spec( R ) can be made explicit in terms of exponents of nilpotence [CM19, Sec. 4.1]. Now F E is a moduleover the hypercomplete sheaf S L K (1) K ( S ) (cf. [CM19, Th. 7.14] for hypercompleteness), hencea hypercomplete sheaf itself, thanks to [CM19, Cor. 4.26]. (cid:3) We also need the following variant of the above with respect to a fixed profinite group.
Construction 6.3 ( F E relative to a profinite group) . Let S be an R -algebra equipped with theaction of a profinite group Γ which is continuous (with respect to the discrete topology on S ), andsuch that R → S := S [1 /p ] is Γ -Galois. Suppose that there exist a cofinal collection of open normalsubgroups N i ⊂ Γ , i ∈ I for which the fixed points S N i form a finite, finitely presented R -algebra.It follows from the above that we obtain a sheaf on the category of finite continuous Γ -sets whichcarries Γ /N i E ( S N i ) . This is just the restriction of F E to the site of finite continuous Γ -sets(which maps to the finite étale site of R ).When is the sheaf of Construction 6.3 hypercomplete? When Γ has finite cohomological dimen-sion, then hypercompleteness is smashing [CM19, Sec. 4.1], so hypercompleteness holds if and onlyif the sheaf of spectra which sends Γ /H L K (1) K ( S H ) (for any cofinal collection of open normalsubgroups H ) is hypercomplete. Example 6.4.
Suppose
Γ = Z np and the Γ -extension of R is obtained by adding compati-ble systems of p -power roots. Explicitly, suppose R is a Z [ ζ p ∞ , t ± , . . . , t ± n ] -algebra and S = R ⊗ Z [ ζ p ∞ ,t ± ,...,t ± n ] Z [ ζ p ∞ , t ± /p ∞ , . . . , t ± /p ∞ n ] , with the evident Γ -action. In this case, the sheaf ofConstruction 6.3 is hypercomplete. Again since hypercompleteness is smashing, this follows becauseany K (1) -local localizing invariant yields a hypercomplete étale sheaf on the site of finite étale (oreven all étale) Z [1 /p, ζ p ∞ , t ± , . . . , t ± n ] -algebras, cf. [CM19, Th. 7.14]. In particular, we take as thelocalizing invariant A L K (1) ( A ⊗ Z [1 /p,ζ p ∞ ,t ± ,...,t ± n ] R ) . Completion of topological Cartier modules.
Here again we use the “decompletion” of thetheory of cyclotomic spectra given by the topological Cartier modules of Antieau–Nikolaus [AN18]and an amplitude property of the completion. We recall that
TR( − ) gives a fully faithful rightadjoint embedding from bounded-below cyclotomic spectra into bounded-below topological Cartiermodules, with image the V -complete objects, cf. [AN18, Th. 3.21]. Proposition 6.5.
Let M be a topological Cartier module which is d -truncated and bounded below.Then the V -completion of M is d + 3 -truncated. If M is p -complete, then the V -completion of M is d + 2 -truncated.Proof. We reduce by dévissage to the case where M is concentrated in degree zero, and and d = 0 .The completion of M is given by cofib(lim ←− M hC pn → M ) where the maps M hC pn → M hC pn − aregiven by V hC pn − , cf. [AN18, Prop. 3.22].Now since M is discrete, the Verschiebung is simply given by a map V : M → M of abelian groups,and the S -action is trivial. We can (as in the proof of [AN18, Lem. 3.25]) form a Z ≥ × Z ≥ -indexeddiagram Y i,j = M ⊗ BC p j + such that the transition maps in the i -direction are given by V and inthe j -direction are given by the canonical projections. By the above, the completion of M is givenby the cofiber of the map lim ←− i,j Y i,j → M .Now a simple computation shows that for any abelian group A , lim ←− j HA ⊗ BC p j + are concentratedin degrees ≤ , and degrees ≤ if A is derived p -complete. This claim will imply the result. Indeed,for the first part, it suffices to show that lim ←− j HA ⊗ BC p j + is concentrated in degrees ≤ when A istorsion-free; this follows because the pro abelian group { H ∗ ( BC p j ; Z ) } j ≥ is pro-zero for ∗ ≥ . Forthe second part, we also observe that the pro-abelian group (cid:8) H ( BC p j ; Z ) (cid:9) j ≥ is simply the tower · · · → Z /p → Z /p → , and our assumption that A is derived p -complete gives A ≃ lim ←− A ⊗ LZ Z /p j is in particular discrete. (cid:3) Remark 6.6.
We can give another proof of Proposition 6.5 using the results from the previoussection in the case where M is annihilated by a power of p . By dévissage, we can reduce to thecase where M is an H F p -module (in topological Cartier modules). Let X = M/M hC p = cofib( V ) be the associated cyclotomic spectrum, so TR( X ) is the derived V -completion of M by the cor-respondence between bounded-below V -complete Cartier modules and bounded-below cyclotomicspectra, cf. [AN18, Th. 3.21]. Then the proof of Proposition 5.2 shows that the cyclotomic Frobe-nius ϕ : X → X tC p is d -truncated, since fib( ϕ ) = fib( F : M → M hC p ) . Using Proposition 5.7, wefind that TR( X ) /p is d + 3 -truncated, whence TR( X ) is d + 2 -truncated. Corollary 6.7.
Let R i , i ∈ I be a filtered system of rings and let R = lim −→ R i . Suppose that the map TR( R i ; F p ) → L K (1) TR( R i ; F p ) is d -truncated for all i ∈ I . Then TR( R ; F p ) → L K (1) TR( R ; F p ) is d + 2 -truncated.Proof. Let v : Σ u ( S /p ) → ( S /p ) be a v -self map (so we can take u = 2 p − for p odd and u = 8 for p = 2 ). Let X be any spectrum. Then we observe that the following are equivalent:(1) The map X/p → L K (1) X/p is d -truncated.(2) The spectrum X ⊗ S / ( p, v ) is d + u + 1 -truncated.The equivalence is proved analogously to Lemma 5.8, noting that L K (1) ( X/p ) = X ⊗ ( S /p )[ v − ] by the telescope conjecture at height one [Mah81, Mil81].Therefore, it suffices to show that TR( R ; Z p ) ⊗ S / ( p, v ) is d + u + 3 -truncated. To this end,let M = lim −→ TR( R i ; Z p ) , so M is a connective topological Cartier module which is not necessarily N K (1) -LOCAL TR derived V -complete; the V -completion of its p -completion is TR( R ; Z p ) since THH( − ) commuteswith filtered colimits (as a functor from rings to CycSp ≥ ). Now the topological Cartier module M ⊗ S / ( p, v ) is d + u + 1 -truncated as a filtered colimit of d + u + 1 -truncated objects. Takingthe V -completion and using Proposition 6.5, we find that TR( R ; Z p ) / ( p, v ) is d + u + 3 -truncated.Therefore, TR( R ; F p ) → L K (1) TR( R ; F p ) is d + 2 -truncated. (cid:3) The main pro-Galois result.
In this subsection, we prove the following pro-Galois descentresult.
Theorem 6.8 (Pro-Galois descent in the generic fiber) . Fix d ≥ . Let R be a p -complete ringsuch that TR( R ; F p ) → L K (1) TR( R ; F p ) is d -truncated. Let S be an R -algebra.Let G be a profinite group of finite p -cohomological dimension which acts continuously on the R -algebra S (given the discrete topology). Suppose that:(1) R [1 /p ] → S [1 /p ] is a G -Galois extension.(2) There is a cofinal collection of open normal subgroups N i ⊂ G, i ∈ I such that S i := S N i is a finite, finitely presented R -algebra and such that TR( S i ; F p ) → L K (1) TR( S i ; F p ) is d -truncated.(3) The induced sheaf of spectra on finite continuous G -sets given by T L K (1) ( K (Fun G ( T, S [1 /p ]))) is hypercomplete. For example, this holds R [1 /p ] has finite Krull dimension and there isa uniform bound on the mod p cohomological dimensions of the residue fields, but also incases such as Example 6.4.Then the map (13) L K (1) TR( R ) → ( L K (1) TR( S )) hG cts := Tot( L K (1) TR( S ) ⇒ L K (1) TR(Fun cts ( G, S )) →→→ . . . ) . is an equivalence and the map TR( R ; F p ) → Tot(TR( S ; F p ) ⇒ TR(Fun cts ( G, S ); F p ) →→→ . . . ) is d + 2 -truncated.Proof. Let F be any product-preserving presheaf from finite continuous G -sets to p -torsion spectra.We let F disc denote the left Kan extension to profinite G -sets, so if S is a profinite G -set which canbe written S ≃ lim ←− S j for some finite continuous G -sets S j , then F disc ( S ) ≃ lim −→ F ( S j ) . We let R Γ( ∗ , F ) = Tot( F disc ( G ) ⇒ F disc ( G × G ) →→→ . . . ) . As in [CM19, Sec. 4.1], R Γ( ∗ , F ) is the value of the hypersheafification or Postnikov sheafificationof F (with respect to the topology on finite continuous G -sets where covering families are jointlysurjective ones) at ∗ . When we work with p -torsion spectra, our assumption that G has finitecohomological dimension implies that R Γ( ∗ , − ) commutes with all colimits.Now we take F to be the presheaf which sends a finite continuous G -set T to TR(Fun G ( T, S ); F p ) ,where Fun G ( T, S ) is the ring of G -equivariant functions T → S . Unwinding the definitions andhypotheses, we find from Corollary 2.5 that if T = G/N j for one of the distinguished normalsubgroups N j , we have that L K (1) F ( ∗ ) ∼ −→ Tot( L K (1) F ( T ) ⇒ L K (1) F ( T × T ) →→→ . . . ) is an equivalence. Note L K (1) F is an example of Construction 6.3 for the profinite group G (thoughwe denote by R the p -complete ring). In particular, our assumptions imply hypercompleteness of L K (1) F , so passing to the limit we find(14) L K (1) F ( ∗ ) ≃ R Γ( ∗ , L K (1) F ) . Note here that F disc is not given by TR( − ; F p ) because TR does not commute with filtered colimits;instead, for instance, F disc ( G ) = lim −→ i ∈ I TR( S i ; F p ) . By assumption (2), F disc ( G ) , F disc ( G × G ) , . . . are spectra with the property that the map to their K (1) -localization is d -truncated; therefore, R Γ( ∗ , F ) → R Γ( ∗ , L K (1) F ) is d -truncated.Now we apply 2-out-of-3 to the sequence of maps F ( ∗ ) → R Γ( ∗ , F ) → R Γ( ∗ , L K (1) F ) . We justshowed that the second map is d -truncated, while the composite map is by (14) identified with TR( R ; F p ) → L K (1) TR( R ; F p ) , which is d -truncated by assumption. Therefore, by 2-out-of-3, wefind that F ( ∗ ) → R Γ( ∗ , F ) is d -truncated. In other words,(15) F ( ∗ ) → Tot( F disc ( G ) ⇒ F disc ( G × G ) →→→ . . . ) , is d -truncated. Here both sides of (15) have the structure of topological Cartier modules since theforgetful functor from topological Cartier modules to spectra commutes with limits and colimits[AN18, Prop. 3.11].Now we take the V -completion of both sides in (15) (considered as topological Cartier modules).The left-hand-side of (15) is already V -complete. To analyze the right-hand-side, observe thatthe totalization in (15) commutes with ( − ) hC pn − because R Γ( ∗ , − ) commutes with colimits. Thelimit over n in computing the V -completion clearly commutes with the totalization. Thus, the V -completion of the right-hand-side of (15) is given by Tot (cid:16)
TR( S ; F p ) ⇒ TR(Fun cts ( G, S ); F p ) →→→ . . . (cid:17) . Note finally that the right-hand-side of (15) is bounded-below by the finiteness of the p -cohomologicaldimension. Taking the V -completion in (15), we find from Proposition 6.5 that TR( R ; F p ) → Tot (cid:16)
TR( S ; F p ) ⇒ TR(Fun cts ( G, S ); F p ) →→→ . . . (cid:17) is ( d + 2) -truncated, whence the last claim of the theorem. Finally, TR( S ; F p ) , TR(Fun cts ( G, S ); F p ) map via ( d + 2) -truncated maps to their K (1) -localizations, via Corollary 6.7, so the K (1) -localdescent statement follows. (cid:3) Example 6.9 (Discrete valuation rings) . Let K be a complete, discretely valued field of charac-teristic zero whose residue field k is of characteristic p with [ k : k p ] ≤ p d . It follows that if k ′ /k isany finite extension, then [ k ′ : k ′ p ] ≤ p d , cf. [GO08, Lem. 2.1.1].By the main result of [GO08] (which is due to [Kat82] in this case), it follows that the Galoisgroup Gal(
K/K ) has p -cohomological dimension ≤ d + 2 . Moreover, by Theorem 4.8, it follows thatif L is any finite extension of K and O L ⊂ L the ring of integers (which is excellent as a completelocal ring), then TR( O L ; F p ) → L K (1) TR( O L ; F p ) is d -truncated.We can now apply Theorem 6.8 to conclude that TR( O K ; F p ) → TR( O K ; F p ) h Gal cts K := Tot(TR( O K ; F p ) ⇒ TR( O K ⊗ K K ; F p ) →→→ . . . ) is d -truncated. Note that the theorem gives a priori that the map is ( d + 2) -truncated, but we canupgrade the conclusion to d -truncated as follows: the analogous comparison map with L K (1) TR( − ) everywhere is an equivalence; the maps TR( O K ⊗ K ···⊗ K K ; F p ) → TR( O K ⊗ K ···⊗ K K ; F p ) are ( − -truncated by Proposition 5.10 since these are p -torsionfree rings whose completions are perfectoid;and the map TR( O K ; F p ) → L K (1) TR( O K ; F p ) is d -truncated (Theorem 4.8). N K (1) -LOCAL TR Suppose in particular that k is perfect and of characteristic > . In this case, the results of[HM03] (recalled in the proof of Theorem 4.1) show that
TR( O K | K ; F p ) is the connective cover of its K (1) -localization, which is L K (1) TR( O K ; F p ) . Again, O K , O K ⊗ K K , . . . have p -completions whichare perfectoid rings, so for them TR( − ; F p ) agrees with the connective cover of its K (1) -localization(Proposition 5.10). It follows that TR( O K | K ; F p ) ≃ τ ≥ TR( O K ; F p ) h Gal cts K . Example 6.10.
Let R be a p -torsionfree perfectoid ring containing a system of primitive p -powerroots of unity. Let R be a formally smooth R -algebra, which is formally étale over the formal torus(i.e., p -completed Laurent polynomial algebra) R (cid:10) t ± , . . . , t ± n (cid:11) .Let G = Z p (1) n and let S = R ⊗ R h t ± ,...,t ± n i lim −→ r R D t ± /p r , . . . , t ± /p r n E with the evident G -action by roots of unity. Using Proposition 5.10, one sees that the hypothesis (2) applies. Hypothesis(3) applies thanks to Example 6.4. It follows that the comparison map (13) is an equivalence. Inparticular, the K (1) -local TR of R is expressed as the inverse limit of a diagram of the K (1) -local TR of various rings whose p -completion is perfectoid.7. The analog of Thomason’s spectral sequence
In this section, we construct in certain cases an analog of Thomason’s étale descent spectralsequence for L K (1) K ( − ) in terms of étale cohomology, cf. [Tho85, TT90], for L K (1) TR( − ) . Ourconstruction splits into two parts. First, we give a formula for TR and its K (1) -localization on thecategory of perfectoid rings. Second, we hypersheafify L K (1) TR( − ) on all rings in the arc p -topologyand compare L K (1) TR( − ) to this hypersheafification.7.1. arc p -cohomology. In this section, we discuss the cohomology with respect to the arc p -topology,mentioned briefly in the introduction (Definition 1.6). This is a variant of the following topology,cf. [ČS19, Sec. 2.2.1]. Definition 7.1 (The p -complete arc -topology) . The p -complete arc -topology or arc ˆ p -topology isthe finitary Grothendieck topology (cf. Remark A.1) on the opposite of the category of derived p -complete commutative rings such that a map R → R ′ is a cover if for every p -complete rank ≤ -valuation ring V and map R → V , there is an extension V → W of p -complete rank ≤ valuation rings and a map R ′ → W fitting into a commutative diagram, R (cid:15) (cid:15) / / R ′ (cid:15) (cid:15) V / / W .
We will also consider restrictions of the arc ˆ p -topology to appropriate subcategories of the categoryof derived p -complete rings, such as the category Perfd of perfectoid rings [BMS18, Sec. 3]. The arc p -topology is defined similarly but we only consider rank p -complete valuation rings where p = 0 . We expect that this works when k has characteristic as well. Throughout, for set-theoretic reasons one should impose cardinality bounds, i.e., assume all the rings one allowsinto the site to be of cardinality less than some uncountable strong limit cardinal κ , so that the sites will be small.However, the choice of κ will not matter in all the constructions we consider and we will consequently suppress it. The arc ˆ p -topology behaves well with respect to perfectoid rings; one knows that the structurepresheaf is a sheaf of rings with respect to this topology, and one even has no higher cohomol-ogy [BS19, Sec. 8]. The arc p -topology is a variant where in some sense we also impose derivedsaturatedness conditions. Note that a functor is an arc p -sheaf if and only if it is an arc ˆ p -sheafand annihilates any F p -algebra. Any derived p -complete ring can be covered in the arc ˆ p -topology(resp. the arc p -topology) by a product of p -complete absolutely integrally closed rank valuationrings (resp. p -complete absolutely integrally closed rank valuation rings where p = 0 ).We now give some examples of arc p -cohomology. To begin with, we consider the simplest case ofconstant sheaves (or p -adically constant ones); the result is that one essentially recovers the p -adicétale cohomology of the generic fiber. Construction 7.2 (Constant sheaves in the arc p -topology) . Given an abelian group M , we canconsider the associated constant sheaf of abelian groups in the arc p -topology; its value on a de-rived p -complete ring R is given by H (Spec( R [1 /p ]) , M ) . To see this, we observe that the R H (Spec( R [1 /p ]) , M ) is a sheaf in the arc p -topology on derived p -complete rings [BM18,Cor. 6.17] (which assumes M torsion; however, this is not necessary since we are only working with H ). It admits a map from the constant presheaf M , and the kernel and cokernel vanish locally inthe arc p -topology. Construction 7.3 ( p -adically constant sheaves and Tate twists in the arc p -topology) . Consider thesheaf of rings Z p in the arc p -topology, defined as the p -completion of the constant sheaf associatedto Z p , or equivalently as the inverse limit of the constant sheaves Z /p n Z ; explicitly, Z p ( R ) = H p (Spec( R ) , Z p ) = H (Spec( R [1 /p ]) , Z p ) . We construct an invertible Z p -module Z p (1) as the arc p -sheafification of the presheaf R Z p (1) pre ( R ) := T p ( R × ) . To check that this is an invertiblemodule, we observe that Z p (1) pre (1)( R ) is an invertible Z p ( R ) -module whenever R is a productof absolutely integrally closed valuation rings of mixed characteristic (0 , p ) . Note that Z p (1) /p can equally be described as the arc p -sheaf associated to the presheaf R µ p ( R ) = R × [ p ] , since arc p -locally all elements have p th roots. Proposition 7.4 ( arc p -cohomology as p -adic vanishing cycles) . Let R be any derived p -completering. Then there is a natural equivalence (16) R Γ arc p (Spec( R ) , Z p ( i )) ≃ R Γ pro´et (Spec( R [1 /p ]) , Z p ( i )) , where Z p ( i ) on the right-hand-side refers to the usual Tate twist on the pro-étale site of Spec( R [1 /p ]) .Proof. We know that the right-hand-side is a D ( Z p ) ≥ -valued sheaf for the arc p -topology by [BM18,Cor. 6.17], and we obtain a map from the presheaf R T p ( R × ) ⊗ i to the right-hand-side. Sheafifyingin the arc p -topology and p -completing, we obtain a map from the left-hand-side to the right-hand-side as in (16). To see that (16) is an equivalence of arc p -sheaves, it suffices to check on productsof rank absolutely integrally closed, p -complete valuation rings of mixed characteristic as theseform a basis (cf. [BM18, Prop. 3.29]), which is handled by the next lemma. (cid:3) Lemma 7.5.
Let R be a product Q t ∈ T V t of absolutely integrally closed, p -complete valuation rings.Then:(1) H j pro´et (Spec( R [1 /p ]) , Z p ( i )) = 0 for j > .(2) The map T p ( R × ) → H (Spec( R [1 /p ]) , Z p (1)) is an isomorphism. N K (1) -LOCAL TR Proof.
It suffices to prove both claims after reducing modulo p . Consider the functor A F j ( A ) := H j (Spec( ˆ A p [1 /p ]; F p ( i )) . This functor commutes with finite products and filtered colimits by theGabber–Fujiwara theorem, cf. [BM18, Th. 6.11] for an account. To prove that F j ( A ) = 0 for j > which implies (1), it suffices as in [BM18, Cor. 3.17] to show that F vanishes on everyultraproduct of the { V t } for each ultrafilter on T . But these ultraproducts are all absolutelyintegrally closed, p -henselian valuation rings, whence (1). The claim (2) is proved similarly usingthe map A × [ p ] → H (Spec( A [1 /p ]) , F p (1)) . (cid:3) Next we consider the cohomology of the structure presheaf.
Construction 7.6 ( arc p -cohomology of the structure presheaf) . For a derived p -complete commu-tative ring R , we let R Γ arc p (Spec( R ) , O ) denote the arc p -cohomology of the structure presheaf.The arc p -topology restricts to a topology on the opposite of Perfd . Our first goal is to identifyconcretely arc p -cohomology on Perfd . Construction 7.7 (Saturation) . Let R be a perfectoid ring. We have the natural quotient R → R ′ := R/ rad(( p )) , which is the universal map from R to a perfect F p -algebra. Note that R ′ ⊗ LR R ′ is discrete (and equivalent to R ′ ) since relative tensor products of perfectoid rings are p -completelydiscrete (and perfectoid).Let J = rad( p ) . We have J ⊗ LR J ∼ −→ J . It follows that one is in the setup of almost math-ematics, and we have a derived almostification functor ( − ) ∗ : D ( R ) → D ( R ) , which is also givenby R Hom R ( J, − ) . We claim that in fact J can be made explicit, and in particular that it hasprojective dimension ≤ , so that for any discrete R -module M , M ∗ ∈ D ( R ) [0 , . Indeed, thereexists an element ω ∈ R such that ω is a unit multiple of p and such that ω admits a compatiblesystem of p -power roots { ω /p n } n ≥ , cf. [BMS18, Lem. 3.9]. We claim that as an R -module, thereis an equivalence(17) J ≃ lim −→ (cid:18) R ω − /p → R ω /p − /p → . . . (cid:19) . To see this, we observe first that the filtered colimit on the right-hand-side is a submodule of R given by the ideal J ′ := S n ( ω /p n ) (i.e., given by multiplication by ω /p n on the n th term); we cansee this by arc ˆ p -descent to reduce to the case where R is a product of rank ≤ -valuation rings,in which case the claim is clear. Now clearly J ′ ⊂ J and R/J ′ is a ring of characteristic p onwhich the Frobenius is surjective. To obtain J ′ = J , we use [BMS18, Lem. 3.10] to see that theFrobenius induces an isomorphism R/ω /p n ∼ −→ R/ω /p n − for n ≥ . This implies that R/J ′ isperfect, whence J = J ′ as desired. Definition 7.8 (Spherically complete fields) . We recall (cf. [vR78, Ch. 4] for an account) that anonarchimedean field is called spherically complete if every decreasing sequence of closed disks hasnonempty intersection; in particular, any such field is complete. Any nonarchimedean field admitsan extension which is spherically complete and algebraically closed.If C is spherically complete and O C ⊂ C is the ring of integers, then any diagram { M i } i ≥ ofcyclic torsion-free O C -modules has vanishing lim ; conversely, this is equivalent to the definitionof spherical completeness. In fact, we may assume the { M i } i ≥ form a descending sequence ofideals { I i } i ≥ ⊂ O C ; using the short exact sequence of inverse systems → { I i } i ≥ → {O C } i ≥ →{O C /I i } i ≥ → (with the middle sequence constant), we see that it suffices to show that O C → lim ←− i O C /I i is surjective. But this is precisely the definition of spherical completeness. Proposition 7.9 ( arc p -cohomology of perfectoids) . The functor R R Γ arc p (Spec( R ) , O ) re-stricted on Perfd agrees with R R ∗ . In particular, for R ∈ Perfd , R Γ arc p (Spec( R ) , O ) ∈ D ( R ) [0 , .Proof. Let R be a perfectoid ring. First, we observe that arc p -cohomology of R with O -coefficientscan be calculated either on all derived p -complete rings or the subcategory of perfectoid rings(endowed with the arc p -cohomology). This is a general argument following from the fact thatperfectoid rings form a basis for the arc p -topology, cf. Proposition A.6 below.Now it suffices to prove that R R ∗ is a D ( Z ) ≥ -valued arc p -sheaf on Perfd and that the map R → R ∗ is locally an equivalence. The fact that it is an arc p -sheaf follows because the structurepresheaf is a D ( Z ) ≥ -valued sheaf on the arc ˆ p -topology on Perfd and R R ∗ annihilates perfect F p -algebras. The fact that R → R ∗ is locally an equivalence follows by taking an arc p -cover bya product of rings of integers in various spherically complete, algebraically closed nonarchimedeanfields of mixed characteristic; for such rings, R = R ∗ . (cid:3) Let R be a perfectoid base ring, and let R be a derived p -complete R -algebra. One has theconstruction of the perfectoidization R perfd of R , a coconnective E ∞ -algebra under R which hasthe property that if R is semiperfectoid, then R perfd is discrete and is the universal perfectoid ringthat R maps to. As shown in [BS19, Sec. 8], R R perfd is the arc ˆ p -cohomology of the structurepresheaf on the category of derived p -complete R -algebras. Proposition 7.10 ( arc p -cohomology as saturated perfectoidization) . For R an R -algebra which isderived p -complete, we have a natural equivalence of E ∞ -algebras in D ( R ) , R Γ arc p (Spec( R ) , O ) ≃ ( R perfd ) ∗ (where almost mathematics is taken relative to R and the ideal rad( p ) ).Proof. This is proved similarly as in Proposition 7.9. We have a natural (in R ) map R → ( R perfd ) ∗ .To show that it is the arc p -sheafification, it suffices to show that the codomain is an arc p -sheaf, andthat the map R → ( R perfd ) ∗ has cofiber which vanishes locally in the arc p -topology; note also thatwe can compute the arc p -cohomology either over all derived p -complete rings or over R -algebras(Example A.7). The codomain is an arc ˆ p -sheaf because of the identification of perfectoidizationwith arc ˆ p -cohomology [BS19, Sec. 8], and hence is an arc p -sheaf since it annihilates F p -algebras.The cofiber of R → ( R perfd ) ∗ vanishes locally in the arc p -topology, as one sees by working withperfectoid R and using the argument of Proposition 7.9. (cid:3) Example 7.11.
Suppose R is the p -completion of a ring which is integral over the perfectoidring R . Then the perfectoidization R perfd is discrete [BS19, Th. 10.11]. It thus follows that R Γ arc p (Spec( R ) , O ) ∈ D ( R ) [0 , . Construction 7.12 (Witt vector cohomology in the arc p -topology) . We consider the presheaf W ( O ) given by R W ( R ) and its arc p -cohomology (with Tate twists) R Γ arc p (Spec( R ) , W ( O )( i )) .Since the Witt vector functor is endowed with Frobenius and Verschiebung operators, so is theconstruction R R Γ arc p (Spec( R ) , W ( O )( i )) .In our setting, we can think of the Witt vector cohomology considered above as a one-parameter(along V ) deformation of the structure presheaf cohomology, especially in light of the next result. Proposition 7.13.
For any ring R , R Γ arc p (Spec( R ) , W ( O )( i )) is p -complete and complete withrespect to the Verschiebung.Proof. By base-changing to Z p [ ζ p ∞ ] , we may assume without loss of generality that i = 0 . Thenthis follows from Proposition A.10 below, since the presheaf W ( O ) commutes with finite productsis complete with respect to ( p, V ) . (cid:3) N K (1) -LOCAL TR Corollary 7.14.
Let R be finite and finitely presented over a perfectoid ring. Then R Γ arc p (Spec( R ) , W ( O )( i )) ∈ D ( Z p ) [0 , for any i .Proof. By p -completeness and V -completeness (Proposition 7.13), it suffices to prove the analogousstatement for R Γ arc p (Spec( R ) , O ( i ) /p ) . Replacing R with R [ ζ p ] and using ( Z /p ) × -Galois descentalong this extension, we may assume that R contains a primitive p th root of unity. This lets usreduce to the case i = 0 , whence the result follows from Example 7.11. (cid:3) We note finally that one can recover the p -adic nearby cycles (cf. Proposition 7.4) as the fixedpoints of Frobenius on Witt vector arc p -cohomology. This will not be used in the sequel. Proposition 7.15.
For any ring R , there is a natural equivalence for each i , (18) R Γ arc p (Spec( R ) , Z p ( i )) ≃ R Γ arc p (Spec( R ) , W ( O )( i )) F =1 . Proof.
We have a natural map Z p → W ( O ) F =1 of presheaves. Twisting and sheafifying in the arc p -topology, we obtain the map from left to right in (18). To see that it is an equivalence, it suffices tocheck that it is an equivalence on products (over some indexing set T ) of rings of the form O C , for C spherically complete and algebraically closed of mixed characteristic (0 , p ) ; now we can trivializethe Tate twists, so can assume i = 0 . Using Lemma 7.5, we find that the left-hand-side is discreteand simply given by Q T Z p . The right-hand-side is also given by Q T Z p by Lemma 7.16 below. (cid:3) Lemma 7.16.
Let C be an algebraically closed complete nonarchimedean field. Then the naturalmap Z p → W ( O C ) F =1 is an equivalence.Proof. It suffices to work modulo p , i.e., to show that the natural map F p → ( W ( O C ) /p ) F =1 isan equivalence. But the Witt vector Frobenius reduces modulo p to the ordinary Frobenius on thering W ( O C ) /p . By the Artin–Schreier sequence, ( W ( O C ) /p ) F =1 is the mod p étale cohomology of W ( O C ) /p . This is unchanged by taking the quotient by the ideal V W ( O C ) /p , which squares tozero; therefore, it is also the étale cohomology of O C /p , or equivalently of the residue field k ; thisin turn is F p as C is algebraically closed and hence so is k . (cid:3) THH C pr of perfectoid rings. Here we calculate the fixed points of
THH of perfectoid rings,generalizing the main result of [Hes06]; these results are known to experts. Recall that for any p -complete ring R , we have π TR( R ; Z p ) ≃ W ( R ) , cf. [HM97, Th. F].Our strategy is to treat the case of a p -torsionfree perfectoid ring by a direct spectral sequenceargument and appeal to known results for a perfect F p -algebra; we will glue both cases togetherusing an excision argument. To begin with, we consider the characteristic p case. For perfect fields,this result appears in [HM97, Sec. 5]. Lemma 7.17.
Let R be a perfect F p -algebra. Then:(1) π ∗ (THH( R ; Z p ) C pr ) ≃ W r +1 ( R )[ σ r ] for | σ r | = 2 .(2) TR( R ; Z p ) ≃ HW ( R ) .Proof. The Segal conjecture holds for R : in fact, ϕ : THH( R ; Z p ) → THH( R ; Z p ) tC p is ( − -truncated, cf. [BMS19, Sec. 6]. Therefore, we have that THH( R ; Z p ) C pr → THH( R ; Z p ) hC pr ϕ hCpr −−−−→ THH( R ; Z p ) tC pr +1 are equivalences on connective covers, cf. [NS18, Cor. II.4.9]. Using the homo-topy fixed point spectral sequence (or [BMS19, Sec. 6]), we find π ∗ (THH( R ; Z p ) tS ) ≃ W ( R )[ σ ± ] .Now THH( R ; Z p ) tC pr +1 ≃ THH( R ; Z p ) tS /p r +1 by [NS18, Lem. IV.4.12]. Combining these factsthe first claim follows as W r +1 ( R ) = W ( R ) /p r +1 . For the second claim, note that
THH( R ) is a module in CycSp over K ( F p ; Z p ) ≃ H Z p . By Propo-sition 5.7, we have that TR( R ; Z p ) → L K (1) TR( R ; Z p ) is -truncated; since the target vanishes, itfollows that TR( R ; Z p ) is -truncated, whence the result as we know π TR( R ; Z p ) . (cid:3) In the following, we will say that an E ∞ -ring A is weakly even periodic if the odd homotopygroups of A vanish, and if for all m, n ∈ Z , the map π m ( A ) ⊗ π ( A ) π n ( A ) → π n + m ) ( A ) is anisomorphism; in particular, π ( A ) is an invertible π ( A ) -module, and Zariski locally π ∗ ( A ) is aLaurent polynomial algebra over π ( A ) on a degree two class. Lemma 7.18.
Let
A, B, C be weakly even periodic E ∞ -rings and fix maps A → C, B → C suchthat π ( A ) → π ( C ) is surjective. Then A × C B is weakly even periodic.Proof. It follows that π ∗ ( A ) ⊗ π ( A ) π ( C ) ∼ −→ π ∗ ( C ) and similarly π ∗ ( A ) ⊗ π ( A ) π ( B ) ∼ −→ π ∗ ( B ) .Now since π ( A × C B ) , π ( A ) , π ( B ) , π ( C ) form a Milnor square, the category of finitely generatedprojective π ( A × C B ) -modules is the homotopy pullback of the categories of finitely generated π ( A ) -modules, π ( B ) -modules, and π ( C ) -modules, cf. [Mil71, Sec. 2]. The result now follows. (cid:3) Construction 7.19 ( p -torsionfree quotients of perfectoid rings) . Let R be a perfectoid ring, andlet I ⊂ R be the ideal of p -power torsion. Then pI = 0 , cf. [BMS19, Prop. 4.19].We have in fact that R/I is perfectoid as well. To see this, we use the theory of perfectoidizations[BS19, Sec. 7–8]. Let R ′ = ( R/p ) perf . For every p -complete valuation ring V with a map R → V which does not annihilate I ⊂ R , we find that V has characteristic p and we obtain a uniqueextension R ′ → V . Therefore, applying [BS19, Cor. 8.11], we obtain a pullback square of perfectoidrings,(19) R (cid:15) (cid:15) / / R ′ (cid:15) (cid:15) ( R/I ) perfd / / ( R ′ /I ) perfd . Since the vertical arrows are surjective (cf. [BS19, Th. 7.4]), this is a Milnor square of rings. Sincethe terms on the right side are both F p -algebras, it follows that the kernel of R → ( R/I ) perfd isannihilated by p ; since it contains I , it must be equal to I and we have ( R/I ) perfd = R/I .In particular, this shows that any perfectoid ring naturally fits into a Milnor square involving a p -torsionfree perfectoid ring and a map of perfect F p -algebras. The diagram (19) is in addition ahomotopy pushout square of E ∞ -rings, i.e., there are no higher Tor terms, since everything involvedis perfectoid. It follows by [LT19] that (19) induces a pullback after applying any localizing invariant.
Proposition 7.20 ( TR n of perfectoid rings) . Let R be a perfectoid ring. Then for each r ≥ , wehave:(1) π ∗ THH( R ; Z p ) C pr is concentrated in even degrees.(2) π THH( R ; Z p ) C pr is an invertible module over π TR r ( R ; Z p ) ≃ W r +1 ( R ) .(3) The multiplication map Sym t ( π THH( R ; Z p ) C pr ) → π t THH( R ; Z p ) C pr is an isomorphismfor all t ≥ .Proof. Recall [BMS19, Sec. 6] that we have a noncanonical isomorphism π ∗ THH( R ; Z p ) tC p ≃ R [ σ ± ] and that ϕ : THH( R ; Z p ) → THH( R ; Z p ) tC p exhibits the source as the connective cover of thetarget. Therefore, we have that THH( R ; Z p ) C pr → THH( R ; Z p ) hC pr ϕ hCpr −−−−→ THH( R ; Z p ) tC pr +1 areequivalences on connective covers, cf. [NS18, Cor. II.4.9]. N K (1) -LOCAL TR Our next claim is that
THH( R ; Z p ) tC pr +1 is a weakly even periodic E ∞ -ring for any r ≥ . In case R is p -torsionfree, this follows from the (degenerate) Tate spectral sequence applied to THH( R ; Z p ) ,whose homotopy groups form a polynomial algebra over R on a class in degree . In case R is an F p -algebra, the claim also follows from Lemma 7.17. We now treat the case of a general perfectoid R . In view of the Milnor square (19) which is also a homotopy pushout of E ∞ -rings and the mainresult of [LT19], we find that THH( − ; Z p ) tC pr +1 carries (19) to a pullback of E ∞ -ring spectra.Moreover, on π ( − ) this square yields a Milnor square since π THH( − ; Z p ) tC pr +1 ≃ W r +1 ( − ) forperfectoids via the previous paragraph and [HM97, Th. F]. Via Lemma 7.18, the claim of weak evenperiodicity follows.Now the above paragraphs show that THH( R ; Z p ) C pr is the connective cover of a weakly evenperiodic E ∞ -ring, and we already know its π is given by W r +1 ( R ) , whence the result. (cid:3) Description of TR of perfectoids. Let C be spherically complete and algebraically closedof mixed characteristic (0 , p ) . Then we recall some of the additional rings attached to C . We havethe tilt C ♭ , the ring of integers O C ♭ ⊂ C ♭ , the Fontaine ring A inf = A inf ( O C ) = W ( O C ♭ ) . Choosinga compatible system (1 , ζ p , ζ p , . . . ) of primitive p n th roots in C , we obtain an element ǫ ∈ O C ♭ andlet [ ǫ ] denote the corresponding element of A inf . We have the map θ : A inf → O C , which exhibitsthe source as the universal p -adically pro-nilpotent thickening of the latter; in particular, this leadsto a map A inf → W ( O C ) , which is a surjection in this case [BMS18, Lem. 3.23]. Lemma 7.21 ( TR in the spherically complete case) . Let C be a spherically complete, algebraicallyclosed nonarchimedean field of mixed characteristic (0 , p ) and let O C ⊂ C be the ring of integers.Let β ∈ π TR( O C ; Z p ) be the Bott element (arising from the image of the cyclotomic trace). Then TR ∗ ( O C ; Z p ) ≃ W ( O C )[ β ] .Proof. Since everything is p -complete, it suffices to see that TR( O C ; F p ) is -truncated. Now foreach n , each homotopy group in π ∗ (TR n ( O C ; F p )) is a cyclic module over W ( O C ) /p (Proposi-tion 7.20), which in turn is a quotient of A inf ( O C ) /p = O C ♭ . Moreover, the homotopy groups areconcentrated in even degrees.We claim that for any N -indexed system of cyclic O C ♭ -modules { M i } i ≥ , one has lim M i = 0 .In fact, writing { M i } as the cokernel of a map { N i } i ≥ → { N ′ i } i ≥ of inverse systems with the N i , N ′ i cyclic and torsion-free (so levelwise isomorphic to O C ♭ ), we reduce to the case where the M i ≃ O C ♭ . But then the vanishing of the lim terms is equivalent to the definition of sphericalcompleteness, and C ♭ is spherically complete, cf. the proof of [BMS18, Lem. 3.23].Using the previous paragraph and the Milnor exact sequence, it follows that π ∗ (TR( O C ; F p )) isconcentrated in even degrees. By Proposition 5.10, we also find that the cofiber of β is -truncated,so π ∗ (TR( O C ; F p ) /β ) is concentrated in degrees and , and is given by W ( O C ) in degree zero.It thus suffices to show that the zeroth Postnikov section TR( O C ; Z p ) /β → HW ( O C ) induces anisomorphism on π ( − ) tS . But we have by [AN18, Cor. 10], TR( O C ; Z p ) tS = THH( O C ; Z p ) tS ,and on homotopy groups this is A inf [ σ ± ] (cf. [BMS19, Sec. 6]). Moreover, with respect to the aboveidentifications, we have that β is a Z × p -multiple of ([ ǫ ] − σ , cf. [HN19, Th. 1.3.6]. Therefore, wehave π (TR( O C ; Z p ) /β ) tS ≃ A inf / ([ ǫ ] − σ ± ] , whence the result since A inf / ([ ǫ ] − ∼ −→ W ( O C ) by [BMS18, Lem. 3.23]. (cid:3) Proposition 7.22 ( TR of perfectoid rings) . For R ∈ Perfd and for i > , we have natural equiva-lences (20) τ [2 i − , i ] TR( R ; Z p ) ≃ R Γ arc p (Spec( R ) , W ( O )( i ))[2 i ] . For R ∈ Perfd , we have for i ∈ Z , (21) τ [2 i − , i ] (cid:0) L K (1) TR( R ; Z p ) (cid:1) ≃ R Γ arc p (Spec( R ) , W ( O )( i ))[2 i ] . Proof.
Taking the inverse limit in Proposition 7.20 over the restriction maps (and accounting forpossible lim terms), we find in particular that R τ [2 i − , i ] TR( R ; Z p ) defines an arc -sheaf ofspectra on Perfd . Since for i > , this functor annihilates perfect F p -algebras, it in fact defines an arc p -sheaf on Perfd .The cyclotomic trace gives a map of Z p -modules T p ( R × ) → π TR( R ; Z p ) . Since π (TR( R ; Z p )) ≃ W ( R ) , we obtain natural maps f i : W ( R )( i ) → π i (TR( R ; Z p )) for i > . To complete the proof, itsuffices to show that:a) f i is an isomorphism for a cofinal (in the arc p -topology) collection of R ∈ Perfd .b) π i − (TR( R ; Z p )) = 0 for a cofinal (in the arc p -topology) collection of R ∈ Perfd .We will take this collection to be the set of products of perfectoid rings of the form O C , for C a spherically complete and algebraically closed nonarchimedean field of mixed characteristic (0 , p ) . First, the above descriptions show that the construction R THH( R ; Z p ) commutes withproducts for R ∈ Perfd ; inductively and taking the limit we find that R TR( R ; Z p ) commuteswith products for R ∈ Perfd . Thus, it suffices to verify the above claims for R = O C itself, andthat follows from Lemma 7.21.For the claim about L K (1) TR( R ; Z p ) , note first that Proposition 5.10 shows easily that R L K (1) TR( R ; Z p ) defines an arc p -sheaf on Perfd (cf. also the remarks at the beginning of the proof ofProposition 5.10, concerning when L K (1) ( − ) commutes with homotopy limits). It then follows fromthe above (by inverting the Bott element over perfectoid rings containing p -power roots of unity) thatthe homotopy groups of the arc p -sheaf L K (1) TR( R ; Z p ) on Perfd are given by π i ≃ W ( O )(2 i ) , andwe obtain a filtration with associated gradeds the right-hand-side of (21) via the Postnikov filtrationas arc p -sheaves. Note that the Postnikov filtration is always exhaustive, and it is complete because itis complete on the arc p -sheaf R τ ≥ L K (1) TR( R ) ≃ τ ≥ TR( R ; Z p ) . Since the relevant associatedgraded pieces are in homological degrees [2 i − , i ] by Corollary 7.14, the result follows. (cid:3) The main results.
In this subsection, we prove Theorem 1.8 from the introduction. Ourstrategy is to compare L K (1) TR( − ) with its arc p -hypersheafication. Construction 7.23 (The invariant (cid:0) L K (1) TR( − ) (cid:1) ♯ ) . We define the functor ( L K (1) TR( − )) ♯ onderived p -complete rings as the arc p -hypersheafification of the functor L K (1) TR( − ) on derived p -complete rings. We have a comparison map(22) L K (1) TR( − ) → (cid:0) L K (1) TR( − ) (cid:1) ♯ . For a ring which is not necessarily derived p -complete, we define (cid:0) L K (1) TR( − ) (cid:1) ♯ as that of itsderived p -completion.To analyze this construction, we will use the results about L K (1) TR of perfectoids proved inthe previous section, as well as some general tools from the appendix. On Perfd , Proposition 7.22implies that L K (1) TR( − ) defines an arc p -hypersheaf. Using Proposition A.6, it follows that therestriction of (cid:0) L K (1) TR( − ) (cid:1) ♯ to Perfd is just L K (1) TR( − ) again, i.e., (22) is an equivalence. Alternatively, one could build (cid:0) L K (1) TR( − ) (cid:1) ♯ using the unfolding construction of Example A.9, by starting withthe arc p -hypersheaf L K (1) TR( − ) on Perfd . N K (1) -LOCAL TR Proposition 7.24.
For any ring R , there is a natural, complete exhaustive Z -indexed filtration on (cid:0) L K (1) TR( R ) (cid:1) ♯ , denoted Fil ≥∗ (cid:0) L K (1) TR( R ) (cid:1) ♯ , with associated graded terms gr i (cid:0) L K (1) TR( R ) (cid:1) ♯ ≃ R Γ arc p (Spec( R ) , W ( O )( i ))[2 i ] .Proof. The filtration in question is the arc p -Postnikov filtration (on the derived p -completion of R ). Note that Postnikov towers converge for hypercomplete arc p -sheaves (Proposition A.10). Forthe identification of the graded pieces (or equivalently, the sheafified homotopy groups), it sufficesby descent (cf. Example A.9) to work with the subcategory Perfd , where the result follows from(21). (cid:3)
We do not know in general for which R the comparison map (22) is an equivalence; for such R ,one obtains a “motivic” filtration on L K (1) TR( R ) with associated graded terms the arc p -cohomologycomplexes R Γ arc p (Spec( R ) , W ( O )( i ))[2 i ] . Here we will show that the comparison map is an equiv-alence in certain formally smooth cases, using the pro-Galois descent result (Theorem 6.8). Theorem 7.25.
Let R be a p -torsionfree perfectoid ring. Let R be a formally smooth p -complete R -algebra. Then L K (1) TR( R ) ∼ −→ ( L K (1) TR( R )) ♯ .Proof. First, we reduce to the case where R admits a compatible system of p -power roots of unity.By André’s lemma [BS19, Th. 7.12], we know that there exists a p -completely flat perfectoid R -algebra R ′ which has this property (e.g., is absolutely integrally closed). Now we have by descent(23) TR( R ; Z p ) ≃ Tot(TR( R ⊗ R R ′ ; Z p ) ⇒ TR( R ⊗ R R ′ ⊗ R R ′ ; Z p ) →→→ . . . ) . Since in high degrees all of the terms in the above totalization agree with their K (1) -localizationby Proposition 5.10, it follows that the descent property (23) holds for L K (1) TR( − ) as well (cf. thebeginning of the proof of loc. cit. ). Moreover, ( L K (1) TR( − )) ♯ satisfies descent for the map R → R ⊗ R R ′ by construction. Therefore, we reduce to the case where R contains \Z p [ ζ p ∞ ] .Working locally on Spf( R ) , we can assume that R receives a map from R (cid:10) t ± , . . . , t ± n (cid:11) whichis étale mod p . We consider the extension R ∞ = R ⊗ Z [ ζ p ∞ ,t ± ,...,t ± n ] Z [ ζ p ∞ , t ± /p ∞ , . . . , t ± /p ∞ n ] andthe evident Z p (1) n -action on R ∞ . As in Example 6.10, we find from Theorem 6.8 that the naturalmap induces an equivalence L K (1) TR( R ) ∼ −→ Tot (cid:16) L K (1) TR( R ∞ ) ⇒ L K (1) TR(Fun cts ( Z p (1) n , R ∞ )) →→→ . . . (cid:17) . Now this is also true for ( L K (1) TR( − )) ♯ , because the above augmented cosimplicial ring is an arc p -hypercover (strictly speaking, for that we replace all rings by their derived p -completions). Butnow this gives L K (1) TR( R ) ∼ −→ ( L K (1) TR( R )) ♯ , since they agree on perfectoids and every term inthe above cosimplicial resolution has perfectoid p -completion. (cid:3) Theorem 7.26.
Suppose R is a formally smooth O K -algebra, where K is a complete discretelyvalued field of mixed characteristic (0 , p ) whose residue field k satisfies [ k : k p ] < ∞ . Then L K (1) TR( R ) ∼ −→ (cid:0) L K (1) TR( R ) (cid:1) ♯ .Proof. We let G = Gal( K/K ) and consider the G -action on S = R ⊗ O K O K . As in Example 6.9,our assumptions imply that G has finite cohomological dimension and moreover that the map (13)is an equivalence. Therefore, it suffices to show that L K (1) TR( A ) ∼ −→ ( L K (1) TR( A )) ♯ when A is oneof S, Fun cts ( G, S ) , . . . . But these are all (up to p -completions) formally smooth over perfectoids,whence the claim by Theorem 7.25 and descent. (cid:3) Remark 7.27.
Suppose L K (1) TR( R ) ∼ −→ (cid:0) L K (1) TR( R ) (cid:1) ♯ . On Frobenius fixed points, one recoversthe Thomason [Tho85, TT90] filtration (i.e., the pro-étale Postnikov filtration) on L K (1) TC( R ) ≃ L K (1) K ( R [1 /p ]) (cf. [BCM20] for this identification), whose associated gradeds are given by gr i ≃ R Γ pro´et (Spec( R [1 /p ]) , Z p ( i ))[2 i ] , cf. Proposition 7.15. Remark 7.28.
The above filtration on L K (1) TR( − ) and the calculations of TR of smooth algebrasover a DVR in [HM03, HM04, GH06] suggest that the cohomology R Γ arc p (Spec( R ) , W ( O )( i )) shouldbe related to the absolute de Rham–Witt complex. The work [Mor18] also suggests that for smoothalgebras over a perfectoid base containing all p -power roots of unity, R Γ arc p (Spec( R ) , W ( O )) shouldbe related to the relative de Rham–Witt complex (over the perfectoid base). Appendix A. Topological preliminaries
In this appendix, we record some basic topological preliminaries about (hyper)sheaves of spectra.
Remark A.1 (Conventions for sites) . We will for simplicity work only with sites of the followingnice form (cf. [Lur18, Sec. A.3.2]). Let C be a category with pullbacks and finite coproducts, suchthat coproducts distribute over pullbacks and are disjoint. Suppose C is equipped with a class ofmorphisms S = S C which contains all equivalences and is stable under composition and pullback.We equip C with the Grothendieck topology where a collection { X i → X } i ∈ I is a covering if thereexists a finite subset I ′ ⊂ I such that F i ∈ I ′ X i → X can be refined by a map belonging to S . Inthis case, a presheaf on C with values in an ∞ -category D with all small limits is a sheaf if and onlyif it carries finite coproducts in C to finite products in D and if it satisfies Čech descent for mapsin S , cf. [Lur18, Sec. A.3.3]. Example A.2. (1) The small or big étale site of a qcqs scheme (where we only allow qcqsschemes) is an example, with S the class of étale surjections.(2) The arc p -topology or arc ˆ p -topology on the opposite of the category of derived p -completerings.(3) The arc p -topology or arc ˆ p -topology on the opposite of the category of perfectoid rings. Remark A.3 (Examples of continuous functors) . Let C , C ′ be sites as in Remark A.1. Let u : C ′ → C be a functor which preserves finite coproducts and pullbacks, as well as morphisms in the respectiveclasses S C , S C ′ . It follows that if F is a sheaf (with values in any ∞ -category D with all small limits)on C , then F ◦ u is a sheaf on C ′ . These are examples of continuous functors; the notion can bedefined for more general sites, cf. [AGV72, Exp. III.1]. Construction A.4 (Sheaves of spectra) . Given a site C (as in Remark A.1), we let PSh( C , Sp) denote the ∞ -category of presheaves of spectra on C and Shv( C , Sp) ⊂ PSh( C , Sp) denote thesubcategory of sheaves of spectra [Lur18, Sec. 1.3]. We equip these both with their canonical t -structures, cf. [Lur18, Sec. 1.3.2].Given u : C ′ → C as in Remark A.3, we obtain a right adjoint and left t -exact functor ( − ) ◦ u : Shv( C , Sp) → Shv( C ′ , Sp) . It has a left adjoint u ! : Shv( C ′ , Sp) → Shv( C , Sp) given by left Kanextension along u followed by sheafification. By adjunction, necessarily u ! is right t -exact. Construction A.5 (Hypersheaves of spectra) . Let C be a site as in Remark A.1. Any presheaf F of spectra on C fits into a unique fiber sequence of spectra(24) F null → F → F ♯ where F null has trivial sheafified homotopy groups and F ♯ is a hypercomplete sheaf of spectra, i.e.,for every presheaf G with trivial sheafified homotopy groups, we have Hom( G , F ♯ ) = 0 . If F is a N K (1) -LOCAL TR sheaf of spectra, then the cofiber sequence shows that F null is also a sheaf of spectra; it is then ∞ -connective with respect to the t -structure on Shv( C , Sp) . We let
Shv hyp ( C , Sp) ⊂ Shv( C , Sp) denote the full subcategory of hypercomplete sheaves. We refer to [CM19, Sec. 2] for an expositionof some of these constructions, which go back to [Jar87, DHI04].
Proposition A.6.
Let u : C ′ → C be a morphism of sites as in Remark A.1 preserving finitecoproducts and pullbacks and carrying the class of arrows S C ′ into S C . Suppose that for any object X ′ ∈ C ′ and a finite covering family { Y i → u ( X ′ ) } i ∈ I in C , there is a refinement which is the imageunder u of a finite covering family { Y ′ i → X ′ } i ∈ I in C ′ .Then the restriction functor ( − ) ◦ u : PSh( C , Sp) → PSh( C ′ , Sp) commutes with hypersheafification(and in particular preserves hypercomplete sheaves).Proof.
Using the cofiber sequence (24), we see that it suffices to prove that ( − ) ◦ u : Shv( C , Sp) → Shv( C ′ , Sp) preserves both the subclasses of objects with trivial homotopy groups and hypercompleteobjects.Our hypotheses imply that if a presheaf of abelian groups on C has trivial sheafification, then itspullback to C ′ has trivial sheafification. Therefore, ( − ) ◦ u preserves objects with trivial homotopygroups. It thus suffices to show that if F ∈
Shv hyp ( C , Sp) , then the sheaf
F ◦ u ∈ Shv( C ′ , Sp) is alsohypercomplete; this will not use the assumption in the second sentence of the statement. Equiva-lently, given
G ∈
Shv( C ′ , Sp) which is ∞ -connective, it suffices to show that Hom
Shv( C ′ , Sp) ( G , F ◦ u ) =0 . By adjointness, this is Hom
Shv( C , Sp) ( u ! G , F ) ; since u ! : Shv( C ′ , Sp) → Shv( C , Sp) is right t -exactand therefore preserves ∞ -connective objects and F is hypercomplete, this is contractible. (cid:3) Example A.7 (Overcategories) . Suppose a ∈ C and C ′ = C /a is the overcategory of a , with theinduced topology. Then the natural forgetful functor C ′ → C clearly satisfies the conditions ofProposition A.6. In particular, the hypersheafification of a presheaf on C when restricted to C /a isthe hypersheafification of the restriction to C /a .A direct consequence is that given an appropriate site with a “basis,” hypersheaves of spectracan be entirely recovered from their values on the basis. The result is a direct analog of [AGV72,Th. 4.1, Exp. III], and appears (for sheaves of spaces) in [Aok20, App. A]. Proposition A.8.
Let C be a site as in Remark A.1. Let C ′ ⊂ C be a full subcategory closed underfinite coproducts and fiber products, and define S C ′ to be the intersection of S C with C ′ . Supposeevery object X ∈ C admits a map Y → X in S C with Y ∈ C ′ . Then the restriction functor Shv( C , Sp) → Shv( C ′ , Sp) restricts to an equivalence on hypercomplete objects.Proof.
Let u : C ′ ⊂ C be the inclusion, so we have a restriction functor ( − ) ◦ u : PSh( C , Sp) → PSh( C ′ , Sp) . By Proposition A.6, it restricts to a functor on hypercomplete sheaves, so we have ( − ) ◦ u : Shv hyp ( C , Sp) → Shv hyp ( C ′ , Sp) . This last functor has a left adjoint L : Shv hyp ( C ′ , Sp) → Shv hyp ( C , Sp) , given by
F 7→ L F := (Lan u F ) ♯ , i.e., L is obtained by applying the left Kan extension Lan u : PSh( C ′ , Sp) → PSh( C , Sp) followed by hypersheafification ( − ) ♯ . We now show that L is fullyfaithful. Indeed, we have for F , G ∈
Shv hyp ( C ′ , Sp) ,(25)
Hom
Shv hyp ( C , Sp) ( L F , L G ) = Hom Shv hyp ( C ′ , Sp) ( F , (Lan u G ) ♯ ◦ u ) . Now since u is fully faithful, since hypersheafification commutes with ( − ) ◦ u by Proposition A.6,and since G is already hypercomplete, we have (Lan u G ) ♯ ◦ u = G . Therefore, the right-hand-side of(25) simplifies to Hom
Shv hyp ( C ′ , Sp) ( F , G ) as desired. Our hypotheses imply that the restriction functor ( − ) ◦ u is conservative on hypercompletesheaves, because it is conservative on sheaves of abelian groups. Since the restriction functor isconservative and has a fully faithful left adjoint, the result follows. (cid:3) Example A.9 (Unfolding in the arc p -topology) . The natural forgetful functor establishes an equiv-alence of ∞ -categories between arc p -hypersheaves of spectra on all derived p -complete rings and arc p -hypersheaves of spectra on perfectoid rings.Finally, we include a basic observation about commuting hypersheafification and certain productswhen the site C is sufficiently large (e.g., the arc p -site). This fact is closely related to the theory ofreplete topoi, cf. [BS15, Sec. 3]. Compare [BS15, Prop. 3.3.3] for the second part of the next resultfor the derived category of abelian sheaves, or equivalently hypercomplete H Z -module sheaves ofspectra. Note in particular that it applies to the arc p -site. This follows because arc -covers in thesense of [BM18] are closed under filtered colimits of rings [BM18, Cor. 2.20] and because a mapof derived p -complete rings R → R ′ is an arc p -cover if and only if R → R ′ × R/p → R [1 /p ] is an arc -cover.In the following, we write PSh ⊔ ( C ) ⊂ PSh( C ) for the subcategory of presheaves which carry finitecoproducts to finite products. Proposition A.10.
Let C be a site as in Remark A.1. Suppose C admits countable filtered limits.Suppose moreover that if · · · → X i → X i − → · · · → X is a sequence of arrows in S , then lim ←− i X i → X belongs to S . Then:(1) The hypersheafification functor, ( − ) ♯ : PSh ⊔ ( C ) → Shv hyp ( C ) commutes with countable prod-ucts and limits along Z ≥ -indexed towers.(2) For any G ∈
Shv hyp ( C ) ⊂ Shv( C ) , the Postnikov tower of G (as a sheaf of spectra) converges.Proof. For (1), since hypersheaves are always closed (inside presheaves) under arbitrary limits andsince Z ≥ -indexed limits can be built from countable products, it suffices to show (via the uniquecofiber sequence (24)) that if {F i } i ∈ N is a countable family of presheaves in PSh ⊔ ( C ) with trivialhypersheafification (that is, trivial sheafified homotopy groups), then the product presheaf Q i ∈ N F i has trivial hypersheafification. But this is just a claim about the presheaves of abelian groups { π j ( F i ) } i ∈ N for each j . Explicitly, given a class α = ( α i ) ∈ Q i ∈ N π j ( F i ( X )) for some X ∈ C , wefind for each i a cover X i → X in S which annihilates α i ∈ π j ( F i ( X )) and then form the cover X × X X × X . . . of X , which annihilates α .Now let G be a hypercomplete sheaf of spectra on C . The Postnikov tower of G ∈
Shv( C ) isobtained by taking the presheaf truncations τ pre ≤ n G and applying the hypersheafification (or sheafi-fication, since these objects are truncated), i.e., one forms { ( τ pre ≤ n G ) ♯ } , which is a tower in PSh ⊔ ( C ) .Since Postnikov towers converge for presheaves of spectra, i.e., G ≃ lim ←− n τ pre ≤ n G , and we have justseen that ( − ) ♯ : PSh ⊔ ( C ) → Shv hyp ( C ) commutes with limits along Z ≥ -indexed towers, we find G ≃ lim ←− n ( τ pre ≤ n G ) ♯ as desired. (cid:3) References [AF15] David Ayala and John Francis,
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