Kaledin's degeneration theorem and topological Hochschild homology
aa r X i v : . [ m a t h . K T ] N ov KALEDIN’S DEGENERATION THEOREM AND TOPOLOGICALHOCHSCHILD HOMOLOGY
AKHIL MATHEW
Abstract.
We give a short proof of Kaledin’s theorem on the degeneration of the noncommu-tative Hodge-to-de Rham spectral sequence. Our approach is based on topological Hochschildhomology and the theory of cyclotomic spectra. As a consequence, we also obtain relative versionsof the degeneration theorem, both in characteristic zero and for regular bases in characteristic p . Introduction
Let X be a smooth and proper variety over a field k . A basic invariant of X arises from the algebraic de Rham cohomology , H ∗ DR ( X ), given as the hypercohomology of the complex Ω ∗ X ofsheaves of algebraic differential forms on X with the de Rham differential. Then H ∗ DR ( X ) is afinite-dimensional graded k -vector space, and is the abutment of the classical Hodge-to-de Rham spectral sequence H i ( X, Ω jX ) = ⇒ H i + j DR ( X ) arising from the naive filtration of the complex ofsheaves Ω ∗ X . It is a fundamental fact in algebraic geometry that this spectral sequence degenerateswhen k has characteristic zero. When k = C and X is K¨ahler, the degeneration arises from Hodgetheory.After 2-periodization and in characteristic zero, the above invariants and questions have non-commutative analogs, i.e., they are defined for more generally for differential graded (dg) categoriesrather than only for varieties. Let C be a smooth and proper dg category over a field k (e.g., C could be the derived category D b Coh( X ) of a smooth and proper variety X/k ). In this case, abasic invariant of C is given by the Hochschild homology
HH( C /k ), a noncommutative version ofdifferential forms for C . Hochschild homology takes values in the derived category D ( k ) of k -vectorspaces; it produces a perfect complex equipped with an action of the circle S , the noncommu-tative version of the de Rham diferential. As a result, one can take the S -Tate construction toform HP( C /k ) def = HH( C /k ) tS , called the periodic cyclic homology of C and often regarded as anoncommutative version of de Rham cohomology. One has a general spectral sequence, arisingfrom the Postnikov filtration of HH( C /k ), HH ∗ ( C /k )[ u ± ] = ⇒ HP ∗ ( C /k ), called the (noncommu-tative) Hodge-to-de Rham spectral sequence.
When C = D b Coh( X ) for X in characteristic zero,this reproduces a 2-periodic version of the Hodge-to-de Rham spectral sequence.The papers [Kal08, Kal16] of Kaledin describe a proof of the following result, conjectured byKontsevich and Soibelman [KS09, Conjecture 9.1.2]. Theorem 1.1 (Kaledin) . Let C be a smooth and proper dg category over a field k of characteristiczero. Then the Hodge-to-de Rham spectral sequence HH ∗ ( C /k )[ u ± ] = ⇒ HP ∗ ( C /k ) degeneratesat E . Date : November 15, 2017.
An equivalent statement is that the S -action on HH( C /k ), considered as an object of the derivedcategory D ( k ), is trivial; thus we may regard the result as a type of formality statement. Using thecomparison between 2-periodic de Rham cohomology and periodic cyclic homology in characteristiczero, one recovers the classical result that the (commutative) Hodge-to-de Rham spectral sequenceH i ( X, Ω jX ) = ⇒ H i + j dR ( X ) from Hodge cohomology to de Rham cohomology degenerates for asmooth proper variety X in characteristic zero.Kaledin’s proof of Theorem 1.1 is based on reduction mod p . Motivated by the approach ofDeligne-Illusie [DI87] in the commutative case, Kaledin proves a formality statement for Hochschildhomology in characteristic p of smooth and proper dg categories which satisfy an amplitude boundon Hochschild cohomology and which admit a lifting mod p . Compare [Kal16, Th. 5.1] and [Kal16,Th. 5.5].In this paper, we will give a short proof of the following slight variant of Kaledin’s characteristic p degeneration results. Analogous arguments as in [Kal08, Kal16] show that this variant also impliesTheorem 1.1. Theorem 1.2.
Let C be a smooth and proper dg category over a perfect field k of characteristic p >
0. Suppose that:(1) C has a lift to a smooth proper dg category over W ( k ).(2) HH i ( C /k ) vanishes for i / ∈ [ − p, p ].Then the Hodge-to-de Rham spectral sequence HH ∗ ( C /k )[ u ± ] = ⇒ HP ∗ ( C /k ) degenerates at E .We will deduce Theorem 1.2 from the framework of topological Hochschild homology and inparticular the theory of cyclotomic spectra as recently reformulated by Nikolaus-Scholze [NS17].We give an overview of this apparatus in Section 2. The idea of using cyclotomic spectra here is,of course, far from new, and is already indicated in the papers of Kaledin.Given C , one considers the topological Hochschild homology THH( C ) as a module over the E ∞ -ring THH( k ), whose homotopy groups are given by k [ σ ] for | σ | = 2. One has equivalences ofspectra:(1) THH( C ) /σ ≃ HH( C /k ).(2) THH( C )[1 /σ ] (1) ≃ HP( C /k ) for smooth and proper C /k . Here the superscript (1) denotesthe Frobenius twist.The first equivalence is elementary, while the second arises from the cyclotomic Frobenius and shouldcompare to the “noncommutative Cartier isomorphisms” studied by Kaledin. These observationsimply that the difference between 2-periodic Hochschild homology and periodic cyclic homology(i.e., differentials in the spectral sequence) is controlled precisely by the presence of σ -torsion inTHH ∗ ( C ). Under the above assumptions of liftability and amplitude bounds, the degenerationstatement then follows from an elementary argument directly on the level of THH. We formulatethis as a general formality statement in Proposition 3.9 below.We also apply our methods to prove freeness and degeneration assertions in Hochschild homologyfor families of smooth and proper dg categories. We first review the commutative version. If S is ascheme of finite type over a field of characteristic zero and f : X → S a proper smooth map, thenone knows by a classical theorem of Deligne [Del68] that the relative Hodge cohomology sheaves R i f ∗ Ω jX/S form vector bundles on S , and that the relative Hodge-to-de Rham spectral sequencedegenerates when S is affine. When S is smooth, this can be deduced by reduction mod p and arelative version of the Deligne-Illusie constructions as in [Ill90]. ALEDIN’S DEGENERATION THEOREM AND TOPOLOGICAL HOCHSCHILD HOMOLOGY 3
There are noncommutative versions of these relative results, too. For example, in characteristiczero, one has the following result.
Theorem 1.3.
Let A be a commutative Q -algebra and let C be a smooth proper dg category over A . Then:(1) The Hochschild homology groups HH i ( C /A ) are finitely generated projective A -modules.(2) The relative Hodge-to-de Rham spectral sequence degenerates.This result can be deduced from Kaledin’s theorem. When A is smooth at least, the freenessof HH i ( C /A ) follows from the existence of a flat connection on periodic cyclic homology, due toGetzler [Get93], together with Theorem 1.1. Compare also [KS09, Remark 9.1.4] for a statement.We will give a short proof inspired by this idea, in the form of the nilinvariance of periodic cyclichomology in characteristic zero and a K¨unneth theorem.In fact, we will formulate the argument as a general formality (and local freeness) criterion for S -actions via cyclotomic spectra. This includes the argument for Kaledin’s theorem as well asadditional input for the relative case. Formality criterion.
Let A be a commutative Q -algebra and let M ∈ Perf( A ) BS be a perfectcomplex of A -modules equipped with an S -action. Suppose that there exists a finitely generated Z -algebra R ⊂ A , a dualizable object M ′ in the ∞ -category Mod THH( R ) (CycSp) of THH( R )-modulesin cyclotomic spectra, and an equivalence M ≃ M ′ ⊗ THH( R ) A in Perf( A ) BS . Then the homologygroups of M are finitely generated projective A -modules and the S -action on M is trivial.In characteristic p , we can approach relative questions as well using the cyclotomic Frobenius,although our methods only apply when the base is smooth. Recent work of Petrov-Vaintrob-Vologodsky [PVV17] has obtained related statements using the methods of Kaledin and the Gauss-Manin connection in periodic cyclic homology. In particular, within the range [ − ( p − , ( p − Theorem 1.4 (Cf. also [PVV17, Theorem 1]) . Let A be a regular noetherian F p -algebra such thatthe Frobenius map A → A is finite. Let e A be a flat lift of A to Z /p . Let C be a smooth and properdg category over A . Suppose that:(1) C lifts to a smooth and proper dg category over e A .(2) HH i ( C /A ) = 0 for i / ∈ [ − ( p − , p − i ( C /A ) are finitely generated projective A -modules andthe relative Hodge-to-de Rham spectral sequence HH ∗ ( C /A )[ u ± ] = ⇒ HP ∗ ( C /A ) degenerates at E . Acknowledgments.
I would like to thank Mohammed Abouzaid, Benjamin Antieau, BhargavBhatt, Lars Hesselholt, Matthew Morrow, Thomas Nikolaus, Alexander Petrov, Nick Rozenblyum,Peter Scholze, and Dmitry Vaintrob for helpful discussions related to this subject. I would also liketo thank Benjamin Antieau for several comments on a draft. This work was done while the authorwas a Clay Research Fellow.
AKHIL MATHEW Cyclotomic spectra
Let C be a k -linear stable ∞ -category over a perfect field k of characteristic p >
0. A basicinvariant of C which we will use essentially in this paper is the topological Hochschild homology THH( C ). The construction THH( C ) is one of a general class of additive invariants of stable ∞ -categories, including algebraic K -theory, and about which there is a significant literature; comparefor example [BGT13].The construction C 7→
THH( C ) is naturally a functor to the homotopy theory of spectra. Bydefinition, THH( C ) is the Hochschild homology of C relative to the sphere spectrum rather than toan ordinary ring. As we show below, THH( C ) contains significant information about the Hochschildhomology HH( C /k ) and the spectral sequence for HP( C /k ). We begin by giving a brief overviewof the relevant structure in this case. Compare also the discussion in numerous other sources, e.g.,[Hes16, BM17, AMN17].A basic input here is the calculation in the case when C = Perf( k ), recalled below (cf. [HM97,Sec. 5]). Theorem 2.1 (B¨okstedt) . THH ∗ ( k ) ≃ k [ σ ] , | σ | = 2 . Theorem 2.1 shows that THH can be controlled in a convenient manner. A more naive variant ofthe construction
C 7→
THH( C ) is to consider the Hochschild homology HH( C / Z ) over the integers.Since (by a straightforward calculation) HH ∗ ( F p / Z ) ≃ Γ( σ ) is a divided power algebra on a degreetwo class, the construction of THH should be regarded as an “improved” version of Hochschildhomology over Z .We now describe more features of topological Hochschild homology. If C is a k -linear stable ∞ -category, then THH( C ) naturally acquires the structure of a module spectrum over the E ∞ -ringTHH( k ). The construction C 7→
THH( C ) yields a symmetric monoidal functor from k -linear stable ∞ -categories to THH( k )-module spectra. If C is smooth and proper over k , then THH( C ) is aperfect module over THH( k ). Furthermore, one has the relation(1) THH( C ) ⊗ THH( k ) k ≃ HH( C /k ) . As a result of (1), THH( C ) can be thought of as a one-parameter deformation of HH( C /k ) over theelement σ .In addition, THH( C ) inherits an action of the circle S . The circle also acts on THH( k ) (consid-ered as an E ∞ -ring spectrum), and THH provides a symmetric monoidal functor { k -linear stable ∞ -categories } → Mod
THH( k ) (Sp BS ) , i.e., into the ∞ -category of spectra with S -action equipped with a compatible THH( k )-action.Using this, one can define the following (which can be thought of as a noncommutative version ofcrystalline cohomology). Definition 2.2 (Hesselholt [Hes16]) . The periodic topological cyclic homology of C is given byTP( C ) = THH( C ) tS .A result of [BMS] (see also [AMN17, Sec. 3]) shows that TP provides a lift to characteristic zeroof the periodic cyclic homology HP( C /k ). For example, TP ∗ ( k ) ≃ W ( k )[ x ± ] for | x | = −
2, and ingeneral one has a natural equivalence of TP( k )-modules(2) TP( C ) ⊗ TP( k ) HP( k ) ≃ HP( C /k ) ≃ TP( C ) /p. In the rest of this paper, we will generally use the language of stable ∞ -categories [Lur14], and in particularwork with k -linear stable ∞ -categories rather than dg categories. We refer to [Coh16] for a comparison. ALEDIN’S DEGENERATION THEOREM AND TOPOLOGICAL HOCHSCHILD HOMOLOGY 5
The construction
C 7→
TP( C ) is another extremely useful invariant one can extract from thismachinery. It naturally provides a lax symmetric monoidal functor { k -linear stable ∞ -categories } → Mod
TP( k ) . At least for smooth and proper k -linear ∞ -categories, the construction TP is actually symmetricmonoidal, i.e., satisfies a K¨unneth theorem, by a result of Blumberg-Mandell [BM17] (see also[AMN17]).In (2), we saw that periodic cyclic homology can be recovered from TP by reducing mod p . Next,we show that we can reconstruct HP from THH in another way. Note first that there is a naturalmap of E ∞ -rings TP( k ) ≃ THH( k ) tS → THH( k ) tC p . Proposition 2.3.
For C a k -linear stable ∞ -category, one has an equivalence of TP( k )-modulespectra THH( C ) tC p ≃ TP( C ) ⊗ TP( k ) THH( k ) tC p ≃ HP( C /k ).For future reference, we actually prove a more general statement. Proposition 2.4.
Let X be an arbitrary object of the ∞ -category Mod THH( k ) (Sp BS ) of modulesover THH( k ) in the symmetric monoidal ∞ -category of spectra equipped with an S -action. Then the natural map of TP( k )-modules(3) X tS ⊗ TP( k ) THH( k ) tC p → X tC p . is an equivalence, and one has a natural equivalence of TP( k )-modules(4) X tS ⊗ TP( k ) THH( k ) tC p ≃ ( X ⊗ THH( k ) k ) tS . Proof.
To see this, we note that there is an S -equivariant map of E ∞ -rings Z → THH( F p ), e.g.,via the cyclotomic trace (cf. [NS17, IV.4]). One obtains a square of E ∞ -rings Z tS (cid:15) (cid:15) / / Z tC p (cid:15) (cid:15) TP( k ) / / THH( k ) tC p , which one easily checks to be a pushout square. Now the equivalence (3) follows from [NS17, LemmaIV.4.12]. To see (4), we use the fact that THH( k ) tC p ≃ TP( k ) /p as TP( k )-modules. This impliesthe result via the formula ( X ⊗ THH( k ) k ) tS ≃ X tS ⊗ TP( k ) k tS ≃ X tS /p , which holds because k = THH( k ) /σ belongs to the thick subcategory generated by the unit in Mod THH( k ) (Sp BS ) (andwhich is a generalization of (2)). (cid:3) In addition to the parameter σ arising from THH( k ), THH comes with another crucial feature:namely, it has the structure of a cyclotomic spectrum. The first feature of the cyclotomic structure isthe S -action on THH( C ). As explained in [NS17], the remaining datum of the cyclotomic structurecan be encoded in a “Frobenius map” (which does not exist for HH( C /k )) ϕ : THH( C ) → THH( C ) tC p , The map ϕ has the structure of an S -equivariant map: S acts on the source, S /C p acts onthe target, and S ≃ S /C p via the p th root. In [NS17], it is shown that in the bounded below(and p -local) setting, the entire datum of a cyclotomic spectrum (studied more classically using Compare the discussion in [AMN17] for a treatment.
AKHIL MATHEW techniques of equivariant stable homotopy theory [BHM93, BM15]) can be constructed from thecircle action and ϕ . Namely, they define: Definition 2.5 (Nikolaus-Scholze [NS17, Def. II.1.6]) . The presentably symmetric monoidal stable ∞ -category CycSp of cyclotomic spectra is the ∞ -category of tuples (cid:8) X ∈ Fun( BS , Sp) , ϕ p : X → X tC p (cid:9) p =2 , , ,... . That is, to specify an object of CycSp amounts to specifying an spectrum X with an action of S ,and S ≃ S /C p -equivariant Frobenius maps ϕ p : X → X tC p for each prime number p . (When X is p -local, we write ϕ = ϕ p and ϕ q = 0 for q = p .)The ∞ -category CycSp agrees with more classical approaches to cyclotomic spectra in thebounded-below case; see also [AMGR17]. Example 2.6 (Cf. [NS17, IV.4] and [HM97]) . Suppose C = Perf( k ). In this case, the map ϕ : THH( k ) → THH( k ) tC p identifies the former with the connective cover of the latter, and π ∗ (cid:0) THH( k ) tC p (cid:1) ≃ k [ t ± ] is aLaurent polynomial ring with | t | = 2. The map ϕ is given by the Frobenius on π and sends σ t .In particular, ϕ induces an equivalenceTHH( k )[1 /σ ] ≃ THH( k ) tC p . This computation was originally done by Hesselholt-Madsen [HM97], and we refer to [NS17, IV.4]for a complete description of THH( k ) as a cyclotomic spectrum.Here THH( k ) ∈ CAlg(CycSp) is a commutative algebra object, and for C a k -linear stable ∞ -category, then THH( C ) is a THH( k )-module. The functor THH yields a symmetric monoidalfunctor { k -linear stable ∞ -categories } → Mod
THH( k ) (CycSp) . Note in particular that for a smooth and proper k -linear stable ∞ -category, THH is therefore adualizable object of Mod THH( k ) (CycSp). In this paper, all our degeneration arguments will takeplace in the latter ∞ -category, and we will often state them in that manner.We saw above that the cyclotomic Frobenius becomes an equivalence on connective covers forTHH( k ). More generally, one can show (cf. [Hes96]) that for a smooth k -algebra, the cyclotomicFrobenius is an equivalence in high enough degrees. For our purposes, we need a basic observationthat in the smooth and proper case, the cyclotomic Frobenius becomes an equivalence after inverting σ . This is a formal dualizability argument once one knows both sides satisfy a K¨unneth formula. Proposition 2.7.
Let C /k be a smooth and proper k -linear stable ∞ -category. In this case, thecyclotomic Frobenius implements an equivalenceTHH( C )[1 /σ ] ϕ ≃ THH( C ) tC p ≃ HP( C /k ) . More generally, if X ∈ Mod
THH( k ) (CycSp) is a dualizable object, then the cyclotomic Frobeniusimplements an equivalence X [1 /σ ] ϕ −→ X tC p ≃ ( X ⊗ THH( k ) k ) tS . The first equivalence is a ϕ -semilinear for the equivalence ϕ : THH( k )[1 /σ ] ≃ THH( k ) tC p , while thesecond equivalence is TP( k )-linear. ALEDIN’S DEGENERATION THEOREM AND TOPOLOGICAL HOCHSCHILD HOMOLOGY 7
Proof.
By Proposition 2.3, it suffices to prove that ϕ is an isomorphism. In fact, both the sourceand target of ϕ are symmetric monoidal functors from dualizable objects in Mod THH( k ) (CycSp)to the ∞ -category of THH( k )[1 /σ ] ≃ THH( k ) tC p -module spectra (cf. [BM17, AMN17]) and thenatural transformation is one of symmetric monoidal functors. Thus the map is an equivalence forformal reasons [AMN17, Prop. 4.6]. (cid:3) Let C be smooth and proper over k . On homotopy groups, it follows that one has isomorphisms ofabelian groups π i THH( C )[1 /σ ] ≃ π i HP( C /k ). Both sides are k -vector spaces, and the isomorphismis semilinear for the Frobenius. In particular, at the level of k -vector spaces, one has a naturalisomorphism ( π i THH( C )[1 /σ ]) (1) ≃ HP i ( C /k ) . Remark 2.8.
Suppose C = Perf( A ) for A a smooth commutative k -algebra. In this case, HP( C /k )is related to 2-periodic de Rham cohomology of A while THH( C )[1 /σ ] is closely related to 2-periodicdifferential forms on C by [Hes96]. The relationship between differential forms and de Rham co-homology arising here is essentially the classical Cartier isomorphism.
In particular, Proposi-tion 2.7 should be compared with the “noncommutative Cartier isomorphism” studied by Kaledin[Kal08, Kal16]. This relationship in the commutative case is made precise in the work of Bhatt-Morrow-Scholze [BMS], where the Cartier isomorphism is an essential feature of their recovery ofcrystalline (and de Rham) cohomology from THH.3.
The degeneration argument
In this section, we give the main degeneration argument. We begin with the following basicobservation and definition.Let R be an E ∞ -ring spectrum (in this section, R will be a field), and let M be an R -modulespectrum equipped with an S -action. Suppose the R -module M is projective. Then the followingare equivalent:(1) The S -Tate spectral sequence for π ∗ ( M tS ) degenerates.(2) The S -action on M (as an R -module) is trivial.Clearly the second assertion implies the first. To see the converse, we observe that if the Tatespectral sequence degenerates, then by naturality, the homotopy fixed point spectral sequence for π ∗ ( M ) must degenerate too, so that the map π ∗ ( M hS ) → π ∗ ( M ) is surjective. Suppose M , as anunderlying R -module, is obtained as the summand F e associated to an idempotent endomorphism e of a free R -module F . If we give F the trivial S -action, the degeneration of the homotopy fixedpoint spectral sequence shows that we can realize the map F → M as an S -equivariant map.Restricting now to the summand F e of F , we conclude that M is equivalent to F e (with trivialaction). This is the way in which we regard the degeneration of the S -Tate spectral sequence as a formality statement. Definition 3.1.
Let k be a field. Let M ∈ Perf( k ) BS . We say that M is formal if the S -Tatespectral sequence for M tS (or equivalently the homotopy fixed point spectral sequence for M hS )degenerates at E . This holds if and only if(5) dim k π even ( M ) = dim k π M tS , dim k π odd ( M ) = dim k π M tS For the rest of this section, k is a perfect field of characteristic p >
0. We will prove a formalitycriterion for objects of Perf( k ) BS . Our main interest, of course, is in the following example; in thissection, we will state our arguments in the more general case of objects in Perf( k ) with S -action, AKHIL MATHEW though. Consider a smooth and proper k -linear stable ∞ -category C /k and its Hochschild homologyHH( C /k ). One has that dim k HH ∗ ( C /k ) < ∞ and that HH( C /k ) inherits a circle action. Definition 3.2.
We say that the
Hodge-to-de Rham spectral sequence degenerates for C /k ifHH( C /k ) ∈ Perf( k ) BS is formal. Equivalently, degeneration holds if and only if one has thenumerical equalities HH even ( C /k ) = HP ( C /k ) , dim k HH odd ( C /k ) = HP ( C /k ).One source of objects of Perf( k ) BS is the ∞ -category of dualizable objects of Mod THH( k ) (CycSp).Given X ∈ Mod
THH( k ) (CycSp), we have X ⊗ THH( k ) k ∈ Mod BS k and if X is dualizable, then X ⊗ THH( k ) k is perfect as a k -module. For such objects, we will translate formality to a statementabout THH( k )-modules. Note that HH( C /k ) ∈ Perf( k ) BS arises in this way, via X = THH( C ).First, we need the following observation about module spectra over THH( k ), which follows fromthe classification of finitely generated modules over a principal ideal domain. Proposition 3.3.
Any perfect THH( k )-module spectrum is equivalent to a direct sum of copies ofsuspensions of THH( k ) and THH( k ) /σ n for various n .The following result now shows that degeneration is equivalent to a condition of torsion-freenesson THH. Proposition 3.4. (1) Let X ∈ Mod
THH( k ) (CycSp) be dualizable. Then X ⊗ THH( k ) k ∈ Perf( k ) BS is formal if and only if X is free (equivalently, σ -torsion-free) as a THH( k )-module.(2) If C is smooth and proper over k , the Hodge-to-de Rham spectral sequence for C degeneratesif and only if THH( C ) is free (equivalently, σ -torsion-free) as a THH( k )-module. Proof.
Clearly, the second assertion is a special case of the first. It suffices to compare with (5). Infact, by the equivalence given by Proposition 2.7, one sees that π ∗ ( X )[1 /σ ] is a finitely generatedgraded free THH( k ) ∗ [1 /σ ]-module. Moreover, one hasdim k π (cid:16) ( X ⊗ THH( k ) k ) tS ) (cid:17) = dim k ( π ( X [1 /σ ])) = rank k [ σ ± ] π even ( X )[1 /σ ] , and similarly for the odd terms. Thus, formality holds if and only if the ranks agree, i.e.,rank k [ σ ± ] π even ( X )[1 /σ ] = dim k π even ( X ⊗ THH( k ) k ) , rank k [ σ ± ] π odd ( X )[1 /σ ] = dim k π odd ( X ⊗ THH( k ) k ) . Note that X ⊗ THH( k ) k ≃ X/σ . It follows (e.g., using Proposition 3.3) that the ranks (over σ = 0 and σ invertible, respectively) agree if and only if X is (graded) free as a THH( k )-module spectrum. (cid:3) It thus follows that, in order to verify degeneration of the Hodge-to-de Rham spectral sequence,one needs criteria for testing σ -torsion-freeness in THH ∗ ( C ). We begin by observing that liftabilityto the sphere allows for a direct argument here. The general idea that liftability to the sphereshould simplify the argument was well-known, and we are grateful to N. Rozenblyum for indicatingit to us. Example 3.5.
Suppose k = F p and suppose C lifts to a stable ∞ -category e C over the sphere S (implicitly p -completed). Note that the map S → THH( F p ) factors through the natural map F p → THH( F p ) given by choosing a basepoint in the circle S via the equivalence THH( F p ) ≃ S ⊗ F p in E ∞ -rings [MSV97]. Then, as THH( F p )-module spectra, one has an equivalenceTHH( C ) ≃ THH( e C ) ⊗ S THH( F p ) ≃ (THH( e C ) ⊗ S F p ) ⊗ F p THH( F p ) . Using the spectral version of the Witt vectors construction, this can be removed.
ALEDIN’S DEGENERATION THEOREM AND TOPOLOGICAL HOCHSCHILD HOMOLOGY 9
Since every F p -module spectrum is (graded) free, this equivalence proves that THH( C ) is free as anTHH( F p )-module. Thus, degeneration holds for C .We will now give the argument for a lifting to W ( k ). If a k -linear stable ∞ -category C liftsto W ( k ), then the THH( k )-module spectrum THH( C ) lifts to THH( W ( k )). By considering themap THH( W ( k )) → THH( k ), we will be able to deduce σ -torsion-freeness (and thus degeneration)in many cases. The argument will require a small amount of additional bookkeeping and relyon an amplitude assumption. The basic input is the following fact about the homotopy ring ofTHH( W ( k )). The entire computation is carried out in [Bru00], at least additively, but we will onlyneed it in low degrees. For the reader’s convenience, we include a proof. Proposition 3.6 (Compare [Bru00]) . Let k be a perfect field.(1) We have π ∗ τ ≤ p − THH( W ( k )) ≃ W ( k )[ u ] /u p , | u | = 2 . (2) The map THH i ( W ( k )) → THH i ( k ) is zero for 0 < i ≤ p −
2. Furthermore, the mapof E ∞ -rings THH( W ( k )) → THH( k ) → τ ≤ p − THH( k ) factors through the map k → THH( k ) → τ ≤ p − THH( k ). Proof.
We compare with Hochschild homology over the integers. The map S p ) → Z ( p ) induces anequivalence on degrees < p −
3. Thus, in the range stated in the theorem, we can compare THHwith Hochschild homology over Z ( p ) or over W ( k ). We haveHH ∗ ( W ( k ) / Z ( p ) ) ≃ Γ ∗ W ( k ) [ u ] , | u | = 2 , i.e., the divided power algebra on a class in degree 2. Indeed, the Hochschild homology is the freesimplicial commutative ring over W ( k ) on a class in degree two.It remains to check that the map THH( W ( k )) → THH( k ) vanishes on π . This, too, followsfrom the comparison with Hochschild homology over Z . For a map of commutative rings A → B ,let L B/A denote the cotangent complex of B over A . Using the classical Quillen spectral sequencefrom the cotangent complex to Hochschild homology (cf., e.g., [NS17, Prop. IV.4.1]), one has toshow that the following map vanishes:(6) π L W ( k ) / Z ( p ) → π L k/ Z ( p ) . Here one can replace the source Z ( p ) with W ( k ) since k is perfect. Recall also that if A is a ringand a ∈ A a regular element, then one has a natural equivalence L ( A/a ) /A ≃ ( a ) / ( a )[1]. In oursetting, one obtains for (6) the map of W ( k )-modules( p ) / ( p ) → ( p ) / ( p ) , which is zero. Finally, the factorization of the map of E ∞ -rings follows because τ ≤ p − THH( W ( k ))is the truncation of the free E ∞ -ring over W ( k ) on a class in degree two. (cid:3) We now give an argument that liftability together with a Tor-amplitude condition implies free-ness. The observation is that if the Tor-amplitude is small, then any torsion has to occur in lowhomotopical degree.
Proposition 3.7.
Let M be a perfect THH( k )-module with Tor-amplitude contained in an inter-val [ a, b ]. Suppose that M lifts to a perfect module over THH( W ( k )). Then multiplication by σ : π i − ( M ) → π i ( M ) is injective for i ≤ a + 2 p − Proof.
Without loss of generality, a = 0. By assumption, here we have M ≃ f M ⊗ THH( W ( k )) THH( k )for some connective and perfect THH( W ( k ))-module f M . Truncating, we find that there is a mapof THH( k )-modules(7) M → τ ≤ p − f M ⊗ τ ≤ p − THH( W ( k )) τ ≤ p − THH( k ) , which induces an isomorphism on degrees ≤ p −
2. However, by Proposition 3.6 and the factthat any k -module spectrum is free, it follows that the right-hand-side is a free module over τ ≤ p − THH( k ) on generators in nonnegative degrees. This shows that multiplication by σ is aninjection in this range of degrees. (cid:3) Proposition 3.8.
Let M be a perfect THH( k )-module with Tor-amplitude concentrated in [ − p, p ].Suppose that M lifts to a perfect module over THH( W ( k )). Then M is free. Proof. M is a direct sum of THH( k )-modules each of which is either free or equivalent to M i,j =Σ i THH( k ) /σ j for − p ≤ i ≤ i + 2 j + 1 ≤ p as M i,j has Tor-amplitude [ i, i + 2 j + 1]. Note that M i,j has an element in π i +2 j − annihilated by σ , so we find i + 2 j − ≥ p − i + 2 j + 1 ≥ p by Proposition 3.7. Therefore, i + 2 j + 1 = p .In particular, we find that if M i,j occurs as a summand, then i + 2 j + 1 = p . We observe nowthat if the hypotheses of the lemma apply to M , then they apply to the THH( k )-linear Spanier-Whitehead dual D M : that is, D M is a perfect THH( k )-module with Tor-amplitude concentrated in[ − p, p ], and such that D M lifts to a perfect module over THH( W ( k )). If M i,j occurs as a summandof M , then its dual, which is given by Σ − i − j − THH( k ) /σ j , occurs as a summand of D M . Applyingthe previous paragraph to D M , it follows also that − i = p . Adding the two equalities, we find that2 j + 1 = 2 p , which is an evident contradiction. (cid:3) Finally, we can state our general degeneration criterion in characteristic p , which will easily implyTheorem 1.2. Proposition 3.9 (General formality criterion, characteristic p ) . Let k be a perfect field of char-acteristic p >
0. Let M ∈ Perf( k ) BS be a perfect k -module with S -action whose amplitude iscontained in [ − p, p ]. Suppose that there exists a dualizable object M ′ ∈ Mod
THH( k ) (CycSp) suchthat, as objects in Perf( k ) BS , we have an equivalence M ≃ M ′ ⊗ THH( k ) k . Suppose furthermorethat the underlying THH( k )-module of M ′ lifts to a perfect module over THH( W ( k )). Then M isformal. Proof.
Combine Propositions 3.4 and 3.8. (cid:3)
Proof of Theorem 1.2.
Let C be a smooth and proper stable ∞ -category over k satisfying the as-sumptions of the theorem. By assumption, there exists a smooth and proper lift e C over W ( k ) suchthat C ≃ e C ⊗ W ( k ) k . Therefore, one has an equivalence of THH( k )-modulesTHH( C ) ≃ THH( e C ) ⊗ THH( W ( k )) THH( k ) . Furthermore, THH( e C ) is a perfect THH( W ( k ))-module. Now, one can apply Proposition 3.9 with M ′ = THH( C ). (cid:3) Remark 3.10.
The slight extension of the dimension range via duality goes back to the work ofDeligne-Illusie [DI87] and appears in the recent work of Antieau-Vezzosi [AV17] on HKR isomor-phisms in characteristic p . Note also that for a smooth and proper k -linear ∞ -category C , the ALEDIN’S DEGENERATION THEOREM AND TOPOLOGICAL HOCHSCHILD HOMOLOGY 11
Hochschild homology HH( C /k ) is always self-dual, cf. [Shk07]. Hence, it is no loss of general-ity to assume that the interval in which the amplitude of Hochschild homology is concentrated issymmetric about the origin.We now describe the deduction of Theorem 1.1 from Theorem 1.2, as in [Kal08, Kal16]. We notethat this is a standard argument and is also used in the commutative case [DI87]. We formulatethe approach in the following formality criterion. Theorem 3.11 (General formality criterion, field case) . Let K be a field of characteristic zero.Let M ∈ Perf( K ) BS be a perfect module equipped with an S -action. Suppose that there existsa finitely generated subring R ⊂ K and a dualizable object M ′ ∈ Mod
THH( R ) (CycSp) such that wehave an equivalence in Perf( K ) BS , M ′ ⊗ THH( R ) K ≃ M. Then M is formal. Proof.
Any finitely generated field extension of Q is a filtered colimit of smooth Z -algebras. There-fore, K is a filtered colimit of its finitely generated subalgebras which are smooth over Z . Enlarging R , we can assume that R is smooth over Z . Enlarging R further, we can assume that the homol-ogy groups of M ′ ⊗ THH( R ) R (which is a perfect R -module spectrum) are finitely generated free R -modules and vanish for i / ∈ [ − p, p ], for every prime p which is noninvertible in R .Suppose that the S -action on M is nontrivial. Therefore, the S -action on M ′ ⊗ THH( R ) R isnontrivial too, and there exists a nontrivial differential in the Tate spectral sequence for ( M ′ ⊗ THH( R ) R ) tS . Then we can find a maximal ideal m ⊂ R such that the first differential (which is a map offinitely generated free R -modules) remains nontrivial after base-change along R → R/ m and thusafter base-change along R → k def = ( R/ m ) perf .Let M ′ k = M ′ ⊗ THH( R ) THH( k ) ∈ Mod
THH( k ) (CycSp), which is a dualizable object. Note that k is a perfect field of characteristic p >
0. Moreover, the map R → k lifts to the length twoWitt vectors because R is smooth over Z . It follows that the underlying THH( k )-module of M ′ k lifts to a perfect THH( W ( k ))-module. It follows that M ′ ⊗ THH( R ) k ∈ Perf( k ) BS is formal byProposition 3.9. This contradicts the statement that there is a nontrivial differential in the Tatespectral sequence for ( M ′ ⊗ THH( R ) k ) tS and proves the theorem. (cid:3) Proof of Theorem 1.1.
Let C be a smooth and proper stable ∞ -category over a field K of char-acteristic zero. By the results of [Toe08], there exists a smooth and proper stable ∞ -category e C over a finitely generated subalgebra R ⊂ K such that C ≃ e C ⊗ R K . Then, one has the dualizableobject THH( e C ) ∈ Mod
THH( R ) (CycSp) and by base-change, one has an equivalence in Perf( K ) BS THH( e C ) ⊗ THH( R ) K ≃ HH( C /K ). Now apply Theorem 3.11. (cid:3) We note that the above arguments actually enable a slight strengthening of Theorem 1.2. Forexample, Theorem 3.11 easily implies that if F : C → D is a functor of smooth and proper stable ∞ -categories over K , then the S -action on the relative Hochschild homology fib(HH( C /K ) → HH( D /K )) is also trivial. More generally, this would work for any appropriately finite diagram.We formulate this as follows.Let K be a field of characteristic zero and let N Mot K denote the presentably symmetric monoidal ∞ -category of noncommutative motives of K -linear stable ∞ -categories introduced by Tabuada[Tab15] (see also [BGT13, HSS17]). Since Hochschild homology is an additive invariant, one has asymmetric monoidal, cocontinuous functorHH( · /K ) : N Mot K → Mod BS K , from N Mot K into the ∞ -category Mod BS K of K -module spectra equipped with an S -action. Let N Mot ωK ⊂ N Mot K denote the thick subcategory generated by the smooth and proper stable ∞ -categories. Recall that if C , D are smooth and proper K -linear stable ∞ -categories, then we haveassociated objects [ C ] , [ D ] ∈ N Mot ωK , and the mapping spectrum is given asHom N Mot K ([ C ] , [ D ]) ≃ K(Fun( C , D )) , i.e., it is the connective algebraic K-theory spectrum of the ∞ -category of ( K -linear) functors C → D . Note for instance that given a functor F : C → D , one can form the fiber of the associatedmap [ C ] → [ D ] of noncommutative motives, so that relative Hochschild homology is given byHochschild homology of an object of N Mot ωK . Corollary 3.12.
For any X ∈ N Mot ωK , HH( X/K ) ∈ Perf( K ) BS is formal. Proof.
By the results of [Toe08], and the fact that K-theory commutes with filtered colimits, itfollows that N Mot ωK is the filtered colimit of the stable ∞ -categories N Mot ωR of dualizable non-commutative motives of smooth and proper R -linear ∞ -categories, as R ranges over the finitelygenerated subrings of K . Thus, there exists R such that X arises via base-change from a dualizableobject e X in the ∞ -category N Mot R . In this case, since THH is an additive invariant of R -linearstable ∞ -categories into cyclotomic spectra (compare [BM12, BGT13, AMGR17] for treatments),we can similarly form the dualizable object THH( e X ) ∈ Mod
THH( R ) (CycSp), which provides a liftingof HH( X/K ). Now we can apply Theorem 3.11 as before. (cid:3) Freeness results and degeneration in families
In this section, we will analyze Hodge-to-de Rham degeneration in families. In particular, wewill give proofs of Theorems 1.3 and 1.4, showing that (under appropriate hypothesis) the relativeHodge-to-de Rham spectral sequence degenerates and that Hochschild homology is locally free. Incharacteristic zero, at least over a smooth base, this result follows from the existence of a connection[Get93] on periodic cyclic homology together with Theorem 1.1.Throughout this section, we will need K¨unneth formulas, as in the form expressed in [AMN17].If ( C , ⊗ , ) is a symmetric monoidal stable ∞ -category with biexact tensor product, then an object X ∈ C is called perfect if it belongs to the thick subcategory generated by the unit. Perfectnessis extremely useful to control objects in C and their behavior. However, it can be tricky to checkdirectly.In [AMN17], the main result is that if k is a perfect field of characteristic p >
0, in the ∞ -categoryMod THH( k ) (Sp BS ) of modules over THH( k ) in the ∞ -category of spectra with an S -action, everydualizable object is perfect. This in particular implies the K¨unneth theorem for periodic topologicalcyclic homology proved by Blumberg-Mandell [BM17]. In this section, we will need variants ofthis result for non-regular rings in characteristic zero (Proposition 4.2) and in the perfect (butnot necessarily field) case in characteristic p (Proposition 4.15). This will enable us to controlHochschild homology of stable ∞ -categories over, respectively, local Artin rings in characteristiczero and large perfect rings in characteristic p .4.1. Characteristic zero.
In this subsection, we explain the deduction of Theorem 1.3, that therelative Hodge-to-de Rham spectral sequence degenerates for families of smooth and proper dgcategories in characteristic zero, and that the relative Hochschild homology is locally free. Weactually prove a result over connective E ∞ -rings and give a strengthening of the general formalitycriterion, Theorem 3.11. ALEDIN’S DEGENERATION THEOREM AND TOPOLOGICAL HOCHSCHILD HOMOLOGY 13
The strategy will be to reduce to the local Artinian case, as is standard. We use the followingdefinition.
Definition 4.1.
A connective E ∞ -ring A is local Artinian if π ( A ) is a local Artinian ring, eachhomotopy group π i ( A ) is a finitely generated π ( A )-module, and that π i ( A ) = 0 for i ≫ k of characteristic zero. Let A be a local Artin E ∞ -ring with residue field k . Notethat A → k admits a section unique up to homotopy by formal smoothness, compare, e.g., [Mat17,Prop. 2.14], and so we will consider A as an E ∞ -algebra over k . Our first goal is to prove K¨unnethformulas for negative and periodic cyclic homology for smooth and proper stable ∞ -categories over A . Following [AMN17], we translate this into the following statement. As in section 2, HH( A/k )defines a commutative algebra object in the ∞ -category Sp BS of spectra with an S -action andwe can consider the symmetric monoidal ∞ -category of modules Mod HH(
A/k ) (Sp BS ). Given an A -linear stable ∞ -category C , the Hochschild homology HH( C /k ) defines an object in Mod HH(
A/k ) (Sp BS ).The homotopy fixed points HH( C /k ) hS are written HC − ( C /k ) and called the negative cyclic ho-mology of C (over k ). See also [Hoy15] for comparisons with more classical definitions. Proposition 4.2.
Any dualizable object in the symmetric monoidal ∞ -category Mod HH(
A/k ) (Sp BS )is perfect. Proof.
Let M ∈ Mod
HH(
A/k ) (Sp BS ) be a dualizable object. We have a lax symmetric monoidalfunctor F : Mod HH(
A/k ) (Sp BS ) → Mod HC − ( A/k ) , N N hS . By general results, this functor is fully faithful. Equivalently, the left adjoint functorMod HC − ( A/k ) → Mod
HH(
A/k ) (Sp BS )is a symmetric monoidal localization. Compare [MNN17, Sec. 7], which implies that Mod k (Sp BS )is identified with the ∞ -category of C ∗ ( BS ; k )-modules complete with respect to the augmentation C ∗ ( BS ; k ) → k .To check the equivalence, it suffices to prove that the functor is strictly symmetric monoidal ondualizable objects by [MNN17, Lemma 7.18]. That is, for dualizable objects M, N ∈ Mod
HH(
A/k ) (Sp BS ),one needs the map(8) F ( M ) ⊗ HC − ( A/k ) F ( N ) → F ( M ⊗ N )to be an equivalence of HC − ( A/k )-module spectra. Note that we have an element x ∈ π − HC − ( A/k )(i.e., a generator of π − HC − ( k/k ) ≃ π − C ∗ ( BS ; k )) such that HC − ( A/k ) /x ≃ HH(
A/k ) and onehas an equivalence of HH(
A/k )-module spectra F ( M ) /x ≃ M for any M ∈ Mod
HH(
A/k ) (Sp BS ) (cf.[MNN17, Sec. 7]). It thus follows that (8) becomes an equivalence after base-change HC − ( A/k ) → HH(
A/k ).It thus suffices to show that (8) becomes an equivalence after inverting x . Now we have( F ( M ) ⊗ HC − ( A/k ) F ( N ))[1 /x ] ≃ M tS ⊗ HP(
A/k ) N tS , F ( M ⊗ N )[1 /x ] ≃ ( M ⊗ HH(
A/k ) N ) tS . In other words, it suffices to show that the functor F ′ : Mod HH(
A/k ) (Sp BS ) → Mod
HP(
A/k ) , N N tS . One could work in the derived ∞ -category D ( k ) in this subsection. is strictly symmetric monoidal on dualizable objects.However, by Lemma 4.3 below, it follows that F ′ can be identified with the functor M ( M ⊗ HH(
A/k ) k ) tS , i.e., F ′ factors through the symmetric monoidal functor Mod HH(
A/k ) (Sp BS ) → Mod k (Sp BS ) given by base-change HH( A/k ) → k . Furthermore, HP( A/k ) ≃ k tS . Since dual-izable objects in Mod k (Sp BS ) are perfect, it follows that F ′ satisfies a K¨unneth formula. Thisimplies the result. (cid:3) Lemma 4.3. If M is an object of Mod HH(
A/k ) (Sp BS ) such that M is bounded below, then thenatural map M → M ⊗ HH(
A/k ) k induces an equivalence on S -Tate constructions. Proof.
Now M ≃ lim ←− τ ≤ n M and M ⊗ HH(
A/k ) k ≃ lim ←− ( τ ≤ n M ⊗ HH(
A/k ) k ). Both of these inverselimits become constant in any given range of dimensions. Therefore, they commute with S -Tateconstructions. Therefore, it suffices to assume that M is n -truncated, and by a filtration argument,discrete. By a further d´evissage, we can assume that M is actually a discrete k -module, consideredas a HH( A/k )-module via the augmentation. We are thus reduced to showing that if N is a discrete k -module, then the map N → N ⊗ HH(
A/k ) k ≃ N ⊗ k ( k ⊗ HH(
A/k ) k ) ≃ N ⊗ k HH( k ⊗ A k/k )induces an equivalence on S -Tate constructions.However, since the homology of k ⊗ A k forms a connected graded, commutative Hopf algebra, itfollows that π ∗ ( k ⊗ A k ) is the tensor product of polynomial algebras on even-dimensional classes andexterior algebras on odd-dimensional classes. Therefore, k ⊗ A k is a free E ∞ - k -algebra Sym ∗ V forsome k -module spectrum V with π i ( V ) = 0 for i ≤
0. Furthermore, HH( k ⊗ A k/k ) ≃ Sym ∗ ( S ⊗ V ).The desired equivalence now follows because for i >
0, Sym i ( S ⊗ V ) is a free module over thegroup ring k [ S ], and so the terms for i > (cid:3) Corollary 4.4.
Let A be a local Artin E ∞ -ring and let C be a smooth and proper stable ∞ -categoryover A . Then the map HP( C /k ) → HP(
C ⊗ A k/k ) is an isomorphism.Note that when A = k itself, this recovers certain cases of the classical theorem of Goodwillie([Goo85, Theorem II.5.1], [Goo86, Lemma I.3.3]) about the nilinvariance of periodic cyclic homology.The corollary follows from Lemma 4.3 because one has an equivalenceHH( C ⊗ A k/k ) ≃ HH( C /k ) ⊗ HH(
A/k ) k. Corollary 4.5.
Let A be a local Artin E ∞ -ring. Let M ∈ Mod
HH(
A/k ) (Sp BS ) be dualizable, andlet M A = M ⊗ HH(
A/k ) A ∈ Mod A (Sp BS ) and M k ∈ M ⊗ HH(
A/k ) k ∈ Perf( k ) BS . Then:(1) M A ∈ Mod A (Sp BS ) belongs to the thick subcategory generated by the unit.(2) M tS A ⊗ A k ≃ M tS k .(3) M tS A is a graded free A tS -module. Proof.
By Proposition 4.2, M belongs to the thick subcategory generated by the unit in Mod HH(
A/k ) (Sp BS ).It follows that M A ∈ Mod A (Sp BS ) belongs to the thick subcategory generated by the unit. Thus,we obtain the first claim. The second claim is implied by the first, as for any perfect object X ∈ Mod A (Sp BS ), one has ( X ⊗ A k ) tS ≃ X tS ⊗ A k by a thick subcategory argument.Finally, one has natural maps M tS → M tS A → M tS A ⊗ A k ≃ M tS k , ALEDIN’S DEGENERATION THEOREM AND TOPOLOGICAL HOCHSCHILD HOMOLOGY 15 such that the composite is an equivalence by Lemma 4.3. Thus, the map M tS A → ( M tS A ) ⊗ A k hasa section of k -module spectra. Lifting a basis, this implies that M tS A is free as an A tS -module. (cid:3) Lemma 4.6.
Let A be an augmented local Artin E ∞ -ring with residue field k . Let M be a perfect A -module. Then(9) dim k ( π ∗ ( M )) ≤ (dim k π ∗ ( A ))(dim k π ∗ ( M ⊗ A k )) , and if equality holds M is free. Proof.
Since A has a filtration (in A -modules) by copies of k , the inequality is evident. If equalityholds, suppose that i ∈ Z is minimal such that π i ( M ) = 0. Choose x ∈ π i ( M ) whose image in π i ( M ⊗ A k ) ≃ π i ( M ) ⊗ π ( A ) k is nonzero. Form a cofiber sequence Σ i A x → M → N of A -modules.It follows thatdim k ( π ∗ ( N ⊗ A k )) = dim k ( π ∗ ( M ⊗ A k )) − , dim k ( π ∗ ( M )) ≤ dim k ( π ∗ ( N )) + dim k π ∗ ( A ) . Combining this with (9), we find that dim k π ∗ ( N ) = (dim k π ∗ ( A ))(dim k π ∗ ( N ⊗ A k )). By an evidentinduction, N is free as an A -module. The long exact sequence in homotopy, which must reduce toa short exact sequence, now shows that M is also free as an A -module. (cid:3) We can now prove the main freeness and degeneration theorems of this section, which providesa substantial strengthening of Theorem 3.11. Let CAlg(Sp ≥ ) denote the ∞ -category of connective E ∞ -rings. In the following argument, one could also work with simplicial commutative rings. Theorem 4.7 (General formality criterion, relative case) . Let A be a connective E ∞ -algebra over Q . Let M ∈ Perf( A ) BS . Suppose that there exists a compact object R ∈ CAlg(Sp ≥ ) with a map R → A , a dualizable object M ′ R ∈ Mod
THH( R ) (CycSp), and an equivalence M ′ R ⊗ THH( R ) A ≃ M ∈ Perf( A ) BS . Then M is a finitely generated projective A -module and the S -action on M is formal. Proof.
We first treat the case where A is a local Artin E ∞ -ring with residue field k . To see that M is free, it suffices to show that equality holds in (9). Our assumptions show that M lifts to adualizable object of Mod HH(
A/k ) (Sp BS ). Using the Tate spectral sequence, one obtains(10) dim k π ( M tS ) + dim k π ( M tS ) ≤ dim k π ∗ ( M ) . Moreover, by Corollary 4.5, we know that M tS is a free A tS -module and that M tS ⊗ A k ≃ ( M ⊗ A k ) tS . Note that π R is a finitely generated Z -algebra. Thus we can apply Theorem 3.11,and we find that M ⊗ A k is formal in Perf( k ) BS . We obtain:dim k π ( M tS ) + dim k π ( M tS ) = (cid:16) dim k π (cid:16) ( M ⊗ A k ) tS (cid:17) + dim k π (cid:16) ( M ⊗ A k ) tS (cid:17)(cid:17) dim k π ∗ ( A )= dim k π ∗ ( M ⊗ A k ) dim k π ∗ ( A ) . Combining the above two inequalities, we obtain dim k π ∗ ( M ⊗ A k ) dim k π ∗ ( A ) ≤ dim k π ∗ ( M ), whichshows that the converse of (9) holds and M is free. Moreover, equality holds in (10), so that the S -Tate spectral sequence for M degenerates and M is formal.We now treat the general case. Clearly it suffices to treat the case where A is a compact objectof the ∞ -category of connective E ∞ -algebras over Q . In this case, π ( A ) is noetherian and thehomotopy groups π i ( A ) are finitely generated π ( A )-modules. We thus suppose A is of this form.To check the above statements, it suffices to replace A by its localization at any prime idealof π ( A ). Thus, we may assume that π ( A ) is local. Let x , . . . , x n ∈ π ( A ) be a system ofgenerators of the maximal ideal. For each r >
0, we let A ′ r = A/ ( x r , . . . , x rn ). Note moreover that A ′ r ≃ lim ←− τ ≤ m A ′ r and that lim ←− r A ′ r is the completion of A , which is in particular faithfully flat over A . By the above analysis, M ⊗ A τ ≤ m A ′ r is a free τ ≤ m A ′ -module for each m, r and the Tate spectralsequence degenerates. Now we can let m, r → ∞ . Since M is perfect as an A -module, it followsthat M is free, as desired, and the S -action is formal. (cid:3) Let A be a connective E ∞ -algebra over Q . Similarly, one can construct [HSS17] the ∞ -category N Mot A of noncommutative motives of A -linear ∞ -categories. We let N Mot ωA denote the thicksubcategory generated by the motives of smooth and proper A -linear ∞ -categories. We have,again, a Hochschild homology functor HH( · /A ) → Mod BS A . The next result gives a basic formalityproperty of this functor; for smooth and proper A -linear ∞ -categories, it includes the degenerationof the relative Hodge-to-de Rham spectral sequence. Corollary 4.8.
Let X ∈ N Mot ωA . Then HH( X/A ) ∈ Mod BS A is a finitely generated projective A -module and the S -action is formal. Proof.
Here we use a refinement of the results of [Toe08] for E ∞ -algebras. Namely, we claim thatthe functor which assigns to an E ∞ -ring spectrum R the ∞ -category of smooth and proper R -linear ∞ -categories commutes with filtered colimits in R . Now, smooth and proper R -linear ∞ -categoriesare compact; in fact, combine [AG14, Props. 3.5, 3.11]. Therefore, it suffices to see that if R is afiltered colimit of E ∞ -algebras R i , then any smooth and proper R -linear ∞ -category C descendsto some R i . To see this, we observe that C is equivalent to Perf( B ) for an associative A -algebra B which is compact [AG14, Prop. 3.11] and we can descend the algebra to a compact algebra oversome R i thanks to [Lur17, Lemma 11.5.7.17]. Moreover, by compactness we can also descend theduality datum to some finite stage.In view of this, we conclude that given X ∈ N Mot ωA , there exists a compact object R ∈ CAlg(Sp ≥ ) mapping to A and a smooth and proper R -linear ∞ -category e C such that C ≃ e C ⊗ R A .Using Theorem 4.7, we can now conclude the proof as before. (cid:3) Characteristic p . The characteristic zero assertion essentially amounts to the idea that pe-riodic cyclic homology should form a crystal over the base which is also coherent, and any such isnecessarily well-known to be locally free. In characteristic p , one can appeal to an analogous argu-ment: given a smooth algebra R in characteristic p , any finitely generated R -module M isomorphicto its own Frobenius twist is necessarily locally free [EK04, Prop. 1.2.3]. In this subsection, we proveTheorem 1.4 from the introduction. In doing so, we essentially use the Frobenius-semilinearity ofthe cyclotomic Frobenius.We first discuss what we mean by liftability. Let A be a regular (noetherian) F p -algebra. Recallthat A is F -finite if the Frobenius map ϕ : A → A is a finite morphism. We refer to [DM17, Sec.2.2] for a general discussion of F -finite rings. Definition 4.9.
Given an F -finite regular noetherian ring A , a lift of A to Z /p will mean a flat Z /p -algebra e A with an isomorphism e A ⊗ Z /p F p ≃ A .Let A be a regular noetherian F p -algebra. By Popescu’s smoothing theorem (see [Sta17, Tag07GC] for a general reference), A is a filtered colimit of smooth F p -algebras. It follows that thecotangent complex L A/ F p is concentrated in degree zero and identified with the K¨ahler differentials;in addition, they form a flat A -module. If A is in addition F -finite, then the K¨ahler differentialsare finitely generated and therefore projective as an A -module. Recall that the cotangent complexcontrols the infinitesimal deformation theory of A [Ill71, Ch. III, Sec. 2]. Therefore, A is formallysmooth as an F p -algebra, and a lift to Z /p exists. Given a lift e A to Z /p , it follows that e A is ALEDIN’S DEGENERATION THEOREM AND TOPOLOGICAL HOCHSCHILD HOMOLOGY 17 formally smooth over Z /p . In particular, it follows that any two lifts to Z /p are (noncanonically)isomorphic. Moreover, if A → B is a map of F -finite regular noetherian F p -algebras and e A, e B arerespective lifts to Z /p , then the map lifts to a map e A → e B .Let A be a regular F -finite F p -algebra. Then the Frobenius ϕ : A → A is a finite, flat morphism.We let A perf denote the perfection of A , i.e., the colimit of copies of A along the Frobenius map.Then we have inclusions A ⊂ A /p ⊂ A /p ⊂ . . . A perf , such that all maps are faithfully flat and the colimit is A perf . Our strategy will essentially be descentto A perf . Unfortunately, A perf is not noetherian. Thus, we will need the following result. Proposition 4.10.
Let A be a regular F -finite F p -algebra.(1) Then the ring A perf is coherent, i.e., the finitely presented modules form an abelian category.(2) Let I ⊂ A be an ideal. Given a finitely presented A perf -module M , the submodule M ′ ⊂ M consisting of those elements annihilated by a power of I is also coherent and its annihilatorin A perf is finitely generated. Proof.
The first assertion follows because A perf is the filtered colimit of copies of the noetherianring A along the Frobenius map, which is flat by regularity. If M is a coherent A perf -module, then M descends to A /p n for some n , i.e., there exists a finitely generated module M n over A /p n suchthat M ≃ A perf ⊗ A /pn A perf . Then M n has an A /p n submodule M ′ n consisting of the I -powertorsion, which is also finitely generated (and hence finitely presented), and such that the quotienthas no I -power torsion. It follows from flatness that M ′ n ⊗ A /pn A perf = M ′ , which is thus coherent.Since M ′ is coherent, its annihilator ideal is also coherent. (cid:3) We will also need to observe that analogs of B¨okstedt’s calculation of THH( k ) hold when k isany perfect F p -algebra, not only a field. Similarly, analogs of Propositions 3.6 and Proposition 3.7hold with analogous arguments. Proposition 4.11.
Let k be a perfect F p -algebra. Suppose M is a perfect THH( k )-module and π i ( M/σ ) = 0 for i / ∈ [ a, b ] for b − a ≤ p −
2. Suppose M lifts to a perfect THH( W ( k ))-module.Then, as π ∗ THH( k ) ≃ k [ σ ]-modules, one has π ∗ ( M ) ≃ π ∗ ( M/σ ) ⊗ k k [ σ ]. Proof.
Without loss of generality, a = 0. For each j , we need to argue that multiplication by σ is a split injection of k -modules π j − ( M ) → π j ( M ). Using the long exact sequence and the assumptionon M/σ , we find that multiplication by σ is an isomorphism for j ≥ p −
1. For j ≤ p −
2, theequivalence in the range [0 , p −
2] as in (7) implies the result. (cid:3)
We can now state and prove the main formality statement in characteristic p over a regular F -finite base. Theorem 4.12 (Formality criterion, relative characteristic p case) . Let A be a regular F -finite F p -algebra. Let e A be a flat lift to Z /p . Let M ∈ Perf( A ) BS . Suppose that:(1) There exists a dualizable object M ′ ∈ Mod
THH( A ) (CycSp) such that M ′ ⊗ THH( A ) A ∈ Perf( A ) BS .(2) π i ( M ) vanishes for i / ∈ [ a, b ] for some a, b with b − a ≤ p − A )-module of M ′ lifts to a perfect THH( e A )-module.Then M is a finitely generated (graded) projective A -module, and the S -action on M is formal. Proof.
First, we can reduce to the case where A is an F -finite regular local ring with maximal ideal m . In this case, we can induct on the Krull dimension d of A . We can assume that the result holdsfor all F -finite regular local rings of Krull dimension less than d . When d = 0, the claim is of courseTheorem 1.2.To verify the claims for A , we can now replace A by its m -adic completion b A , which is faithfullyflat over A . Note that b A is also an F -finite regular local ring of Krull dimension d . Since b A iscomplete, it contains a copy of its residue field k and is identified with b A ≃ k [[ x , . . . , x n ]]. We canconsider the faithfully flat map b A → k perf [[ x , . . . , x d ]]. Replacing A with k perf [[ x , . . . , x d ]], we willnow simply assume that A is in addition complete and has perfect residue field. By the inductivehypothesis, all the differentials in the Hodge-to-de Rham spectral sequence are m -power torsion andthat HH( C /A ) is locally free away from m .Let A perf denote the (colimit) perfection of A , so one has a faithfully flat map A → A perf .We form the base-changes M ′ perf def = M ′ ⊗ THH( A ) THH( A perf ) ∈ Mod
THH( A perf ) (CycSp) (which isa dualizable object) and M perf = M ⊗ A A perf ∈ Perf( A perf ) BS . We claim that the cyclotomicFrobenius ϕ : M ′ perf [1 /σ ] → ( M ′ perf ) tC p ≃ ( M perf ) tS is an equivalence. This follows using the same arguments as in [AMN17, Sec. 4]; again, one needsto know that both sides are symmetric monoidal functors in M ′ perf . For this, it suffices to showthat M ′ perf belongs to the thick subcategory generated by the unit in Mod THH( A perf ) (Sp BS ). Wewill check this in Proposition 4.15 below.Note that M ′ perf is an THH( A perf )-module, and M ′ perf /σ ≃ M perf . Under the liftability hypothe-ses, we conclude using Proposition 4.11 that there is an isomorphism of A perf [ σ ]-modules π ∗ ( M ′ perf ) ≃ π ∗ ( M perf )[ σ ] . Combining, we find an isomorphism of A perf -modules(11) π ∗ ( M perf ) [ σ ± ] (1) ≃ π ∗ (cid:16) M tS perf (cid:17) . In addition, we have the Hodge-to-de Rham spectral sequence, which shows that π ( M tS perf ) isa subquotient of π even ( M perf ) and is a coherent A perf -module. Since the differentials are m -powertorsion, it follows that the m -power torsion in π ( M tS perf ) is a subquotient of the m -power torsion in π even ( M perf ).Let I be the annihilator of the m -power torsion in π even ( M perf ), which by Proposition 4.10 is afinitely generated ideal. Then combining the above observations and (11), we find that I [ p ] (i.e.,the ideal generated by p th powers of elements in I ) is the annihilator of the m -power torsion in π ( M tS perf ). Since this is a subquotient of π even ( M perf ), it follows that I ⊂ I [ p ] , which is onlypossible for a finitely generated proper ideal if I = (0). Therefore, π even ( M perf ) (and similarly forthe odd-dimensional Hochschild homology) is torsion-free.Finally, it suffices to prove freeness. We have proved that π ∗ ( M ) consists of finitely generated,torsion-free A -modules. Let x ∈ m \ m , so that A/x is a regular local ring too. It follows that π ∗ ( M/x ) is x -torsion-free and that, by induction on the Krull dimension, π ∗ ( M ) / ( x ) is a free A/ ( x )-module. This easily implies that π ∗ ( M ) is free as an A -module. By comparing with the base-changefrom A to the perfection of its fraction field, it also follows that M is formal. (cid:3) ALEDIN’S DEGENERATION THEOREM AND TOPOLOGICAL HOCHSCHILD HOMOLOGY 19
In the course of the above argument, we had to check a perfectness statement. In [AMN17], suchresults are proved when A perf is a field, but they depend on noetherianness hypotheses. One cancarefully remove the noetherianness hypotheses in this case, but for simplicity, we verify this byusing the technique of relative THH (also discussed in [AMN17, Sec. 3]). The starting point is arelative version of B¨okstedt’s calculation. We denote by S [ q , . . . , q n ] the E ∞ -ring Σ ∞ + ( Z n ≥ ). Theidea of considering THH relative to such E ∞ -rings is known to experts, and will play an importantrole in the forthcoming work [BMS]. Proposition 4.13.
Let A be an F -finite regular local ring with system of parameters t , . . . , t n and perfect residue field k . Consider the map of E ∞ -rings S [ q , . . . , q n ] → A, q i t i . ThenTHH(
A/S [ q , . . . , q n ]) ∗ ≃ A [ σ ] , | σ | = 2 . Proof.
Compare also the treatment in [AMN17, Sec. 3]. Since A is F -finite and regular, the cotan-gent complex L A/ F p is a finitely generated free module in degree zero. By the transitivity sequence, L A/ Z p [ t ,...,t n ] is a perfect A -module. Thus, by the Quillen spectral sequence, the homotopy groupsof HH( A/ Z [ q , . . . , q n ]) and thus THH( A/ Z [ q , . . . , q n ]) are finitely generated A -modules. Comparealso [DM17] for general finite generation results.Moreover, after base-change S [ q , . . . , q n ] → S sending q i
0, one obtains B¨okstedt’s calcula-tion THH( k ) ∗ ≃ k [ σ ]. Since the homotopy groups of THH( A/S [ q , . . . , q n ]) are finitely generated A -modules, and A is local, the result follows. (cid:3) Let A be as above. Given a smooth and proper A -linear stable ∞ -category C , one can considerthe invariant THH( C /S [ q , . . . , q n ]), which naturally takes values in the symmetric monoidal ∞ -category Mod THH(
A/S [ q ,...,q n ] (Sp BS ). This produces a one-parameter deformation of Hochschildhomology over A , and it is particularly well-behaved (at least for smooth and proper A -linear stable ∞ -categories) by the following result. Proposition 4.14.
Let A be an F -finite regular local ring with system of parameters t , . . . , t n and perfect residue field k . Any dualizable object in Mod THH(
A/S [ q ,...,q n ]) (Sp BS ) is perfect. Proof.
This follows by regularity from [AMN17, Theorem 2.15]. (cid:3)
Proposition 4.15.
Let A be an F -finite regular local ring. Let N ∈ Mod
THH( A ) (Sp BS ) bedualizable. Then N ⊗ THH( A ) THH( A perf ) is perfect. Proof.
In fact, we have a factorization of E ∞ -rings with S -actionTHH( A ) → THH(
A/S [ q , . . . , q n ]) → THH( A perf /S [ q /p ∞ , . . . , q /p ∞ n ]) ≃ THH( A perf ) . We have just seen that N ⊗ THH( A ) THH(
A/S [ q , . . . , q n ]) is perfect in Mod THH(
A/S [ q ,...,q n ]) (Sp BS );base-changing up to THH( A perf ), the result follows. (cid:3) Once more, we make the statement for Hochschild homology of categories, or more generally fornoncommutative motives. Let A be an F -finite regular noetherian ring with lift e A to Z /p . We use,again, the ∞ -category N Mot A , its subcategory N Mot ωA generated by the motives of smooth andproper A -linear ∞ -categories, and the Hochschild homology functor HH( · /A ) : N Mot A → Mod BS A . Corollary 4.16.
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