aa r X i v : . [ m a t h . K T ] A p r The May filtration on THH and faithfully flat descent
Liam KeenanApril 28, 2020
Abstract
In this paper, we prove that both topological Hochschild homology and topological cyclic ho-mology are sheaves for the fpqc topology on connective commutative ring spectra, by exploitingthe May filtration on topological Hochschild homology.
Contents ( Sp ) , Tow ( Sp ) , and Day convolution . . . . . . . . . . . . . . . . . . 32.2 The evaluation functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 The associated graded functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 The Whitehead tower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 THH hC p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Topological Hochschild homology (THH) and topological cyclic homology (TC), along with theirDennis and cyclotomic trace maps, have proved to be very useful tools for calculating algebraic K -theory. Algebraic K-theory has the nice property of being a sheaf for the Nisnevich topology;in practice, this allows one to decompose calculations into more manageable pieces. However,K-theory fails to satisfy the more computationally useful étale descent. Fortunately, THH andTC do not have this deficiency; they have fpqc descent for commutative rings, by a theorem ofBhatt-Morrow-Scholze in [BMS19]. Additionally, they have étale descent for E ∞ -ring spectra bya theorem of Clausen-Mathew in [CM19]. At this stage, it is reasonable to wonder whether it ispossible to promote fpqc descent to ring spectra. Our work answers this in the affirmative, providedwe work with connective commutative ring spectra, and our main theorem is the following: Theorem 1.1.
Let F = THH , THH (−) h T , THH (−) hC p , THH (−) hC p , THH (−) tC p , or TC viewed asfunctors CAlg ≥ → Sp . Then F satisfies descent for the fpqc topology on CAlg ≥ . One would hope that our proof might follow the same strategy as for commutative rings, butworking with spectra excludes this possibility. In [BMS19], THH is shown to have fpqc descent bya series of reductions. First, the authors reduce the problem to the case of Hochschild homology, by1sing the Postnikov tower of S , and then reduce to the case of wedge powers of the cotangent com-plex, via the Hochschild-Konstant-Rosenberg (HKR) filtration on Hochschild homology. Finally,the case of the cotangent complex is handled by a theorem of Bhatt, a proof of which appears as[BMS19, 3.1]. Since many of the tools used in [BMS19] are unique to working with (simplicial) com-mutative rings, different methods became necessary. Our strategy exploits structural properties ofthe May filtration on THH, as introduced in [AKS18]. In particular, this filtration is Hausdorff, andits associated graded can be calculated via a canonical equivalence gr May ∗ THH ( A ) ≃ THH ( Hπ ∗ A ) .By proving Theorem 1.1 in the case of generalized Eilenberg-Mac Lane spectra, we bootstrap tothe general case using our control of the May filtration.We now provide a summary of our work. In Section 2, we carefully prove several folklore resultson filtered spectra, all of which are undoubtedly known to experts. In particular, we prove that the"evaluate at zero" functor and the associated graded functor Fun ( Z op ≥ , Sp ) → Sp are symmetricmonoidal for the Day convolution, and that the Whitehead tower functor Sp → Fun ( Z op ≥ , Sp ) isalso symmetric monoidal. In Section 3, we review some facts about ˇCech conerves and provea special case of our main theorem for generalized Eilenberg-Mac Lane spectra. In Section 4, wepresent an ∞ -categorical treatment of the May filtration on THH and use this to construct a similarfiltration on the C p -homotopy fixed points of THH. Before doing so, we give a brief review of thesymmetric monoidal envelope construction which appears in [Lur17, 2.2.4], which is crucial tothe ∞ -categorical treatment of the May filtration. In Section 5, we turn to the proof of our maintheorem, and use the May filtration in conjunction with our work in Section 3 to deduce the generalresult.Throughout this work, we freely use the theory of ∞ -categories, incarnated via quasicategories,as in [Lur09] and [Lur17]. For consistency, wherever possible, we also follow the notation therein.Throughout, we refer to the ∞ -category of spectra by Sp and use ⊗ to denote the smash product.Additionally, we will use the notation Sp ≥ to denote connective spectra, and CAlg ≥ to denoteconnective commutative ring spectra. We will fix the following convention that a spectrum X is n -connective provided that π m X (cid:27) m < n , and a map f : X → Y of spectra is n -connectiveprovided that π m f is an isomorphism in degrees m < n and a surjection when n = m . We will makefrequent use of the groundbreaking work of Nikolaus-Scholze, providing an ∞ -categorical treat-ment of topological Hochschild homology, topological cyclic homology, and cyclotomic spectra,[NS18, III.2.3, II.1.8, II.1.6]. Acknowledgements:
The author extends his greatest thanks to his Ph.D. advisor, Tyler Law-son. His insights, suggestions, and patience have been invaluable throughout the course of thiswork. The author would also like to thank John Rognes, for his suggestion to use the work of[DR18], and Martin Speirs and Joel Stapleton, for providing helpful feedback on a draft. Addition-ally, conversations with Micah Darrell and David Mehrle helped improve the author’s understand-ing of this material.
Our main technical tool is the May filtration on THH, as defined in [AKS18], which we must importinto the ∞ -categorical context. To do so, we establish several folklore results about filtered (resp. N -filtered) spectra. While the May filtration we consider is indexed by Z op ≥ , it will be technicallyuseful to work with Z op -indexed filtrations as well. Unless otherwise specified, Z op and Z op ≥ carrythe usual ordering, and by abuse, we use the same notation to refer to the nerves of these posets. Definition 2.1.
The ∞ -category of filtered spectra is Fil ( Sp ) = Fun ( Z op , Sp ) , and the ∞ -categoryof N -filtered spectra is Tow ( Sp ) = Fun ( Z op ≥ , Sp ) .It is readily checked that both of these ∞ -categories are presentably symmetric monoidal andstable where the symmetric monoidal product in both cases is Day convolution; see [Lur09, 5.5.3.6],[Lur17, 1.1.3.1], and [Gla13, 2.13]. Throughout, we will let ⊛ denote the Day convolution productand allow context to dictate exactly which product we mean. The unit object in Fil ( Sp ) , denoted2y Fil , is the filtered spectrum · · · → → → S → S → · · · which is S in degrees n ≤ ( Sp ) ,denoted by Tow , is the N -filtered spectrum which is S in degree 0, and 0 otherwise: · · · → → → S Because Fil ( Sp ) and Tow ( Sp ) are symmetric monoidal, we may consider the ∞ -categories of asso-ciative (resp. commutative) algebra objects, which we denote by Alg Fil (resp. CAlg
Fil ) and Alg
Tow (resp. CAlg
Tow ). These categories will appear later when we work with the May filtration on THH.The following are the technical claims we seek to prove:1. The evaluation functor ev : Tow ( Sp ) → Sp is symmetric monoidal;2. the associated graded functor gr ∗ : Tow ( Sp ) → Sp is colimit-preserving and symmetricmonoidal functor; and3. the Whitehead tower functor Sp → Tow ( Sp ) is lax monoidal.Before commencing, we will record several general results regarding Fil ( Sp ) , Tow ( Sp ) , and theirrelationship to one another. Fil ( Sp ) , Tow ( Sp ) , and Day convolution Recall, that for C a stable ∞ -category, an object X ∈ C is said to generate C , provided that π Map C ( X , Y ) ≃∗ implies that Y ≃
0. We will say that a collection of objects of C , { X i } i ∈ I , jointly generates C , pro-vided that π Map ( X i , Y ) ≃ ∗ for all i ∈ I implies that Y ≃ Lemma 2.2.
Let K be a simplicial set and let ev k : Fun ( K , Sp ) → Sp denote evaluation at k ∈ K .For all objects k ∈ K , the functors ev k admit left adjoints, L k : Sp → Fun ( K , Sp ) given on vertices by L k X : k ′ X ⊗ Map K ( k , k ′ ) . Additionally, the objects S ( k ) = L k S have the following properties:1. S ( k ) is compact for all k ∈ K ;2. the S ( k ) ’s jointly generate Fun ( K , Sp ) ; and3. the collection { Σ n S ( k )} n ∈ Z , k ∈ K generates Fun ( K , Sp ) under small colimits.Proof. Since ev k preserves small limits and colimits, it admits a left adjoint L k : Sp → Fun ( K , Sp ) .The compactness of S ( k ) follows from the fact that S is compact in Sp and ev k preserves colimits.To explicitly identify L k , we use the following chain of natural equivalences obtained from the endformula, which appears for example, in [GHN17, 5.2]Map Fun ( K , Sp ) ( X ⊗ Map K ( k , −) , E • ) ≃ lim ←−− i → j ∈ Tw ( K ) Map Sp (cid:0) lim −−→ Map K ( k , i ) X , E j (cid:1) ≃ lim ←−− i → j ∈ Tw ( K ) lim ←−− Map K ( k , i ) Map Sp ( X , E j )≃ Map Sp ( X , lim ←−− i → j ∈ Tw ( K ) E Map K ( k , i ) j ) As the exponential object E Map K ( k , i ) j is equivalent to the mapping spectrum Map Sp ( Σ ∞ + Map K ( k , i ) , E j ) ,an application of the end formula and the spectral co-Yoneda lemma yield:Map Sp ( X , lim ←−− i → j ∈ Tw ( K ) E Map K ( k , i ) j ) ≃ Map Sp (cid:16) X , Map
Fun ( K , Sp ) ( Σ ∞ + Map K ( k , −) , E • ) (cid:17) ≃ Map Sp ( X , E k ) . L k is given on vertices as claimed.To prove the final assertions, note that as Map Fun ( K , Sp ) ( S ( k ) , E • ) ≃ E k , the collection { S ( k )} k ∈ K jointly generates the stable ∞ -category Fun ( K , Sp ) . This allows us to mimic the proof of [Lur17,1.4.4.2] to show that the objects Σ n S ( k ) generate Fun ( K , Sp ) under small colimits. (cid:3) Example . In the case where K = Z op , the object S ( k ) is the tower with S in degrees ≤ k and 0otherwise. The maps S ( k ) n → S ( k ) n − are the identity if both objects are S . Similarly, if K = Z op ≥ ,the objects S ( k ) are given by S with identity maps in the range [ , k ] and 0 otherwise.This next lemma will give us a convenient way to calculate Day convolution products of ( N -)filtered spectra. Lemma 2.4.
Let ⊕ : Z op × Z op → Z op denote the monoidal product and let A k ⊆ B k = ( Z op × Z op ) × Z op Z op / k denote the full subcategory of those pairs ( n , m ) such that k + ≥ n + m ≥ k . Then theinclusion A k ⊆ B k is cofinal.Proof. This is a straightforward application of the ∞ -categorical version of Quillen’s Theorem A,[Lur09, 4.1.3.1]. For ( r , s ) ∈ B k , where r + s > k , the nerve of A k × B k ( B k ) ( r , s )/ is equivalentto finitely many copies of Λ glued together along their initial and terminal vertices in a zig-zagpattern, hence contractible. If r + s = k , then the nerve of A k × B k ( B k ) ( r , s )/ is contractible. In eithercase, the nerve is contractible, so we may conclude by Quillen’s Theorem A. (cid:3) Remark . A virtually identical proof holds for Z op ≥ as well.Using Lemma 2.4, we can now determine how the L k S ’s interact through Day convolution. Lemma 2.6.
Let K denote either Z op or Z op ≥ . There is a natural equivalence S ( n ) ⊛ S ( m ) ≃ S ( n + m ) . Proof.
This is a straightforward, though tedious, computation. By Lemma 2.4, we can compute ( S ( n ) ⊛ S ( m )) k as the colimit of the diagram S ( n ) s ⊗ S ( m ) k + − s S ( n ) s + ⊗ S ( m ) k − s S ( n ) s + ⊗ S ( m ) k − s − · · ·· · · S ( n ) s ⊗ S ( m ) k − s S ( n ) s + ⊗ S ( m ) k − s − S ( n ) s + ⊗ S ( m ) k − s − ←→ ←→ ←→ ←→ ←→ ←→ ←→ The objects are either of the form S ( n ) s ⊗ S ( m ) k − s or S ( n ) s ⊗ S ( m ) k + − s . When k > n + m , both S ( n ) s ⊗ S ( m ) k − s and S ( n ) s ⊗ S ( m ) k + − s are trivial, and hence the colimit of the diagram above is 0.In the case where k ≤ n + m , the diagram is nonzero in a range: between the objects S ( n ) k − m ⊗ S ( m ) k −( k − m ) and S ( n ) n ⊗ S ( m ) k − n . In fact, in this range, all objects are given by S and all mapsbetween nonzero objects are equivalent to id : S → S , by Example ?? . This implies that the colimitof the relevant diagram is given by S . In summary, S ( n ) ⊛ S ( m ) is zero in degrees k > n + m andand given by S with identity maps in degrees k ≤ n + m . (cid:3) Proposition 2.7.
Let E ⊆ Fil ( Sp ) denote the full subcategory of those filtered spectra with the prop-erty that the maps X n → X n − are equivalences for n ≤ . Then, E is a symmetric monoidal subcate-gory of Fil ( Sp ) , and the restriction map Fil ( Sp ) → Tow ( Sp ) induced by the inclusion Z op ≥ ⊆ Z op is asymmetric monoidal equivalence E → Tow ( Sp ) .Proof. By [Lur09, 4.3.2.15], there is an equivalence of ∞ -categories θ : E → Tow ( Sp ) given byprecomposing with the inclusion Z op ≥ ⊆ Z op . We will prove that E is a symmetric monoidal sub-category of Fil ( Sp ) and that θ is symmetric monoidal.Note that S ( n ) in E restricts to S ( n ) in Tow ( Sp ) . Since the restriction map θ : E → Tow ( Sp ) is anequivalence and compatible with the formation of iterated suspensions, we find that the desuspen-sions of the objects S ( n ) , where n ≥
0, generate E under small colimits. Thus, to show that E is asymmetric monoidal subcategory, we can immediately reduce to showing that Σ s S ( n ) ⊛ Σ t S ( m ) ∈ E ⊛ commutes with colimits separately in each variable. Additionally, since Σ r : Fil ( Sp ) → Fil ( Sp ) is an equivalence for all r ∈ Z , the result follows from Lemma 2.6.It remains to prove that the canonical map θ E , F : θ ( E • ) ⊛ θ ( F • ) → θ ( E • ⊛ F • ) is an equivalence for all objects in E . Fix E • and let X denote the full stable subcategory of E consisting of those towers F • for which θ E , F is an equivalence. Since this category is closed withrespect to colimits, we can reduce to checking the claim for objects of the form S ( n ) where n ≥ θ ( S ( n )) ⊛ θ ( S ( m )) → θ ( S ( n ) ⊛ S ( m )) ≃ θ ( S ( n + m )) is an equivalence by Lemma 2.6. This implies that θ is symmetric monoidal and hence Tow ( Sp ) → E ⊆ Fil ( Sp ) is symmetric monoidal. (cid:3) For the remainder of Section 2, we proceed with the proofs of the desired folklore claims.
It will be convenient to follow the notation appearing in [Gla13], the relevant parts of which wenow recall.
Definition 2.8.
Let C ⊗ and D ⊗ be two symmetric monoidal ∞ -categories. Define Fun ( C , D ) ⊗ tobe the simplicial set over Fin ∗ defined by the universal propertyFun Fin ∗ ( K , Fun ( C , D ) ⊗ ) = Fun
Fin ∗ ( C ⊗ k , D ⊗ ) where C ⊗ k is the pullback of C ⊗ → Fin ∗ along the given structure map k : K → Fin ∗ . This simplicialset is an ∞ -category by [Gla13, 2.3]. Observation . If S ∈ Fin ∗ , we let S o denote S \ {∗} . Let C ⊗ be a symmetric monoidal ∞ -catego-ry and let ( Fin ∗ ) act / S denote the full subcategory of ( Fin ∗ ) / S spanned by the active morphisms to S .Then for any S ∈ Fin ∗ there is a canonical product decomposition ( Fin ∗ ) / S ≃ Ö s ∈ S o ( Fin ∗ ) act /{ s } + ! × Fin ∗ This implies that for any morphism f : S → T in Fin ∗ , there is a canonical decomposition C ⊗ f ≃ © « Ö ∆ , t ∈ T o C ⊗ µ f − ( t ) + ª®¬ × C ⊗ β f − (∗) . For V ∈ Fin ∗ , the map µ V is given by the active map V → h i if V is nonempty, and the inclusion ∗ → h i if V is empty. The map β V denotes the unique map V → ∗ . One can check that thisdecomposition of C ⊗ f is compatible with the decompositions of C ⊗ S and C ⊗ T . Definition 2.10. [Gla13, 2.8] Let C ⊗ and D ⊗ be symmetric monoidal ∞ -categories so that thetensor product of D ⊗ preserves colimits separately in each variable. The Day convolution sym-metric monoidal ∞ -category is the largest simplicial subset of Fun ( C , D ) ⊗ whose vertices over S correspond to functors F : C ⊗ S → D ⊗ S which lie in the essential image of the inclusion Ö s ∈ S o Fun ( C , D ) → Fun ( C ⊗ S , D ⊗ S ) f : S → T correspond to functors F : C ⊗ f → D ⊗ f in the essential image ofthe inclusion Ö t ∈ T o Fun ∆ ( C ⊗ µ f − ( t ) + , D ⊗ µ f − ( t ) + ) × Fun ∆ ( C ⊗ β f − (∗) , D ⊗ β f − (∗) ) → Fun ∆ ( C ⊗ f , D ⊗ f ) By [Gla13, 2.10], this is a symmetric monoidal ∞ -category, and in the case where C ⊗ also preservescolimits separately, Day convolution preserves colimits separately in each variable [Gla13, 2.13].With our notation set, we are ready to prove that ev : Fun ( Z op ≥ , Sp ) → Sp is symmetric monoidal.
Proposition 2.11.
Let C ⊗ , D ⊗ and E ⊗ be symmetric monoidal ∞ -categories where the tensor productof E ⊗ preserves colimits separately in each variable, and let F ⊗ : C ⊗ → D ⊗ be a symmetric monoidalfunctor. Then there is an induced "pullback" functor F ∗ : Fun ( D , E ) ⊗ → Fun ( C , E ) ⊗ given by precomposition with F ⊗ . The "pullback" functor ev ⊗ , which is induced by ∆ → Z op ≥ , issymmetric monoidal.Proof. The universal property of Fun ( D , E ) ⊗ from [Gla13, 2.1], immediately implies that F ⊗ inducesa functor of ∞ -categories F ∗ : Fun ( D , E ) ⊗ → Fun ( C , E ) ⊗ . It now suffices to check that the restriction of F ∗ to Fun ( D , E ) ⊗ factors through Fun ( C , E ) ⊗ .The vertices of Fun ( D , E ) ⊗ over S ∈ Fin ∗ are precisely those functors D ⊗ S → E ⊗ S in the essentialimage of the inclusion Fun ( D , E ) S → Fun ( D ⊗ S , E ⊗ S ) . Since F ⊗ is symmetric monoidal, F ⊗ S ≃ F S , sothe vertices of Fun ( D , E ) ⊗ are sent to the vertices of Fun ( C , E ) ⊗ . Similarly, the edges are functors F ⊗ f : D ⊗ f → E ⊗ f in the essential image of the map Ö t ∈ T o Fun ∆ ( D ⊗ µ S + t , E ⊗ µ S + t ) ! × Fun ( D ⊗ β S ∗ , E ⊗ β S ∗ ) → Fun ∆ ( D ⊗ f , E ⊗ f ) , where S + t = f − ({ t }) + and S ∗ = f − ({∗}) . As before, F ⊗ f ≃ (cid:16)Î ∆ , t ∈ T o F ⊗ µ S + t (cid:17) × F ⊗ β S ∗ , which meansthe edges in Fun ( D , E ) ⊗ are sent to the edges in Fun ( C , E ) ⊗ . This implies we have a well-definedfunctor F ∗ as claimed.Let K ⊗ denote the nerve of the category of operators of Z op ≥ (this construction appears in [Lur17,2.0.0.2], for example). The map 0 : ∆ → Z op ≥ induces a functor e : Fin ∗ → K ⊗ which induces a"pullback" ev ⊗ : Tow ( E ) ⊗ = Fun ( Z op ≥ , E ) ⊗ → Fun ( ∆ , E ) ⊗ ≃ E ⊗ . To show that ev ⊗ is symmetricmonoidal, we must show that it takes cocartesian edges to cocartesian edges. By [Gla13, 2.10], thecocartesian edges over f : S → T in Tow ( E ) ⊗ are precisely those functors F f : K ⊗ f → E ⊗ f whosedecomposition components are p -left Kan extensions. As ev ⊗ ( F f ) is the composite ∆ (cid:27) Fin ∗ × Fin ∗ ∆ K ⊗ × Fin ∗ ∆ E ⊗ f E ⊗ , ← → ← → F f ← → by [Lur09, 4.3.1.4] and [Lur09, 4.3.1.15] it suffices to show that the diagram ∆ K ⊗ S E ⊗ f ∆ K ⊗ f ∆ ← → e S ←→ ←→ ← → F S ←→ p ← → e f ← → ← → F f ⊗ ( F f ) as the p -colimit of F S ◦ e S . By assumption, we may decompose F f into a productof active and inert pieces, all of which are relative left Kan extensions. Furthermore, since theformation of overcategories commutes with pullbacks, we only need show that ev ⊗ ( F д ) is a p -colimit, where д is one of the active or inert parts of f , as in Observation 2.9. Therefore, the resultwill follow if we can prove the following claim – Let f : S → T be a morphism in Fin ∗ , and let F f : K ⊗ f → E ⊗ f be p -left Kan extension of F f | K ⊗ S , exhibited by the diagram: K ⊗ S E ⊗ f K ⊗ f ∆ ←→ ← → ←→ p ← → ← → Then, F f ◦ e f : ∆ → E ⊗ f is a p -left Kan extension of F f | K ⊗ S ◦ e S : ∆ → E ⊗ f along the sourceinclusion ∆ ⊆ ∆ . To prove this, note that the edge Fin ∗ × Fin ∗ ∆ → K ⊗ f is a lift of f : S → T , with source ( , . . . , ) ∈ N ( Z op ≥ ) S ≃ K ⊗ S . The structure of K ⊗ forces the target of ˜ f to be ( , . . . , ) ∈ N ( Z op ≥ ) T ≃ K ⊗ T . The functors ∆ / → ( K ⊗ S ) / ˜ f ( ) and ∆ / → ( K ⊗ S ) / ˜ f ( ) are cofinalby [Lur09, 4.1.2.6], as everything in sight is a contractible Kan complex. This shows that all therelevant induced diagrams are p -colimits, so that ∆ → E ⊗ f is a p -left Kan extension, and hence acocartesian arrow. (cid:3) In this section, we establish that the associated graded functor gr ∗ : Tow ( Sp ) → Sp is symmetricmonoidal by exploiting the result for Fil ( Sp ) , established in [Lur15, 3.2.1]. Let ( Z op ) disc be Z op endowed with the discrete topology, and let Gr ( Sp ) = Fun (( Z op ) disc , Sp ) denote the symmetricmonoidal ∞ -category of graded spectra equipped with the Day convolution product Proposition 2.12. [Lur15, 3.2.1]
There exists a functor, gr : Fil ( Sp ) → Gr ( Sp ) , given on vertices by X •
7→ ( X i / X i − ) i ∈ Z , which is symmetric monoidal for the Day convolution and colimit preserving. As Gr ( Sp ) ≃ Î i ∈ Z Sp, there is a colimit-preserving symmetric monoidal functor É : Gr ( Sp ) → Sp given on vertices by ( X i ) i ∈ Z É i ∈ Z X i . Now define gr Fil ∗ : Fil ( Sp ) → Sp as É ◦ gr, andlet gr ∗ : Tow ( Sp ) → Sp denote the composition of gr
Fil ∗ with the symmetric monoidal functor,Tow ( Sp ) → Fil ( Sp ) , from Proposition 2.7. Proposition 2.13.
The associated graded functor gr ∗ : Tow ( Sp ) → Sp , is colimit preserving andsymmetric monoidal.Proof. As Tow ( Sp ) → Fil ( Sp ) is symmetric monoidal, it is immediate that gr ∗ is symmetric monoidal.The fact that gr ∗ is colimit-preserving follows from the fact that gr Fil ∗ is colimit-preserving and thatthe functor Tow ( Sp ) → Fil ( Sp ) is given by left Kan extension. (cid:3) Corollary 2.14. gr ∗ induces functors gr ∗ : Alg Tow → Alg and gr ∗ : CAlg Tow → CAlg whichpreserve sifted colimits.Proof.
As gr ∗ is symmetric monoidal, it induces maps functors on commutative and associativealgebras by definition ([Lur17, 2.1.3.1]). Additionally, [Lur17, 3.2.3.1] guarantees that gr ∗ preservessifted colimits of associative and commutative algebras. (cid:3) Recall, that for all n ∈ Z , there are functors τ ≥ n : Sp → Sp ≥ n , which are right adjoint to theinclusions i n : Sp ≥ n ⊆ Sp [Lur17, 1.2.1.7]. The composites i n ◦ τ ≥ n : Sp → Sp are colocalizationfunctors, and the inclusions · · · ⊆ Sp ≥ n ⊆ Sp ≥ n − ⊆ · · · ⊆ Sp ≥ ⊆ Sp induce a diagram, Z op ≥ → Fun ( Sp , Sp ) , given by · · · i n ◦ τ ≥ n → i n − ◦ τ ≥ n − → · · · → i ◦ τ ≥ . T : Tow ( Sp ) → Tow ( Sp ) which is given on verticesby { X n } n ≥
7→ { τ ≥ n X n } n ≥ . Definition 2.15.
The
Whitehead tower functor is the composite W = T ◦ δ : Sp → Tow ( Sp ) , where δ is the constant tower functor. Proposition 2.16.
The Whitehead tower functor Sp → Tow ( Sp ) is lax symmetric monoidal.Proof. Let E ⊆ Tow ( Sp ) denote the full subcategory of N -filtered spectra, X • , with the propertythat X n ≃ τ ≥ n X n . Additionally, note that the essential image of T is exactly E ; this yields a rightadjoint R : Tow ( Sp ) → E , which is a colocalization. It remains to check the hypotheses of [Lur17,2.2.1.1] are satisfied, and by [Lur17, 2.2.1.2] it suffices to show that E contains the unit and is closedwith respect to the Day convolution for Tow ( Sp ) . Certainly Tow ∈ E , and if X • , Y • ∈ E , ( X • ⊛ Y • ) n = lim −−→ p + q ≥ n X p ⊗ Y q ∈ Sp ≥ n since Sp ≥ n is stable under colimits in Sp. We conclude that E ⊆ Tow ( Sp ) is symmetric monoidaland R : Tow ( Sp ) → E is lax monoidal, meaning that T is also lax monoidal. As the constant towerfunctor δ : Sp → Tow ( Sp ) is symmetric monoidal, we may conclude the result. (cid:3) Remark . Since W is lax monoidal, we obtain functors Alg → Alg
Tow and CAlg → CAlg
Tow .In particular, for A ∈ CAlg ≥ , W produces a multiplicative tower · · · → τ ≥ n A → τ ≥ n − A → · · · → τ ≥ A ≃ A , where the k -th graded term is given by Σ k Hπ k A . In this section, we prove a special case of our main theorem. Before doing so, we establish somefacts about ˇCech conerves and recall a few definitions.
Definition 3.1.
Let C be an ∞ -category which admits finite colimits, and let f : X → Y be anarrow in C . Then the augmented ˇ Cech conerve of f , denoted by C • ( f ) + : N ( ∆ + ) → C is the leftKan extension of f : ∆ → C along the inclusion N ( ∆ ≤ + ) ⊆ N ( ∆ + ) . The ˇ Cech conerve , denoted by C • ( f ) is the restriction of C • ( f ) + to N ( ∆ ) . In what follows, we may abuse notation and conflate C • ( f ) + and C • ( f ) . Example . Let f : A → B be a morphism in CAlg. As the coproduct in CAlg is given by therelative smash product, the augmented ˇCech conerve is simply the augmented cobar complex of f : A B B ⊗ A B · · ·← → ← →← → ← →← →← → We will be primarily interested in this cobar complex when f is faithfully flat. Definition 3.3.
Let f : A → B be a morphism in CAlg. We say that f is faithfully flat , providedthe following conditions hold:1. The map π f : π A → π B is a faithfully flat map of commutative rings; and2. the induced map ( π B ) ⊗ π A ( π ∗ A ) → π ∗ B is an isomorphism of graded rings.As in the discrete case, faithfully flat morphisms determine a Grothendieck topology on bothCAlg and CAlg ≥ , see [Lur18, B.6.1.3, B.6.1.7]. Lemma 3.4.
Let C be an ∞ -category which admits finite colimits and let F : ∆ × ∆ → C be acommutative square, such that α = F | { }× ∆ and β = F | { }× ∆ are equivalences in C , which we depictas YX ′ Y ′ ← → f ←→ α ←→ β ← → д Then, the induced map C • ( f ) + → C • ( д ) + is an equivalence of augmented cosimplicial objects.Proof. Since C admits finite colimits we can take the left Kan extension of F along the inclusion ∆ ≤ + × ∆ ⊆ ∆ + × ∆ , yielding a morphism of augmented cosimplicial objects e F : ∆ + × ∆ → C . Itsuffices to check that e F | { n }× ∆ is an equivalence for all n ∈ ∆ + . Since e F | { n }× ∆ is given by the map Y Ý X · · · Ý X Y → Y ′ Ý X ′ · · · Ý X ′ Y ′ induced by α and β , it is an equivalence. (cid:3) Lemma 3.5.
Let C be an ∞ -category which admits finite colimits and let D : ∆ × ∆ → C be acoCartesian square, depicted as X YX ′ Y ′ ← → f ←→ ϕ ←→ ← → f ′ Then, the natural map C • ( f ) Ý X X ′ → C • ( f ′ ) is an equivalence of cosimplicial objects of C .Proof. First, we construct a pushout functor. Let E be the full subcategory of Fun ( ∆ × ∆ , C ) consisting of pushout diagrams. The restriction functor E → Fun ( Λ , C ) is an acyclic fibrationby [Lur09, 4.3.2.15]. However, since Λ (cid:27) ( ∂ ∆ ) ⊳ (cid:27) ∆ Ý ∆ { } ∆ we know that Fun ( Λ , C ) (cid:27) Fun ( ∆ , C ) × C Fun ( ∆ , C ) . Pulling back along the inclusion of the vertex { ϕ : X → X ′ } , we obtainan acyclic fibration q : E × Fun ( ∆ , C ) { ϕ } → Fun ( ∆ , C ) × C { X } . By choosing a section of q and restricting to the opposite edge, we obtain a functor (−) Þ X X ′ : Fun ( ∆ , C ) × C { X } → E × Fun ( ∆ , C ) { ϕ } → Fun ( ∆ , C ) × C { X ′ } which takes X → Y to X ′ → Y Ý X X ′ and preserves colimits on account that the section of q is a left adjoint and colimits in functor categories are computed pointwise. By taking left Kanextensions, we obtain the following commutative diagram: ∆ ≤ + Fun ( ∆ , C ) × C { X } Fun ( ∆ , C ) × C { X ′ } ∆ + ←→ ← → f ′ ← → f ← → ← → C • ( f ) + ← → C • ( f ′ ) + By the universal property, we obtain a natural map C • ( f ) + Ý X X ′ → C • ( f ′ ) + . This is an equiva-lence, since left Kan extensions are calculated pointwise as colimits and (−) Ý X X ′ preserves them.Finally, by projecting down to C , we obtain the desired equivalence of augmented cosimplicialobjects of C . (cid:3) We now record two key results used in our proof of Proposition 3.8.
Theorem 3.6. [BMS19, Corollary 3.4, Remark 3.5]
The functors
THH (−) and
THH (−) hC p are Sp -valued sheaves for the fpqc topology on the category of commutative rings. Theorem 3.7. [DR18, Theorem 1.2]
Let A → B be a 1-connected morphism in CAlg ≥ , and let F = THH or THH (−) hC p . Then, the natural map ( A ) lim ←−− (cid:18) F ( B ) F ( B ⊗ A B ) · · · (cid:19) ← → ← →← → ← →← →← → is an equivalence in Sp . Proposition 3.8.
Let f : A → B be a faithfully flat map in CAlg ≥ , which induces a faithfully flatmap Hπ ∗ A → Hπ ∗ B . The natural map THH ( Hπ ∗ A ) lim ←−− (cid:18) THH ( Hπ ∗ B ) THH ( Hπ ∗ B ⊗ H π ∗ A Hπ ∗ B ) · · · (cid:19) ← → ← →← → ← →← →← → is an equivalence in Sp .Proof. For brevity, we will write A ∗ (resp. B ∗ ) for Hπ ∗ A (resp. Hπ ∗ B ) and A (resp. B ) for Hπ A (resp. Hπ B ). Since f is faithfully flat, we find that the commutative square A ∗ B ∗ A B ← → f ←→ ϕ A ∗ ←→ ϕ B ∗ ← → д is a pushout in CAlg ≥ by a Künneth spectral sequence calculation. Viewing this square as afunctor ∆ ≤ + × ∆ ≤ + → CAlg ≥ we can take its left Kan extension along the inclusion ∆ ≤ + × ∆ ≤ + ⊆ ∆ + × ∆ + , which we denote by X • , • + : ∆ + × ∆ + → CAlg ≥ . By [Lur09, 4.3.2.8] and the dual of [Lur09,6.1.2.11], we can realize this diagram as the degreewise ˇCech conerves of either C • ( f ) + → C • ( д ) + or C • ( ϕ A ∗ ) + → C • ( ϕ B ∗ ) + . In virtue of this, we may identify the n -th row in the bisimplical diagram, X • , n + , as the augmented ˇCech conerve of the map A ⊗ A ∗ · · ·⊗ A ∗ A → B ⊗ B ∗ · · ·⊗ B ∗ B , and the m -thcolumn, X m , • + , as the augmented ˇCech conerve of the map B ∗ ⊗ A ∗ · · · ⊗ A ∗ B ∗ → B ⊗ A · · · ⊗ A B .Note that the diagram, THH ◦ X • , • + , induces the following commutative squareTHH ( X − , − + ) lim ←−− m ∈ ∆ THH ( X m , − + ) lim ←−− n ∈ ∆ THH ( X − , n + ) lim ←−− n ∈ ∆ lim ←−− m ∈ ∆ THH ( X m , n + )← → ←→ ←→ ← → (3.1)Since X • , − + is the augmented ˇCech conerve of the map A ∗ → B ∗ , the result will follow if wecan prove all the other arrows in Diagram 3.1 are equivalences. As the maps X m , − + → X m , + are1-connected for all m ≥ −
1, Theorem 3.7 implies that the mapsTHH ( X m , − + ) → lim ←−− n ∈ ∆ THH ( X m , n + ) are equivalences for all m ≥ −
1. Thus, the vertical maps in the commutative square are equiva-lences, by commuting the limits in the bottom right corner. It remains to show thatTHH ( X − , n + ) → lim ←−− m ∈ ∆ THH ( X m , n + ) is an equivalence for all n ≥
0, which we prove by induction. The case n = n . As noted above, the n -th row of X • , • + is given bythe augmented ˇCech conerve of the map A ⊗ A ∗ · · · ⊗ A ∗ A → B ⊗ B ∗ · · · ⊗ B ∗ B . However, thisaugmented cobar construction is equivalent to the augmented ˇCech conerve of the map id ⊗ · · · ⊗ id ⊗ f : A ⊗ A ∗ · · · ⊗ A ∗ A → A ⊗ A ∗ · · · ⊗ A ∗ A ⊗ A ∗ B . Since the following diagram is a retract in CAlg,10 A ∗ A B B ∗ B ←→ f ← → ←→ f ∗ ← → ←→ f ← → ← → we obtain a retract diagram of augmented ˇCech conerves C • ( id ⊗ n ⊗ f ) + → C • ( id ⊗( n − ) ⊗ f ) + → C • ( id ⊗ n ⊗ f ) + and hence a retract diagramTHH (cid:0) C • ( id ⊗ n ⊗ f ) + (cid:1) → THH (cid:0) C • ( id ⊗ n − ⊗ f ) + (cid:1) → THH (cid:0) C • ( id ⊗ n ⊗ f ) + (cid:1) . By the inductive hypothesis, we may conclude that THH (cid:0) C • ( id ⊗ n ⊗ f ) + (cid:1) is a limit diagram. Thiscompletes the proof. (cid:3) Proposition 3.9.
Let f : A → B be a faithfully flat map in CAlg ≥ . Then the natural map THH ( Hπ ∗ A ) hC p → lim ←−− (cid:16) THH ( Hπ ∗ B ⊗ Hπ ∗ A • ) hC p (cid:17) is a limit diagram.Proof. This argument is identical to the one in Proposition 3.8 since we can apply Theorems 3.7and 3.6 for THH (−) hC p . (cid:3) THH
Let A be a connective algebra spectrum and let A • be its Whitehead tower. The May filtration onTHH ( A ) , as defined in [AKS18], is the geometric realization of the following diagram; A • A • ⊛ A • A • ⊛ A • ⊛ A • · · ·←→ ←→ ←→ ←→ ←→ ←→ ←→ ←→ ←→ i.e. the cyclic bar construction of A • with respect to the Day convolution product on Tow ( Sp ) . Inorder to make this construction precise in the ∞ -categorical language, we mimic the constructionof THH in [NS18]. One prerequisite therein, is the symmetric monoidal envelope of [Lur17, 2.2.4.1].After a brief discussion of the monoidal envelope, we present a ∞ -categorical description of theMay filtration, and establish several useful properties. Remark . The cyclic bar construction can be performed for any associative algebra in a pre-sentably symmetric monoidal ∞ -category. Since we are primarily interested in the Whiteheadtowers of algebra spectra and the Day convolution product, we will not work in such generalterms, though many of the results in this section will hold in greater generality. Definition 4.2. [Lur17, 2.2.4.1] Let p : C ⊗ → Fin ∗ be a symmetric monoidal ∞ -category. The monoidal envelope of C ⊗ , is the fiber productEnv ( C ) ⊗ = C ⊗ × Fun ({ } , Fin ∗ ) Act ( Fin ∗ ) , where Act ( Fin ∗ ) ⊆ Fun ( ∆ , Fin ∗ ) is the full subcategory spanned by the active morphisms.By [Lur17, 2.2.4.4], evaluation at { } ⊆ ∆ induces a map q : Env ( C ) ⊗ → Fin ∗ , exhibitingEnv ( C ) ⊗ as a symmetric monoidal ∞ -category. If we let Env ( C ) = Env ( C ) ⊗h i denote the fiber of q over h i , by unwinding the definition, we may identify Env ( C ) with the subcategory C ⊗ act ⊆ C ⊗ spanned by objects and active morphisms between them. In other words, C ⊗ act is a symmetricmonoidal ∞ -category, and its monoidal product, ⊕ : C ⊗ act × C ⊗ act → C ⊗ act , may be described asfollows: for X ∈ C ⊗h n i and Y ∈ C ⊗h m i corresponding to sequences ( X i ) ni = and ( Y j ) mj = , the product X ⊕ Y corresponds to the concatenation ( X i ) ni = ∪ ( Y j ) mj = ∈ C ⊗h n + m i .11here is a diagonal embedding Fin ∗ → Act ( Fin ∗ ) , and the pullback along this embedding in-duces a lax symmetric monoidal inclusion i C : C ⊗ ⊆ Env ( C ) ⊗ , [Lur17, 2.2.4.4, 2.2.4.16]. Thefollowing proposition provides a useful characterization of symmetric monoidal functors out ofEnv ( C ) ⊗ . Proposition 4.3. [Lur17, 2.2.4.9]
Let C ⊗ and D ⊗ be symmetric monoidal ∞ -categories. The inclusion i C : C ⊗ ⊆ Env ( C ) ⊗ induces an equivalence of ∞ -categories Fun ⊗ ( Env ( C ) , D ) ≃ Alg C ( D ) . Here,
Fun ⊗ ( Env ( C ) , D ) denotes the ∞ -category of symmetric monoidal functors Env ( C ) ⊗ → D ⊗ and Alg C ( D ) is the ∞ -category of lax symmetric monoidal functors C ⊗ → D ⊗ . Under this equivalence, the identity functor C → C , which is lax monoidal, corresponds toa symmetric monoidal functor, ⊗ C : Env ( C ) = C ⊗ act → C , given on vertices by ( X , . . . , X n ) 7→ X ⊗ · · · ⊗ X n . In order to establish several key properties of the May filtration on THH, weneed a lemma on the interaction of ⊗ C with symmetric monoidal functors. Suppose we are givena symmetric monoidal functor, F ⊗ : C ⊗ → D ⊗ . Pulling this morphism back along Act ( Fin ∗ ) ,produces a functor Env ( F ) ⊗ : Env ( C ) ⊗ → Env ( D ) ⊗ which is also symmetric monoidal by [Lur17,2.2.4.15]. Env ( F ) ⊗ fits into the following diagram, C ⊗ D ⊗ Env ( C ) ⊗ Env ( D ) ⊗ ← → F ⊗ ←→ i C ←→ i D ← → Env ( F ) ⊗ which is commutative in virtue of the fact that i C and i D are induced by the diagonal. Lemma 4.4.
Let F ⊗ : C ⊗ → D ⊗ be a symmetric monoidal functor. There is an equivalence ⊗ D ◦ Env ( F ) ⊗ ≃ F ⊗ ◦ ⊗ C . Proof.
As the equivalence in Proposition 4.3 is given by precomposing by the lax symmetric monoidalinclusion i C : C ⊗ ⊆ Env ( C ) ⊗ , we can check the desired functors are equivalent by showing that ⊗ D ◦ Env ( F ) ⊗ ◦ i C ≃ F ⊗ ◦ ⊗ C ◦ i C . This is immediate since Env ( F ) ⊗ ◦ i C ≃ i D ◦ F ⊗ , ⊗ C ◦ i C ≃ id , and ⊗ D ◦ i D ≃ id . (cid:3) Remark . In the case where C = Tow ( Sp ) , we will denote ⊗ C by ⊛ , and in the case where C = Sp,we will denote ⊗ C by ⊗ . THH
Definition 4.6.
Let A • ∈ Alg
Tow , given by a section A ⊗• : Assoc ⊗ → Tow ( Sp ) ⊗ . Let THH Tow ( A • ) ∈ Tow ( Sp ) B T denote the colimit of the diagram N ( Λ op ) Assoc ⊗ act Tow ( Sp ) ⊗ act Tow ( Sp ) , ← → V o ← → A ⊗• ← → ⊛ where V o is the map appearing in [NS18, B.1]. Remark . This is essentially the definition of THH as given in [NS18], the only salient differencebeing that we are using the Day convolution of towers to form the cyclic bar construction as op-posed to the smash product. Additionally, we know THH
Tow ( A • ) ∈ Tow ( Sp ) B T since the geometricrealization of a cyclic object admits a T -action, [NS18, B.5]. Proposition 4.8.
The functor
THH
Tow : Alg
Tow → Tow ( Sp ) B T is symmetric monoidal. roof. This essentially follows from the fact that Day convolution preserves sifted colimits. Thecareful reader should consult [Lur17, 3.2.4.3, 3.2.4.3] to explicitly construct the functor ( THH
Tow ) ⊗ : ( Alg
Tow ) ⊗ → ( Tow ( Sp ) B T ) ⊗ which exhibits THH Tow as symmetric monoidal. (cid:3)
Corollary 4.9.
THH
Tow induces a functor
CAlg
Tow → (
CAlg
Tow ) B T .Proof. Since THH
Tow is symmetric monoidal, it induces a functorCAlg ( Alg
Tow ) →
CAlg ( Tow ( Sp ) B T ) . As CAlg ( Alg
Tow ) ≃
CAlg ( Tow ( Sp )) and CAlg ( Tow ( Sp ) B T ) ≃ ( CAlg
Tow ) B T , we conclude the result. (cid:3) For connective commutative ring spectra, the May filtration on THH can be given an alternativeconstruction, following [AKS18]. Since CAlg
Tow is presentable, it is automatically tensored overspaces [Lur09, 4.4.4]; we denote this by X ⊗ A • . Using the simplicial model, ∆ / ∂ ∆ , for S , astandard argument shows we may identify S ⊗ A • in CAlg Tow with the cyclic bar construction of A • , where A • is the Whitehead tower of A ∈ CAlg ≥ . More generally, given A • ∈ CAlg
Tow and X afinite space, the tower X ⊗ A • ∈ CAlg
Tow gives a filtration on X ⊗ A , which is also called the Mayfiltration in [AKS18, 3.3.3].The remainder of this subsection is devoted an ∞ -categorical treatment of results in [AKS18].In particular, Proposition 4.10 allows us to define the May filtration on THH in the ∞ -categoricallanguage. Proposition 4.10.
Let A ∈ Alg ≥ and let A • denote the Whitehead tower of A . There is a canonicalequivalence of spectra with T -action, ev THH
Tow ( A • ) ≃ THH ( A ) . In the case where A ∈ CAlg ≥ ,this is an equivalence in CAlg B T .Proof. Proposition 2.11 states that ev : Tow ( Sp ) → Sp is symmetric monoidal so we may applyLemma 4.4 to extract the following commutative square.Tow ( Sp ) ⊗ act Tow ( Sp ) Sp ⊗ act Sp ← → ⊛ ←→ ( ev ) ⊗ act ←→ ev ← → ⊗ Additionally, note that ev commutes with colimits and that ev A • ≃ A . We obtain the followingchain of canonical equivalences in Sp B T :ev THH
Tow ( A • ) = ev lim −−→ (cid:0) ⊛ ◦( A ⊗• ) act ◦ V o (cid:1) ≃ lim −−→ (cid:0) ev ◦ ⊛ ◦( A ⊗• ) act ◦ V o (cid:1) ≃ lim −−→ (cid:0) ⊗ ◦ A ⊗ act ◦ V o (cid:1) = THH ( A ) . In the case where A ∈ CAlg ≥ , a similar argument works, since the induced functor ev : CAlg Tow → CAlg preserves sifted colimits, by [Lur17, 3.2.3.2]. (cid:3)
Remark . The proof of Proposition 4.10 yields a slightly more general result; there is a canonicalequivalence of spectra with T -action, ev THH
Tow ( A • ) ≃ THH ( A ) , for any A • ∈ Alg
Tow . Definition 4.12. [AKS18, 3.3.3, 3.4.9] Let A ∈ Alg ≥ and let A • denote the Whitehead tower of A .The May filtration of THH ( A ) is the tower THH Tow ( A • ) . We will let Fil May i denote ev i THH
Tow ( A • ) and let gr May i denote Fil May i / Fil
May i + . While our notation makes no reference to A , the context of ourusage should indicate the underlying connective ring spectrum.13f we view THH as a functor, THH : Alg → Sp B T , we obtain a tower of functors · · · → Fil
May i → · · · → Fil
May1 → THH . We call this tower the
May filtration of the functor
THH, and by taking cofibers, it induces anothertower · · · →
THH / Fil
May i → · · · → THH / Fil
May1 → . The following is the ∞ -categorical version of [AKS18, 3.3.10], which is crucial for the sequel. Proposition 4.13.
For A • ∈ Alg
Tow , there is a canonical equivalence of spectra with T -action, gr ∗ THH
Tow ( A • ) ≃ THH ( gr ∗ A • ) . In the case where A • ∈ CAlg
Tow , this is an equivalence in
CAlg B T .Proof. By Proposition 2.13 and Lemma 4.4, it follows thatTow ( Sp ) ⊗ act Tow ( Sp ) Sp ⊗ act Sp ← → ⊛ ←→ ( gr ⊗∗ ) act ←→ gr ∗ ← → ⊗ commutes. Additionally, since gr ∗ preserves,gr ∗ THH
Tow ( A • ) = gr ∗ lim −−→ (cid:0) ⊛ ◦( A ⊗• ) act ◦ V o (cid:1) ≃ lim −−→ (cid:0) gr ∗ ◦ ⊛ ◦( A ⊗• ) act ◦ V o (cid:1) ≃ lim −−→ (cid:0) ⊗ ◦ ( gr ∗ A • ) ⊗ act ◦ V o (cid:1) = THH ( gr ∗ A • ) . In the case where A ∈ CAlg ≥ , a similar argument works, since the induced functor gr ∗ : CAlg Tow → CAlg preserves sifted colimits, by [Lur17, 3.2.3.2]. (cid:3)
An immediate consequence of the previous result is the following:
Corollary 4.14. [AKS18, 3.4.11]
For A ∈ CAlg ≥ and A • the Whitehead tower of A , there is acanonical equivalence in CAlg B T , gr May ∗ THH ( A ) ≃ THH ( Hπ ∗ A ) . This result, combined with several others, is used to construct the THH-May spectral sequencesequence E ∗ , ∗ (cid:27) π ∗ THH ( Hπ ∗ A ) ⇒ π ∗ THH ( A ) ; see [AKS18, 3.4.8]. Proposition 4.15. [AKS18, 3.5.4] If A • ∈ Alg
Tow ≥ with the property that π m A n (cid:27) for all m < n ,then π m Fil
May n (cid:27) for all m < n as well.Proof. First, recall that if E and F are n and m -connective spectra, respectively, then E ⊗ F is ( n + m ) -connective. Next, note that Fil May n is the geometric realization of a simplicial object whose k -th termis given by ev n ( A ⊛ k • ) = lim −−→ i + ··· + i k ≥ n A i ⊗ · · · ⊗ A i k . As each A i m is i m -connective, by our recollection, each term in the simplicial object is n -connective,and hence Fil May n is n -connective. (cid:3) Corollary 4.16.
For A ∈ Alg ≥ , the May filtration has the property that lim ←−− n ≥ (cid:16) Fil
May n (cid:17) ≃ i.e. the filtration is Hausdorff. Additionally, the May filtration on the functor THH is also Hausdorff. roof. By Proposition 4.15 and a Bousfield-Kan spectral sequence calculation, Fil
May n is n -connective.Since the connectivity grows linearly in the tower, we conclude lim ←−− n ≥ (cid:16) Fil
May n (cid:17) ≃ (cid:3) Corollary 4.17.
Let A ∈ Alg ≥ . Then the tower · · · → THH ( A )/ Fil
May i → · · · → THH ( A )/ Fil
May2 → THH ( A )/ Fil
May1 converges to
THH ( A ) .Proof. For n ≥ May i → THH ( A ) → THH ( A )/ Fil
May i . Passing to thelimit and applying Corollary 4.16, we conclude. (cid:3) Remark . As limits are calculated pointwise in functor categories, the lemma above impliesthat the tower of functors { THH / Fil
May i } i ≥ converges to THH. Remark . While working with Tow ( Sp ) suffices for our purposes, by Remark 4.7, we can defineTHH Fil ( A • ) for A • ∈ Alg
Fil , satisfying many of the same properties. For example, analogues ofPropositions 4.10 and 4.13 hold for THH
Fil ; instead of using the symmetric monoidality of ev , weuse the symmetric monoidality of lim −−→ : Fil ( Sp ) → Sp, and instead of the associated graded functorgr ∗ : Tow ( Sp ) → Sp, we use the functor gr
Fil ∗ from Section 2.3. THH hC p In order to prove our main theorem, we need to establish variants of Proposition 4.10 and Corollar-ies 4.14, 4.16, and 4.17 for THH (−) hC p . We proceed by applying C p -homotopy fixed points to thetower THH Tow ( A ) , and mimicking the proofs in the previous subsection. Proposition 4.20.
Let A ∈ Alg ≥ , and let A • denote the Whitehead tower of A . There is an equiva-lence of spectra with T -action, ev (cid:16) THH
Tow ( A • ) hC p (cid:17) ≃ THH ( A ) hC p . In the case where A ∈ CAlg ≥ ,this is an equivalence in CAlg B T .Proof. Since the functor ev is colimit-preserving, we have:ev (cid:16) THH
Tow ( A • ) hC p (cid:17) ≃ (cid:16) ev THH
Tow ( A • ) (cid:17) hC p ≃ THH ( A ) hC p . As in Proposition 4.10, similar considerations yield the commutative variant. (cid:3)
Definition 4.21.
Let A ∈ Alg ≥ and let A • denote the Whitehead tower of A . We define the May filtration on THH ( A ) hC p as THH Tow ( A • ) hC p . We will denote the i -th piece of this filtration byFil May i ( p ) , and the i -th graded piece by gr May i THH
Tow ( A • ) hC p . Proposition 4.22.
Let A ∈ Alg ≥ , and let A • denote the Whitehead tower of A . There is an equiv-alence of spectra with T -action, gr May ∗ (cid:16) THH
Tow ( A • ) hC p (cid:17) ≃ THH ( Hπ ∗ A ) hC p . In the case where A ∈ CAlg ≥ , this is an equivalence in CAlg B T .Proof. Since the functor gr ∗ is colimit-preserving, we have:gr ∗ (cid:16) THH
Tow ( A • ) hC p (cid:17) ≃ (cid:16) gr ∗ THH
Tow ( A • ) (cid:17) hC p ≃ THH ( Hπ ∗ A ) hC p . As in Proposition 4.13, similar considerations yield the commutative variant. (cid:3) roposition 4.23. If A • ∈ CAlg
Tow ≥ with the property that π m A n (cid:27) for all m < n , then π m Fil
May n ( p ) (cid:27) , for all m < n as well.Proof. It is standard that homotopy orbits preserve connectivity, so we may conclude by Proposi-tion 4.15. (cid:3)
Corollary 4.24.
For A ∈ CAlg ≥ , the May filtration of THH ( A ) hC p is Hausdorff.Proof. By Proposition 4.23, the connectivity of the tower grows linearly, solim ←−− n ≥ Fil
May n ( p ) ≃ . (cid:3) In this section we prove our main result. Using Proposition 3.8, we deduce the graded pieces ofthe May filtration are fpqc sheaves and conclude the result for THH by induction up the towerin Corollary 4.17. By similar methods, we prove the claim for THH hC p , and use the structure ofCycSp to prove it for TC as well. Lemma 5.1.
Let f : A → B be a faithfully flat morphism in CAlg ≥ . Let f • : A • → B • denote theinduced map between their Whitehead towers and let f ∗ : Hπ ∗ A → Hπ ∗ B denote gr ∗ ( f • ) . There is acanonical equivalence of augmented cosimplicial commutative ring spectra gr ∗ ( C ( f • ) + ) ≃ C ( f ∗ ) + .Proof. As ˇCech conerves are computed by left Kan extensions, there is a natural transformation C • ( f ∗ ) + → gr ∗ C • ( f • ) + since gr ∗ C • ( f • ) + | ∆ ≤ + = f ∗ . For n >
0, we have a canonical map C n ( f ∗ ) + ≃ Hπ ∗ B ⊗ H π ∗ A · · · ⊗ H π ∗ A Hπ ∗ B → Hπ ∗ ( B ⊗ A · · · ⊗ A B ) ≃ gr ∗ C n ( f ) + . Since A → B is faithfully flat, the usual Künneth spectral sequence calculation shows that this mapis an equivalence. Thus, the morphism of ˇCech conerves is an equivalence. (cid:3) Corollary 5.2.
The functors gr May ∗ THH
Tow , gr May ∗ THH
Tow (−) hC p : CAlg ≥ → Sp are sheaves forthe fpqc topology.Proof. By Proposition 4.13, there is an equivalence of functors gr
May ∗ THH ≃ THH ( gr ∗ (−)) . Nowlet f : A → B be a faithfully flat morphism in CAlg ≥ and observe that by Lemma 5.1, C • ( f ∗ ) ≃ gr ∗ C • ( f ) . Thus, Proposition 3.8 applied to the commutative diagramgr May ∗ THH ( A ) lim ←−− ∆ gr May ∗ THH ( C • ( f )) THH ( Hπ ∗ A ) lim ←−− ∆ THH ( C • ( f ∗ ))← → ←→ ←→ ← → proves the claim for gr May ∗ THH
Tow . An identical proof works for gr
May ∗ THH
Tow (−) hC p as well. (cid:3) Proposition 5.3.
For all i ≥ , the functors gr May i THH (−) are fpqc sheaves.Proof.
By definition, gr
May ∗ THH ≃ É i ≥ gr May i THH, so that gr
May i THH is a retract of gr
May ∗ THH.This implies that gr
May i THH is an fpqc sheaf for all i ≥ (cid:3) Corollary 5.4.
For all i ≥ , the functors THH / Fil
May i : CAlg ≥ → Sp are fpqc sheaves. roof. Induct on i using Proposition 5.3 and the fiber sequencesgr May i THH → THH / Fil
May i + → THH / Fil
May i . (cid:3) Theorem 5.5.
The functors
THH , THH hC p : CAlg ≥ → Sp are sheaves for the fpqc topology.Proof. Since limits of Sp-valued sheaves can be computed in the category of Sp-valued presheaves,the equivalence THH ≃ lim ←−− i ≥ (cid:16) THH / Fil
May i (cid:17) , guarantees THH is an fpqc sheaf. Similarly, sinceTHH (−) hC p ≃ lim ←−− i ≥ (cid:16) THH (−) hC p / Fil
May i ( p ) (cid:17) , THH (−) hC p is an fpqc sheaf as well. (cid:3) Corollary 5.6.
The functors
THH (−) h T , THH (−) hC p , and THH (−) tC p are fpqc sheaves on CAlg ≥ .Proof. The first two functors are fpqc sheaves because homtopy fixed points are calculated vialimits. The third is an fpqc sheaf because there is a cofiber sequenceTHH (−) hC p → THH (−) hC p → THH (−) tC p and THH (−) hC p and THH (−) hC p are fpqc sheaves. (cid:3) Using the results above, we will show that THH : CAlg ≥ → CycSp is an fpqc sheaf. As aCorollary, we will show TC is as well. We now recall the construction of CycSp and the definitionof topological cyclic homology.
Definition 5.7. [NS18, II.1.6] Let P denote the set of all primes. The ∞ -category of cyclotomicspectra is the pullback CycSp Fun ( ∆ , Î p ∈ P Sp B T ) Sp B T Î p Sp B T × Î p ∈ P Sp B T ← → ←→ ←→ ( ev ;ev ) ← → F where the functor F is given by the formula F = (cid:16)Î p ∈ P id , Î p ∈ P (−) tC p (cid:17) . By [NS18, II.1.5], CycSpis stable and presentable, hence enriched over Sp. The topological cyclic homology of a cyclotomicspectrum, X , is the mapping spectrumTC ( X ) = Map
CycSp ( S triv , X ) , where S triv is the sphere spectrum with the trivial T -action and Frobenius maps given by the com-posite T -equivariant maps S → S hC p → S tC p ; see [NS18, I.2.3.i, II.1.2.ii] for further details. AsTHH ( A ) admits the structure of a cyclotomic spectrum for all A ∈ Alg ([NS18, III.2]), we defineTC ( A ) = TC ( THH ( A )) . Proposition 5.8.
THH : CAlg ≥ → CycSp is a fpqc sheaf.Proof.
Because CycSp is defined as as lax equalizer, which is a strict pullback of ∞ -categories, itsuffices to check that the functors obtained from THH by post-composing with the projections areall fpqc sheaves. Indeed, we obtain the following three functors:17. THH : CAlg ≥ → Sp B T ;2. Î p ( φ p (−)) : CAlg ≥ → Î p Fun ( ∆ , Sp B T ) ; and by identifying composites,3. (cid:16)Î p THH (−) , Î p THH (−) tC p (cid:17) : CAlg ≥ → Î p Sp B T × Î p Sp B T .The functor in (1) is an fpqc sheaf because THH : CAlg ≥ → Sp is and limits in Sp B T are calculatedpointwise. Note that the functor in (2) is a sheaf if the functor in (3) is as well. To show the functorin (3) is an fpqc sheaf, we can project onto the different factors and reduce to checking that THHand THH tC p are fpqc sheaves on CAlg ≥ . Thus, the claim is proved. (cid:3) Corollary 5.9.
TC : CAlg ≥ → Sp is an fpqc sheaf.Proof. As TC ( A ) = Map
CycSp ( S triv , THH ( A )) , and THH is an fpqc sheaf valued in CycSp, for A → B faithfully flat, we have an equivalenceTC ( A ) → Map
CycSp ( S triv , lim ←−− THH ( B ⊗ A • )) ≃ lim ←−− Map
CycSp ( S triv , THH ( B ⊗ A • )) = lim ←−− TC ( B ⊗ A • ) (cid:3) References [AKS18] Gabe Angelini-Knoll and Andrew Salch,
A May-type spectral sequence for higher topolog-ical Hochschild homology , Algebraic & Geometric Topology (2018), no. 5, 2593–2660.[BMS19] Bhargav Bhatt, Matthew Morrow, and Peter Scholze, Topological Hochschild homologyand integral p-adic Hodge theory , Publications mathématiques de l’IHÉS (2019), no. 1,199–310.[CM19] Dustin Clausen and Akhil Mathew,
Hyperdescent and étale K-theory , 2019.[DR18] Bjørn Ian Dundas and John Rognes,
Cubical and cosimplicial descent , Journal of the Lon-don Mathematical Society (2018), no. 2, 439–460.[GHN17] David Gepner, Rune Haugseng, and Thomas Nikolaus, Lax colimits and free fibrations in ∞ -categories , Documenta Mathematica (2017), 1255–1266.[Gla13] Saul Glasman, Day convolution for ∞ -categories , arXiv preprint arXiv:1308.4940 (2013).[Lur09] Jacob Lurie, Higher topos theory , Princeton University Press, 2009.[Lur15] ,
Rotation invariance in algebraic K-theory du/~lurie (2015).[Lur17] ,
Higher algebra
Spectral algebraic geometry
On topological cyclic homology , Acta Mathematica221