THH and TC are (very) far from being homotopy functors
aa r X i v : . [ m a t h . K T ] J u l THH AND TC ARE (VERY) FAR FROM BEING HOMOTOPY FUNCTORS
ELDEN ELMANTO
Abstract.
We compute the A -localization of several invariants of schemes namely, topolog-ical Hochschild homology (THH), topological cyclic homology (TC) and topological periodiccyclic homology (TP). This procedure is quite brutal and kills the completed versions ofmost of these invariants. The main ingredient for the vanishing statements is the vanish-ing of A -localization of de Rham cohomology (and, eventually, crystalline cohomology) inpositive characteristics. Introduction and vista
In this short paper, we compute the A -localization of several invariants relevant to recentdevelopments in p -adic Hodge theory. They turn out to be mostly zero. In some sense, theseare “no-go theorems” which state that, at least in characteristic p , the motivic perspective ofMorel and Voevodsky [MV99], which is based on the notion of A -invariance, is incompatible in a strong way with the motivic perspective of [BMS19], which is based on descent propertiesof “trace invariants” (e.g. topological Hochschild and cyclic homology) of schemes.This is perhaps unsurprising: Ayoub in [Ayo14, Lemme 3.10] and Cisinski-D´eglise in [CD16,Proposition A.3.1] have proved that versions of ´etale motives are Z [ p ]-linear while Bachmannand Hoyois have upgraded these to a spectral version (unpublished). On the other hand theaforementioned “trace invariants” are p -adic ´etale sheaves. This paper adds another collectionof results along these lines (though, a priori unrelated).The key actor among these “trace invariants” is arguably topological cyclic homology TCof [BHM93], recently revisited by Nikolaus and Scholze [NS18]. This theory is not, in general, A -invariant. In fact, as explained in Remark 4.0.6, it must not be in for it to be of any use forK-theory. We prove: Theorem 1.0.1. L A TC vanishes after profinite completion. In other words, L A TC is purelyrational. This is stated more precisely in Theorem 4.1.1, and says that the profinitely completed versionof this invariant vanishes after A -localization. One should contrast this to the situation foralgebraic K-theory which is also not A -invariant in general but whose A -localization, Weibel’sKH-theory [Wei89], is still a vastly interesting invariant (in any characteristic).1.1. Summary.
The title of this paper is an homage of the results of Geller and Weibel [GW89]in characteristic zero. We give a reformulation of what they did in Theorem 2.1.1, whichimmediately adapts to the contractibility statement for L A THH in Theorem 2.2.1. We thenproceed to compute these invariants in characteristic p and later over Z .One would like to prove that the A -localization of TP is zero. The problem, and thisis ultimately the technical crux of the present paper, is that the Tate construction does notcommute with colimits in general.Therefore, the result for THH does not immediately boostrap to TP. Instead we exploit theBhatt-Morrow-Scholze (BMS) filtration [BMS19] and reduce the problem to vanishing state-ments for A -localization of crystalline (Proposition 3.2.5) and, eventually, de Rham cohomol-ogy (Lemma 3.2.1). This gives us the vanishing result for A -localization of TC first (The-orem 3.1.1) and, later, of TP (Corollary 3.3.1). Eventually, we prove profinite vanishing of L A TC over the integers (Theorem 4.1.1) using the previous results and the observation thatL A TC is a truncating invariant.1.2.
Vista.
This paper is a “contained experiment” about the interaction between A -invarianceand invariants derived from THH: the results, as one can see, are negative. However we dobelieve that these two perspectives are complementary and can be useful as long as they arenot mixed together. In other words, we should separate them.One concrete way in which “separating them” is a good idea is the following cartesiansquare which lets us break down algebraic K-theory into constituent pieces (when restricted tonoetherian schemes of finite dimension):(1.2.1) K TCL A K L cdh TC . Here L cdh is the sheafification functor with respect to Suslin and Voevodsky’s cdh topology[SV00]. The cartesian-ness of this square is deep: it requires knowing that 1) L A K ≃ L cdh Kwhich was first proved by Haesemeyer in characteristic zero [Hae04] and by Kerz-Strunk-Tammein [KST17, Theorem 6.3] in general, and 2) that K inv is a cdh sheaf, which is a result of [GH10]over perfect fields and assuming resolution of singularities, and [LT19] in general. This last resultultimately boils down to the celebrated theorem of Dundas-Goodwillie-McCarthy [DGM13].We see that the key idea is to find a bridge between the homotopy invariant and the traceperspectives — in this case this is given by cdh-sheafification. We are currently conductingfurther investigations in and around the square (1.2.1).1.3.
Convention.
We use standard ∞ -categorical terminology. A little note on possible con-fusion: our functor HH is the derived version, i.e., left Kan extended from polynomial algebras.Hence, so are the functors HC − , HP , HC and so on. Any kind of “affine line” appearing in thisnote is the flat affine line so that π ∗ (R[ t ]) = π ∗ (R)[ t ].1.4. Acknowledgements.
I would like to thank Joseph Ayoub, Akhil Mathew, Matthew Mor-row, Jay Shah, Zijian Yao and Allen Yuan for useful conversations, Benjamin Antieau, SanathDevalpurkar, Arpon Raksit and Chuck Weibel on comments on an earlier draft and Lars Hes-selholt who suggested that the vanishing result over the integers should be true. I would alsolike to thank Benjamin Antieau, Tom Bachmann, Lars Hesselholt, Marc Hoyois and MatthewMorrow for informing my perspective on “motives” over the years. Lastly I would like to thankVitoria the cat for constant distractions.2.
Topological Hochschild Homology R denote the ∞ -category of derived R-algebras,concretely presented as the ∞ -category obtained from simplicial commutative R-algebras andinverting weak equivalences or the sifted-colimit completion of polynomial R-algebras. We havethe exact localization endofunctorL A : PSh(CAlg op , Spt) → PSh(CAlg op , Spt) , reflecting presheaves of spectra into A -invariant presheaves: those that convert the canonicalmap R → R[ t ] to equivalences. Also note that since homotopy invariant presheaves are stableunder colimits, the endofunctor L A preserves colimits. A concrete formula for this functor is HH AND TC ARE (VERY) FAR FROM BEING HOMOTOPY FUNCTORS 3 given object-wise by the formula :(2.0.2) (L A F )(R) = colim ∆ op F (R[∆ • ]) . One consequence of (2.0.2) is that L A is strong symmetric monoidal since the (pointwise)symmetric monoidal structure on presheaves of spectra commutes with colimits and the colimitappearing in (2.0.2) is sifted. Therefore L A preserves algebras and modules over them.2.1. Hochschild homology.
We now present the main result of Geller and Weibel’s paper[GW89, Theorem 2.1]; it is morally the same proof as theirs.
Theorem 2.1.1 (Geller-Weibel) . L A of HH , HC are zero, while the canonical map HC − → HP is an L A -equivalence.Proof. The second statement follows from assertion about HC by the norm-cofiber sequence:ΣHC → HC − can −−→ HP , and the fact that L A is exact.To prove the assertion about HC, we first prove the assertion about HH. The homotopygroups of the spectrum (L A HH)(R) are modules over the ring π ((L A HH)(R) ≃ Eq( π (HH(R[ t ])) = π (R)[ t ] ⇒ π (HH(R)) = π (R)) . This ring is actually zero since the coequalizer instructs us to set t = 0 = 1. Therefore sincethe homotopy groups of L A HH(R) are modules over the zero ring, they are all zero and hencethe spectrum itself is contractible.Now, we need to show that L A (HC( − )) ≃ . Evaluating this on a ring R, we need to compute the geometric realization of the simplicialobject (in D (R), say) n HH(R[∆ n ]) h S . To compute this, note that the diagram n HH(R[∆ n ])is a diagram in S -spectra since the transition maps are induced by ring maps, whence thegeometric realization is an S -spectrum. Therefore from the fact that geometric realizationsand taking S -orbits commute (they are both colimits) we get that | HH(R[∆ • ]) h S | ≃ | HH(R[∆ • ]) | h S ≃ | | h S ≃ . (cid:3) Q , hence we are looking at derived rings in characteristiczero. In this situation a result of Kassel [Kas87, Corollary 3.13] proves thatL A HP ≃ HP , by way of comparison with de Rham cohomology (see also [Goo85, Theorem III.5.1] for a directproof of a more general “homotopy invariance statement”). As a result we deduce [GW89,Theorem 4.1]: We learned from Weibel some history behind this formula and we take this opportunity to record it. On π , this was introduced by Swan and Gersten in [Ger71]. The simplicial ring Z [∆ • ] was then considered by D.Anderson in [And73]. Weibel took the conceptual leap of taking the geometric realization of K(R[∆ • ]) in spectrato construct his KH-theory. In his Luminy talk, Suslin then suggested this construction as a recipe for motiviccohomology, details are in a joint paper with Voevodsky [SV96]. Of course, this construction has since beencentral to workers in A -homotopy theory as a formula for A -localization, beginning with the introduction ofthe subject [MV99]. Actually, in this case, an object of D (R). E. ELMANTO
Corollary 2.1.3 (Geller-Weibel) . Let R be a ring of characteristic zero. Then we have acanonical equivalence (L A HC − )(R) ≃ HP(R) . Topological Hochschild homology.
The story for THH is similar and the proof ofTheorem 2.1.1 goes through in the topological setting.
Theorem 2.2.1. L A of THH and
THH h S are zero. On the other hand, the canonical map can : TC − → TP is an L A -equivalence.Proof. Indeed, the key observation of Theorem 2.1.1 is that π ( L A HH(R)) ≃
0. But then π (THH(R)) ≃ π (HH(R)) ≃ π (R) and the same argument follows through. (cid:3) Characteristic p Suppose now that R is a derived ring of characteristic p > ∈ CAlg F p ).3.0.1. We begin with a discussion of L A . We will be looking at functors CAlg R → C where C is the derived ∞ -category of some abelian category; most likely it will be D ( Z p ) or the ∞ -category of spectra. These functors will land inside the more manageable category of “derived p -complete objects” denoted by C ∧ p which admits a completion functor C → C ∧ p , an exact leftadjoint . In general, this functor does not preserve colimits; equivalently p -complete objectsare not stable under colimits; see [Stacks, Tag 0ARC] for an example in the case of C = D ( Z p ).3.0.2. Fortunately, many of our invariants are bounded below and the A -localization functoris computed by a geometric realization (2.0.2). Lemma 3.0.3. If R is a connective E -ring, I a finitely generated ideal. Let M • be a simplicialobject in R -modules which are uniformly bounded below with respect to the standard t -structure.Then if each M n is I -complete, so is | M • | .Proof. We might as well assume that each M n is connective. We use the criterion (b) ⇒ (a) in[Lur18, Theorem 7.3.4.1]. Indeed, it suffices to prove that Ext iπ (R) ( π (R)[ x − ] , π k ( | M • | )) = 0for i = 0 ,
1. But then, since each M n is connective, π k ( | M • | ) on depends on a finite skeleton ofM • which is a finite colimit, whence I-complete. The result follows from the other direction of[Lur18, Theorem 7.3.4.1]. (cid:3) In this light, our L A -functor will regarded as in the previous section. In other words we arestill studying the effects of the endofunctorL A : PSh(CAlg opR , C ) → PSh(CAlg opR , C ) , and indicate so when an equivalence is only know after p -completion by writing L A (X) ≃ p L A (Y).3.1. Topological cyclic homology.
With this technical discussion out of the way, we prove
Theorem 3.1.1. If R ∈ CAlg F p , then (L A TC)(R) ≃ . We begin with some preliminaries. Let R be a smooth (or, more generally, quasisyntomic[BMS19, Definition 4.10]) k -algebra where k is a perfect field of characteristic p . Then, Bhatt-Morrow-Scholze constructed in [BMS19] complete, multiplicative descending filtrations(3.1.2) Fil ∗ BMS
TC(R) → TC(R) Fil ∗ BMS TC − (R) → TC − (R) Fil ∗ BMS
TP(R) → TP(R) , and also identified the associated graded pieces. Hence a localization.
HH AND TC ARE (VERY) FAR FROM BEING HOMOTOPY FUNCTORS 5 p , the answers are particularly nice. For example we have [BMS19,Theorems 1.10, 1.12] gr TC − (R) ≃ RΓ crys (R / W( k )) , which periodizes in TP to gr q BMS
TP(R) ≃ RΓ crys (R / W( k ))[2 q ] . Here RΓ crys (R / W( k )) is the (object in D (W( k )) computing) crystalline cohomology of R; notethat there is a canonical representative of this object in the D (W( k )) given by the de Rham-Wittcomplex of Bloch-Deligne-Illusie [Ill79]:WΩ • R = WΩ → WΩ → · · · . • ∈ PSh(CAlg opR , D (W( k )) ∧ p ) , and the localization thereof. First, we study the analogous question for the derived de Rhamcomplex: LΩ • ∈ PSh(CAlg opR , D ( k )) . Lemma 3.2.1.
Let k be a perfect field of characteristic p , then L A LΩ • ≃ .Proof. Since L A LΩ • (R) is a module over L A LΩ • ( k ), it suffices to prove the following result:(L A LΩ • )( k ) ≃ colim ∆ op Ω • k [∆ • ] ≃ . Indeed, the above object is a module over H (L A Ω • ( k )) which is the coequalizer of(3.2.2) H (Ω • k [T] ) ≃ k [T p ] ⇒ H (Ω • k ) ≃ k, where one of the maps sends T p to 0 and the other to 1. Therefore the coequalizer is the zeroring. (cid:3) Remark 3.2.3.
The vanishing phenomenon described in this paper can be attributed to the factthat H (Ω • k [T] ) is rather large in characteristic p . In contrast, in the presence of A -invariance,i.e. in characteristic zero, this group is just the base field k . In this case, the equalizer (3.2.2)reads as id , id : k ⇒ k Proposition 3.2.5.
Let k be a perfect field of characteristic p , then L A LWΩ • ≃ p , i.e., iszero after p -completion.Proof. This follows from Lemma 3.2.1 since(L A LWΩ • ) /p ≃ L A (LWΩ • /p ) ≃ L A LΩ • ≃ . (cid:3) A -invariant crystalline and de Rham cohomology also vanish as theyare modules over their values on F p where the derived and non-derived versions coincide. E. ELMANTO
Proof of Theorem 3.1.1.
It suffices to prove that (L A TC)( F p ) is zero. We employ the BMSfiltration on TC. First, according to [AMMN20, Theorem 5.1(1)] for a smooth F p -algebra R,Fil > i BMS
TC(R) is ( i − A Fil > i BMS
TC)( F p )are all smooth F p -algebras, this spectrum is again at least ( i − ∞ , we get that lim i (L A Fil > i BMS
TC)( F p ) ≃ A -BMS filtrationis complete on TC( F p ).To leverage this fact, we note that we have a diagram of presheaves of spectra where eachcolumn is a cofiber sequence(3.2.8) · · · Fil > TC Fil > TC Fil > TC · · · TC TC TC · · ·
Fil
BMS TC Fil
BMS TC Fil
BMS TC ≃ gr TCwhich induces the same diagram after applying L A again with column-wise cofiber sequences.We have proved that taking limit along the top row results in a contractible spectrum. Therefore,to prove that the middle term is contractible, we only need to prove that(L A gr i BMS
TC)( F p ) ≃ i >
0. Since the BMS filtration is multiplicative, the (presheaf of) graded E ∞ -algebra(s) ⊕ gr i BMS
TC is naturally a (presheaf of) E ∞ -algebra(s) over gr TC. Ditto their L A -versions.Hence we only need to prove that the zero-th graded pieces vanish: (L A gr TC)( F p ) ≃ TC → gr TC − can − ϕ p −−−−−→ gr TP , where the can map is an equivalence in this case. Therefore, after identifying gr TP withcrystalline cohomology on smooth k -schemes (of course we only need polynomial) we get acofiber sequenceL A gr TC → L A RΓ( − / W( F p )) id − ϕ p −−−−→ L A RΓ( − / W( F p )) . where ϕ p is the composite of ϕ p with the inverse of can. Evaluating the above on F p andusing Proposition 3.2.5, we conclude the desired result after p -completion. Since the terms inTC( F p [∆ • ]) are all uniformly bounded below, we conclude by Lemma 3.0.3, that (L A TC)( F p )is already p -complete and hence actually zero. (cid:3) Topological periodic cyclic homology.
The point of going to TC in the above argu-ment is that the diagram (3.2.8) is N op -indexed so we can do some sort of induction. This isnot the case for TP. We use the above result to deduce vanishing for TP after p -completion: Corollary 3.3.1. L A TP ≃ p .Proof. Using the formula [NS18], we view TC as the equalizer of can , ϕ p : TC − ⇒ TP andsimilarly for the L A -local version as this functor preserves finite limits. But now Theorem 3.1.1proves that L A TC ≃
0, whence the maps can and ϕ p are homotopic and, in particular, ϕ p is invertible since can is by Lemma 2.2.1. Since everything in sight is p -completed suffices toprove that L A TP is also Z [ p ]-linear, i.e., p acts invertibly. HH AND TC ARE (VERY) FAR FROM BEING HOMOTOPY FUNCTORS 7
According to [NS18, Section IV.4] (see also [LB, Proposition 12]) the homotopy groups ofTC − ( F p ) and TP( F p ) are as follows: π ∗ (TC − ( F p )) = Z p [ u, v ] / ( uv − p ) π ∗ (TP( F p )) = Z p [ σ, σ − ] . Furthermore, since can( v ) = σ − and ϕ p ( v ) = pσ − we have the following commutative diagramof presheaves of TC − ( F p )-modules:(3.3.2) Σ − TP TPΣ − TC − TC − Σ − TP TP TP , σ − v can ϕ p can ϕ p σ − p But now, after applying L A , we see that can and ϕ p are invertible by the previous discussionso that the endomorphism p is invertible after applying L A . (cid:3) Remark 3.3.3.
Alternatively, we can prove Corollary 3.3.1 by showing that L A TP /p ≃ L A HP ≃
0. The latter equivalence is just a version of periodized de Rham cohomology whichvanishes in characteristic p by Lemma 3.2.1. We thank Antieau for pointing this out.4. Profinite vanishing over the integers
We finish off with vanishing of profinite TC after A -localization. Let us recall what wemean by integral TC and its various localizations.4.0.1. If E is a spectrum, then the profinite completion, denoted by E ∧ is modeled as thecofiber of the map Maps( Q , E) → E . The model for integral topological cyclic homology is given by Nikolaus-Scholze in [NS18, SectionII.1]:(4.0.2) TC(R) = Eq(id , ϕ h S : TC − (R) ⇒ TP(R) ∧ ) , using the implicit identification [NS18, Lemma II.4.2](X t S ) ∧ p ≃ (X t C p ) h S . We will show that the A -localization of profinitely-completed TC ∧ is zero.4.0.3. Following [LT19, Definition 3.13], given a Z -linear localizing invariant, i.e., a functor F : Cat perf Z → Spt we can define their A -localization by taking | F ( C ⊗ Z Perf ( Z [∆ • ]) | =: (L A F )( C ) . Following the conventions of the above sections, this A -localization functors are of the formL A : PSh((Cat perf Z ) op , Spt) → PSh((Cat perf Z ) op , Spt) , and we will indicate when equivalences are only true after further profinite completion byL A (X) ≃ ∧ L A (Y). E. ELMANTO Z -linear categories, is to make sense of the next lemma; see also 4.1.3. Lemma 4.0.5.
The functor L A TC is truncating.Proof. We have a fiber sequenceL A K inv → L A K =: KH → L A TC . The presheaf, KH is truncating by [LT19, Proposition 3.14]. On the other hand, the Dundas-Goodwillie-McCarthy theorem informs us that K inv is a truncating invariant. But since π (R[ t ]) ≃ π (R)[ t ] for R an Z p - E -algebra, we see that it remains truncating. Thus we conclude thatL A TC is truncating. (cid:3)
Remark 4.0.6.
Lemma 4.0.5 gives a structural explanation to the non- A -invariance of TC.Indeed, if it were, then TC ≃ L A TC and so it would be nilinvariant over any base ring. Thisis certainly not true over F p , by looking at the example F p [ x ] / ( x ) → F p and the calculationsof [HM97, Spe20]. The non-nilinvariance of TC is indeed one of the main desiderata for itsinvention — as an approximation to the non-nilinvariant part of algebraic K-theory.4.1. Integral topological cyclic homology.
Finally:
Theorem 4.1.1.
For any derived ring R , (L A TC ∧ )(R) ≃ . Therefore, the rationalizationmap TC → TC Q is an L A -equivalence.Proof. We note that the terms TC ∧ ( Z [∆ • ]) are uniformly bounded below: since TC ∧ p converts p -adic equivalences to p -adic equivalences we need only prove that the terms of TC ∧ p ( Z p [∆ • ])are uniformly bounded below for each p . But this follows, for example, from the connectivityestimates in [AMMN20, Theorem 5.1(1)].Hence, after Lemma 3.0.3, we need only prove that (L A TC ∧ )( Z ) ≃ ∧
0. It then suffices toprove that (L A TC ∧ p )( Z ) ≃ p p . Hence we need only prove that(L A TC ∧ p )( Z ) ≃ p (L A TC ∧ p )( Z p ) ≃
0. To this end, we claim: • the canonical map L A TC ∧ p ( Z p ) → lim s L A TC ∧ p ( Z p /p s )is an equivalence.To see that the claim implies the desired vanishing, note that since L A TC is truncating, it isnilinvariant by [LT19, Corollary 3.5]. Therefore the limit above is stabilizes as L A TC ∧ p ( F p )which is zero by Theorem 3.1.1.To prove the desired claim, we note that the limit above commutes with L A since the termsin TC ∧ p ( Z p /p s [∆ • ]) are uniformly bounded below and geometric realization behaves as a finitecolimit in a range of degrees (just like the argument in Lemma 3.0.3). Thus it suffices to provethat for each n >
0, the mapTC( Z p [T , · · · , T n ]) → lim s TC( Z p /p s [T , · · · , T n ])is a p -adic equivalence. Since p -adic TC preserves p -adic equivalences, we may p -adically com-plete the rings inside and prove that the mapTC( Z p [T , · · · , T n ] ∧ p ) → lim s TC( Z p /p s [T , · · · , T n ])is a p -adic equivalence. This then follows by continuity of THH as in [CMM18, Proposition5.4], noting that Z p [T , · · · , T n ] ∧ p /p ≃ F p [T , · · · , T n ] is F-finite since it is finite type over F p and the continuity of TC as in [CMM18, Remark 2.8]. HH AND TC ARE (VERY) FAR FROM BEING HOMOTOPY FUNCTORS 9
For the last statement we look at the fracture square (which is cartesian)TC TC Q TC ∧ TC ∧ Q . We have proved that, after applying L A , the bottom left corner is zero. This means that thethe bottom right corner is zero as well after applying L A since it is a ring admitting a ringmap from the zero ring. Hence the bottom map is an equivalence after applying L A , whencethe top map is an equivalence after applying L A as well since L A preserves finite limits. (cid:3) Rational situation.
Let us consider these functors on CAlg Q . In this situation (4.0.2)tells us that TC ≃ TC − = HC − since TP ∧ ≃
0. By the Geller-Weibel theorem, Corollary 2.1.3,we further have that L A HC − ≃ HP. Hence we get that L A TC ≃ HP which is a non-zero, butfamiliar, invariant.4.1.3. We further note that our arguments also show that, as a Z -linear localizing invariant,L A TC vanishes after profinite completion. Hence we obtain an analogous “purely rational”result for L A TC regarded in the noncommutative setting.4.1.4. In the proof of Theorem 4.1.1, we can avoid continuity results, by exhibiting a “Gysinsequence” L A TC( F p ) → L A TC( Z p ) → L A TC( Q p ) , and noting that the last term vanishes after profinite completion. Gysin sequences in thenoncommutative world appears to interact well with A -invariance as indicated by the workof Tabuada and Van den Bergh [TVdB18] in geometric settings; the author thanks Mathewfor pointing this out. We are working on a sequel establishing these Gysin sequences in the A -invariant, noncommutative world. References [AMMN20] B. Antieau, A. Mathew, M. Morrow, and T. Nikolaus,
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