A Universal HKR Theorem

TL;DR

This paper investigates the limitations of the HKR theorem in positive and mixed characteristic rings by constructing a filtered circle that interpolates between Hochschild, cyclic homology, and derived de Rham cohomology, recovering known filtrations.

Contribution

It introduces a novel filtered circle construction based on affine stacks and Witt vectors, providing a new perspective on the HKR theorem's failure and its relation to existing filtrations.

Findings

Constructed a filtered circle interpolating between topological and formal circles.

Revealed a natural filtration connecting Hochschild, cyclic homology, and derived de Rham cohomology.

Recovers the filtrations of Antieau and Bhatt-Morrow-Scholze.

Abstract

In this work we study the failure of the HKR theorem over rings of positive and mixed characteristic. For this we construct a filtered circle interpolating between the usual topological circle and a formal version of it. By mapping to schemes we produce this way an interpolation, realized in practice by the existence of a natural filtration, from Hochschild and (a filtered version of) cyclic homology to derived de Rham cohomology. In particular, we show that this recovers the filtration of Antieau and Bhatt-Morrow-Scholze. The construction of our filtered circle is based on the theory of affine stacks and affinization introduced by the third author, together with some facts about schemes of Witt vectors.

Equations

$$\text{circle}missing{\text{Mod}}_k^{\otimes }\simeq ϵ-{\text{Mod}}_k^{\otimes },$$

$$\text{HH}(A)=A{\otimes }_k\text{circle}missing{\text{Sym}}_{\text{rmA}}({\text{cotangent}}_{\text{rmA}/k}[1])=\text{DR}(A),$$

$$\text{loops}X\simeq \mathsf{T}[-1]X$$

$${\text{Sym}}_k^{\mathsf{co}\Delta }(k[-1])\simeq k\oplus k[-1]:=k[\eta ]$$

$$\text{Map}(\text{B}\text{Ga},k,X)\simeq \text{Map}(\text{Spec}(k[\eta ]),X)=:\mathsf{T}[-1]X$$

$$\text{Affinization}(\text{circle}missing\underset{B)}{\underbrace{\simeq }}\text{B}\text{Ga}\underset{A)}{\underbrace{\simeq }}\text{Spec}(k\oplus k[-1])$$

$$ϵ-{\text{Mod}}_k^{\otimes }\simeq k[\eta ]-{\mathrm{CoMod}}_k^{\otimes }$$

$$k[\eta ]-{\mathrm{CoMod}}^{\otimes }\simeq \text{circle}missing{\text{Mod}}_k^{\otimes }$$

$$\text{Map}(\text{circle}missingX)\simeq \text{Map}(\text{B}\text{Frobfixed},X)\text{ and }\mathrm{T}[-1]X\simeq \text{Map}(\text{B}\text{Frobkernel},X)$$

$$\HCminFil(A)\to\HCmin(A):=\HH(A)^{\mathrm{h}\circle{missing}\end{equation}\vskip 6.0pt plus 2.0pt minus 2.0pt\noindent andtheassociatedgradedpiecesof\HCminFil(A)arethetruncatedcompletederiveddeRhamcomplexes\mathbb{L}\widehat{\DR}^{\,\,\geq p}(A/k).}$$

$$\text{coSCRings}k\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

Full text

1. 1 Introduction

2. 1.1 Circle action and de Rham differential in characteristic zero

3. 1.2 What we do in this paper

1. Introduction

\mylabel

Introduction

The purpose of the present paper is to investigate the failure of the Hochschild-Kostant-Rosenberg theorem in positive and mixed characteristic situations. For this, we construct a filtered circle $\Filcircle$, an object of an algebro-homotopical nature, which interpolates between the usual homotopy type of a topological circle and a degenerate version of it called the formal circle. Given an arbitrary derived scheme $X$, we take the mapping stack, in the sense of derived algebraic geometry, from $\Filcircle$ to $X$; this provides an interpolation between Hochschild and (a filtered version of) cyclic homology on one side and derived de Rham cohomology of $X$ on the other. The existence of such an interpolation, realized concretely in terms of a filtration, is the main content of this work.

1.1. Circle action and de Rham differential in characteristic zero

\mylabel

background Over any commutative ring $k$, the HKR theorem [MR0142598] identifies $\Omega_{\dderham}^{\ast}(X)$ - the graded commutative algebra of differential forms on a smooth $k$-scheme $X=\Spec\,A$, with $\HH_{\ast}(A)$ - the graded algebra of Hochschild homology. When $k$ is of characteristic zero \personal [MR1269324, 9.8.12] this lifts to the level of chain complexes, identifying the de Rham complex of differential forms $\DR(A)$ with the Hochschild complex $\HH(A)$. More is true; the de Rham complex comes equipped with a natural differential which arises, via this identification, from the natural $S^{1}$ action on the Hochschild complex. A precise implementation of this fact requires a further enhanced version of the HKR theorem, combining the intervention of the homotopical circle action and the multiplicative structure on the Hochschild complex on one side, and the full derived de Rham algebra with its natural grading and de Rham differential on the other. This is established in [MR2862069] as a consequence of an equivalence of symmetric monoidal \icategories

[TABLE]

where $\circle{missing}\Mod_{k}^{\otimes}$ denotes equivariant $k$-modules and $\epsilon-\Mod_{k}^{\otimes}$ is the category of mixed complexes. Here $\epsilon$ is a generator of homological degree 1 that makes $k[\epsilon]\simeq k\oplus k[1]\simeq\mathrm{H}_{\ast}(\circle{missing}k)$. On both sides we have symmetric monoidal structures: on the l.h.s using the diagonal of $\circle{missing}andonther.h.susingtheco-multiplicationgivenby$ϵ↦ϵ⊗1+ 1⊗ϵ.

Via this identification, one obtains a multiplicative equivalence

[TABLE]

and the agreement of the symmetric monoidal structures in \eqformulaequivalencecircleepsiloncharzero guarantees that the circle action on the left matches the de Rham differential on the right. Geometrically as in [MR2928082], this can also be interpreted as an identification of the derived stack of free loops on an affine $k$-scheme $X=\Spec(\rmA)$, $\loops X:=\Map(\circle{missing}X)$, with the shifted tangent stack $\mathsf{T}[-1]\,X:=\Map(\Spec(k[\eta],X)$ where this time $k[\eta]:=k\oplus k[-1]\simeq\mathrm{H}^{\ast}(\circle{missing}k)$ is the differential graded algebra given by the cohomology of the circle. Here $\eta$ is a generator of homological degree $-1$. In this language, the HKR theorem reads as an equivalence of derived stacks

[TABLE]

and passing to global functions, recovers the isomorphism in (2).

Our first observation is that \eqformulaequivalencecircleepsiloncharzero is no longer a symmetric monoidal equivalence when we abandon the hypothesis that $k$ be a field of characteristic zero; indeed, the proof of \eqformulaequivalencecircleepsiloncharzero uses two essential facts about $\B\Ga{k}$ - the classifying stack of the group $\Ga{k}$:

1. A)

In any characteristic, the stack $\B\Ga{k}$ is equivalent to $\Spec^{\Delta}(\Sym^{\mathsf{co}\Delta}_{k}(k[-1]))$ where $\Sym^{\mathsf{co}\Delta}_{k}(k[-1])$ is the free cosimplicial commutative $k$-algebra over one generator in (cosimplicial) degree $1$ (see LABEL:notationcosimplicialandsimplicial below and [MR2244263, Lemma 2.2.5]). This can be checked at the level of the functor of points. But when $k$ is a field of characteristic zero, since the cohomology of the symmetric groups with coefficients in $k$ vanishes, we recover an equivalence of commutative differential graded algebras

[TABLE]

where on the r.h.s we have the split square zero extension. In particular, we have

[TABLE] 2. B)

For any ring $k$ the complex of singular cochains $\C^{\ast}(\circle{missing}k)$ is given by $k\oplus k[-1]$. The canonical map of groups $\bbZ\to\Ga{k}$ produces a map of group stacks $\circle{missing}=\B\bbZ\to\B\Ga{k}$. As in A), because the cohomology of symmetric groups with coefficients in a field of characteristic zero vanishes, the pullback map in cohomology $\C^{\ast}(\B\Ga{k},\structuresheaf)\to\C^{\ast}(\circle{missing}k)$ is an equivalence. This fact exhibits the abelian group stack $\B\Ga{k}$ as the affinization of the constant group stack \circle{missing}inthesenseof\cite[cite]{[\@@bibref{}{MR2244263}{}{}]}(seealso\cite[cite]{[\@@bibref{}{lurie-DAGVIII}{}{}]},\cite[cite]{[\@@bibref{}{MR2928082}{}{}, Lemma 3.13]}andour\lx@cref{creftype~refnum}{reviewaffinestacks}).ItfollowsfromtheuniversalpropertyofaffinizationandfromZariskidescentthat$$\Map(\circle{missing}X)\simeq\Map(\B\Ga{k},X)

The accident that allows A) and B) in characteristic zero also makes the equivalences

[TABLE]

compatible with the group structures111Essentially, by the uniqueness of the abelian group structure on the stack $\B\Ga{k}$. See LABEL:corollaryequivalenceasHopfalgebras for a similar argument in the context of this paper.. From here it is easy to recover the symmetric monoidal equivalence \eqformulaequivalencecircleepsiloncharzero: $k[\epsilon]\simeq\mathrm{H}_{\ast}(\circle{missing}k)$ is dual to $k[\eta]:=\mathrm{H}^{\ast}(\circle{missing}k)\simeq\Sym^{\mathsf{co}\Delta}_{k}(k[-1]))$ and this gives us a symmetric monoidal equivalence

[TABLE]

where on the r.h.s we now have the tensor product induced by convolution with the Hopf algebra structure on $k[\eta]=\mathrm{H}^{\ast}(\circle{missing}k)$ induced by the group structure on the circle. Finally the equivalences of groups in (4) gives a symmetric monoidal equivalence

[TABLE]

1.2. What we do in this paper

\mylabel

thispaper

Away from characteristic zero, equivalence \eqformulaequivalencecircleepsiloncharzero still holds at the level of \icategories. However, the compatibility of the two symmetric monoidal structures fails dramatically. As a consequence, we can no longer identify the circle action with the de Rham differential, the latter which fundamentally requires a notion of a mixed complex.

A key observation we make in this paper is that although these symmetric monoidal structures are not equivalent, there is a natural degeneration between the two. More precisely, we equip the symmetric monoidal category of complexes with a circle action with a filtration, whose “associated graded" is the symmetric monoidal category of mixed graded complexes, by which we mean complexes equipped with a strict co-action of the trivial square zero extension $k\oplus k[-1]$. See LABEL:proprepmonoidal and LABEL:splitfiltration2.

As a glimpse to our construction, we remark that the two copies of $\B\Ga{\,,k}$ appearing in A) and B) play distinct roles. What we propose, working over $\integerslocalp$, is a construction that interpolates between the two. It is inspired by an idea of [MR2244263] of using the group scheme $\Wittp$ of $p$-typical Witt vectors as a natural extension of the additive group $\Ga{}$. The group $\Wittp$ is an abutment of infinitely many of copies of $\Ga{}$ and comes canonically equipped with a Frobenius map $\Frob_{p}$. The abelian subgroup $\Frobfixed$ of fixed points of the Frobenius map has a natural filtration whose associated graded is the kernel of the Frobenius, $\Frobkernel$. After base change from $\integerslocalp$ to $\bbQ$ both $\Frobfixed$ and $\Frobkernel$ are isomorphic to $\Ga{}$ (see the LABEL:remark-characteristiczerocase below ) but over $\integerslocalp$ they are very different. Without further ado, our first main theorem is the following:

Theorem 1.1** (See LABEL:propositionaffinizationunderlying and LABEL:theorem-splitsquarezero).**

\mylabel

filteredcircletheorem

1. (a)

The abelian group stack $\B\Frobfixed$ is the affinization of $\circle{missing}over$\integerslocalp$.\par\par\vskip 6.0pt plus 2.0pt minus 2.0pt\par\enumerate@item@Theabeliangroupstack$\B\Frobkernel$hascohomologyringgiventhe(cosimplicial)splitsquarezeroextension(see\lx@cref{creftype~refnum}{splitsquarezeronotation})$\integerslocalp⊕\integerslocalp[-1]$givenbythecohomologyofthecircle.\noindent\par\par\vskip 6.0pt plus 2.0pt minus 2.0pt\par\enumerate@item@Thegroupstack$\B\Frobfixed$isequippedwitha\emph{filtration},compatiblewiththegroupstructure,whoseassociatedgradedstackis$\B\Frobkernel$.\par\vskip 6.0pt plus 2.0pt minus 2.0pt\par\enumerate@item@Afterbase-changealong$\Spec*(\bbQ)→\Spec(\integerslocalp)*,wehave$$\B\Frobfixed\otimes\bbQ\simeq\B\Ga{\,,\bbQ}$$\noindent Moreover,thefiltrationsplitsandwehave\par$$(\B\Frobfixed)^{\gr}_{\bbQ}\simeq\B\Frobkernel\otimes\bbQ\simeq\B\Ga{\,,\bbQ}**

Definition 1.2**.**

\mylabel

Definition-filteredcirclefirst The group stack $\B\Frobfixed$, equipped with the filtration of LABEL:filteredcircletheorem - (iii), will be called the filtered circle and denoted as $\Filcircle$.

Remark 1.3** (Filtrations).**

*In order to define filtrations on stacks, we will follow the point of view of C. Simpson [MR1159261, Lemma 19] and [MR1492538] which identifies filtered objects with objects over the stack $\Filstack$ - see [tasos] and LABEL:definition-filteredstacks. The content of LABEL:filteredcircletheorem and LABEL:Definition-filteredcirclefirst can then be reformulated as the construction of an abelian group stack $\Filcircle$ over $\Filstack$ whose fiber at [math] has the property in A); at $1$ has the property in B); and whose pullback to $\bbQ$ is the constant family with values $\B\Ga{\,,\bbQ}$. The construction of $\Filcircle$ is the subject of LABEL:filtrationskernelfixed, after reviewing the basics of Witt vectors in LABEL:section-remindersWitt. The proof that $\Filcircle$ satisfies (i) and (ii) will be discussed later in LABEL:associatedfilteredcircle and LABEL:underlyingfilteredcircle. The proof of (iv) is explained in the LABEL:remark-characteristiczerocase.

Having in mind the well known interpretation of cyclic homology in terms of derived loop spaces, our main theorem can now be stated as follows:

Theorem 1.4** (See LABEL:thmhkr-stackversion and LABEL:thmhkr).**

\mylabel

*corollaryHKRpositive Let $X=\Spec(\rmA)$ be a derived affine scheme over $\integerslocalp$. Then:

1. (a)

We have canonical equivalences of derived mapping stacks

[TABLE]

In particular, the derived mapping stack $\Map(\circle{missing}X)$ admits the structure of a filtration compatible with an action of the filtered circle $\Filcircle$ and whose associated graded is $\mathsf{T}X[-1]$; 2. (b)

Passing to global functions, (i) produces a filtration on $\HH(A)$, compatible with the circle action and the multiplicative structure, and whose associated graded is the derived de Rham algebra $\DR(A)$. 3. (c)

Since the filtration is compatible with the circle action, by taking homotopy fixed points of $\HH(A)$ seen as a filtered object, get a new filtered object which we call $\HCminFil(A)$. It has a canonical map to the usual fixed points

[TABLE]

Remark 1.5**.**

\mylabel

remark-comparisonfiltrationBMS As we will see later (LABEL:filteredglobalsectionsfixedpoints) the map (7) measures the difference between the underlying object of fixed points of a filtration and the fixed points of the underlying object. When $A$ is discrete and quasi-smooth, the two processes coincide - see [1808.05246, Lemma 4.10] and LABEL:thmhkr-(d).

In LABEL:section-comparison-Antieu we show that the point (iii) extends previously known filtrations constructed in [1808.05246] for discrete rings and in Bhatt-Morrow-Scholze in [MR3949030] for $p$-adic rings. The construction presented in our work is very different in nature.

The main emphasis of our result is that this filtration exists on the algebraic circle itself, before the HKR theorem.

*In LABEL:lastsection we will discuss several applications, generalizations and possible research directions suggested by LABEL:filteredcircletheorem and LABEL:corollaryHKRpositive.

{application}

[Shifted Symplectic Structures in positive characteristic]\mylabelapplicationshiftedsymplectic In LABEL:section-shiftedsymplectic we discuss the extension of the notion of shifted symplectic structures of [MR3090262] to derived stacks in positive characteristic. For this we use our HKR theorem in order to produce certain classes in the second layer of the filtration induced on negative cyclic homology. This is achieved by analyzing the Chern character map at the first two graded pieces of the filtration on $\HCminFil$. We also suggest a possible definition of $n$-shifted symplectic structures and show that the universal $2$-shifted symplectic structure on $\B G$ exists essentially over any base ring $k$. By the techniques developed in [MR3090262] we obtain this way extensions of various previously known $n$-shifted symplectic structures over non-zero characteristic bases.

{application}

[Generalized Cyclic Homology and Formal groups]\mylabelapplicationellipticcurves The application discussed in LABEL:section-generalizedcyclicformalgroups comes from the observation that the degeneration from $\Frobfixed$ to $\Frobkernel$ of LABEL:filteredcircletheorem-(iii) is Cartier dual to the degeneration of the multiplicative formal group $\formalGm{}$ to the additive formal group $\formalGa{}$ (see LABEL:recoverviaCartierduality). In LABEL:section-generalizedcyclicformalgroups we will discuss how to generalize LABEL:filteredcircletheorem and LABEL:corollaryHKRpositive replacing $\formalGm{}$ by a more general formal group law $\rmE$, in particular, one associated to an elliptic curve.

{application}

[Topological and q-analogues]\mylabelapplicationqanalogue In LABEL:topoqanalogue we briefly present topological and $q$-deformed possible generalizations of our filtered circle. We investigate two related ideas, a first one that predicts the existence of a topological, non-commutative version of $\Filcircle$ as a filtered object in spectra. A second one, along the same spirit, predicting the existence of a $q$-deformed filtered circle $\Filcircle(q)$ possibly related to q-deformed de Rham complex (see for instance [MR1792063]) in a similar fashion that $\Filcircle$ is related to de Rham theory. Again, such a quantum circle can only exist if one admits non-commutative objects in some sense.

Relation with other works:\mylabelsection-futureworks The object $\Filcircle$ and the constructions behind it can be placed in a general context. For instance, this construction is of homotopical significance, as the underlying object of $\Filcircle$ is the affinization of the topological circle over $\integerslocalp$ in the sense of [MR2244263]. In [1911.05509] the third author studies the integral version of the affinization and its relation to our group scheme $\Frobfixed$. The result there may be used in order to extend our HKR theorem over $\bbZ$. Concerning, the filtration, we believe that our construction is much more general and that for any finite CW homotopy type $X$ the affinization $(X\otimes\bbZ)=\Spec\,\C^{*}(X,\bbZ)$ comes equipped with a canonical filtration whose associated graded is $\Spec\,\mathsf{H}^{*}(X,\bbZ)$ (at least when $X$ has torsion free cohomology groups). This is in a way the canonical filtration that degenerates a homotopy type over $\bbZ$ to a formal homotopy type.

In a different direction, we would like to mention the work [Arpon], in which the author also constructs a filtered circle, as filtered Hopf algebra object in a suitable $\infty$-category of derived commutative rings. As far as the authors understand [Arpon], this Hopf algebra is a model for the Hopf algebra of functions on our filtered circle $\Filcircle$.

Acknowledgements.

\mylabelacknowledgementsThe idea of a filtered circle has been inspired by the construction of the Hodge filtration in Carlos Simpson’s non-abelian Hodge theory, and we thank Carlos Simpson, Tony Pantev, Gabriele Vezzosi and Mauro Porta for several conversations on these themes along the years. We also thank Arpon Raksit for discussions related to his approach to the filtered circle. We are very thankful to Pavel Etingof for having pointed to us the possible existence of the q-deformed circle (presented in LABEL:topoqanalogue), this has also lead the authors to realize the existence of various circle associated to any commutative formal group law. We are also thankful to the anonymous referee for encouraging us to include a new section comparing our filtration with the one of Antieau. This paper was written while the first two authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2019 semester, supported by the National Science Foundation under Grant No. DMS-1440140. Tasos Moulinos and Bertrand Toën are supported by the grant NEDAG ERC-2016-ADG-741501. Marco Robalo was supported by the grant ANR-17-CE40-0014. *

{notation}

[Simplicial, Cosimplicial and $\Einfinity$] \mylabelnotationcosimplicialandsimplicial Unless mentioned otherwise, all higher categorical notations are borrowed from [lurie-ha, lurie-htt]. Let $k$ be a discrete commutative ring. Throughout the paper we will denote by

1. a)

We use homological conventions. We write $\Mod_{k}$ for the \icategoryof chain complexes of $k$-modules; $\Mod_{k}^{\geq 0}$, resp. $\Mod_{k}^{\leq 0}$ the categories of connective and coconnective complexes. 2. b)

The notation $\CAlg$ will always be used to denote $\Einfinity$-algebras. In particular $\CAlg_{k}$ will denote $\Einfinity$-algebras in $\Mod_{k}$; $\CAlg_{k}^{\mathsf{cn}}\simeq\CAlg(\Mod_{k}^{\geq 0})$ the full subcategory of connective algebras [lurie-ha, 2.2.1.3, 2.2.1.8, 7.1.3.10] and $\CAlg_{k}^{\mathsf{ccn}}$ the category of $\Einfinity$-algebras in $\Mod_{k}^{\leq 0}$ for the symmetric monoidal structure induced from the fact $\tau_{\leq 0}$ is a monoidal localization. In particular, as the inclusion $\Mod_{k}^{\leq 0}\subseteq\Mod_{k}$ is lax monoidal, we have an induced map at the level of algebras $\CAlg_{k}^{\mathsf{ccn}}\to\CAlg_{k}$ 3. c)

$\SCRings{k}$* the \icategoryof simplicial commutative rings over $k$. This is the sifted completion of the discrete category of polynomial algebras $\Nerve(\Poly{k})$. See [lurie-sag, 25.1.1.5] and [lurie-htt, 5.5.9.3]. The universal property of sifted completion gives us the normalized Dold-Kan functor $\theta:\SCRings{k}\to\CAlg_{k}^{\mathsf{cn}}$. By [lurie-sag, 25.1.2.2, 25.1.2.4] this is both monadic and comonadic and if $k$ is of characteristic zero it is an equivalence. Given $A\in\SCRings{k}$, $\theta(A)$ will be called the underlying $\Einfinity$-algebra of $A$.* 4. d)

By $\Sym$ we will always mean the simplicial version $\Sym^{\Delta}$ as a monad in $\Mod_{k}^{\geq 0}$; 5. e)

$\coSCRings{k}$* the \icategoryof cosimplicial commutative rings over $k$ (See Definition LABEL:cosimplicialcommalgebra below.) We also denote by $\theta:\coSCRings{k}\to\CAlg_{k}$ the conormalized Dold-Kan construction (see [MR2244263, §2.1]). This functor is conservative, commutes with tensor products and preserves limits by the discussion in Section LABEL:hopfalgebrasandcomodules. It can be factored by a functor $\theta^{\mathsf{ccn}}$*

[TABLE]

where $\theta^{\mathsf{ccn}}$ is the co-dual Dold-Kan construction of [MR1243609]. 6. f)

By $\Sym^{\mathsf{co}\Delta}$ we will mean the free cosimplicial commutative algebra on $\Mod_{k}^{\mathsf{\leq 0}}$. 7. g)

$\Stacks{k}$* the \icategoryof stacks over the site of discrete commutative $k$-algebras and $\dSt_{k}$ the \icategoryof derived stacks, ie, stacks over $\SCRings{k}$.* 8. h)

$\dSch_{k}$* the $\infty$-category of derived schemes and $\Spec:\SCRings{k}^{\op}\to\dSch_{k}$ the affine derived scheme associated to of a simplicial commutative ring.* 9. i)

$\cospec:\coSCRings{k}^{\op}\to\Stacks{k}$* the*