This is the second post in a series on my favorite work of 2020.

Two recent papers, those of Moulinos, Robalo, and Toën  and Raksit , explain the existence of the Hochschild–Kostant–Rosenberg filtration on Hochschild homology via the existence of a filtered circle.

We will let $k$ be a base commutative ring and $R$ be any commutative $k$-algebra. In fact, it is possible to allow $k$ and $R$ to be more generally animated commutative rings or even so-called derived commutative rings in the sense of Bhatt and Mathew. The theory of animated commutative rings (terminology of Clausen) is the $\infty$-category underlying the homotopy theory of simplicial commutative rings and weak equivalences, while derived commutative rings are a nonconnective generalization. These notions are important for expressing the correct universal properties of the theories that arise here.

# Hochschild homology

Let $\aCAlg_k$ denote the $\infty$-category of animated commutative rings and let $\aCAlg_k^{BS^1}\simeq\Fun(BS^1,\aCAlg_k)$ denote the $\infty$-category of animated commutative rings with a circle action. Choosing a basepoint $x\in BS^1$, there is an evident forgetful functor

$x^*\colon\aCAlg_k^{BS^1}\rightarrow\aCAlg_k.$

Hochschild homology is defined to be the left adjoint

$x_!\colon\aCAlg_k\rightarrow\aCAlg_k^{BS^1}.$

Often, this left adjoint is written $x_!R\simeq S^1\otimes_k R$, but I prefer to use the copower $x_!R\simeq {^{S^1}R}$. In any case, we will write

$\HH(R/k)=x_!R.$

The universal property is evident: given an animated commutative ring $S$ with circle action, there is a natural equivalence

$\{\text{maps R\rightarrow S}\}\simeq\{\text{S^1-equivariant maps \HH(R/k)\rightarrow S}\}.$

# The HKR filtration

When $R$ is a smooth commutative $k$-algebra, there are canonical isomorphisms $\Omega^i_{R/k}\cong\HH_i(R/k).$ From this, we can build for any animated commutative $k$-algebra $R$ a complete decreasing filtration $\F^\star\HH(R/k)$ on Hochschild homology with

$\gr^i\HH(R/k)\simeq\Lambda^i\L_{R/k}[i]$

by left Kan extending the Whitehead tower on $\HH(R/k)$ in the polynomial case.

Question. Where does the HKR filtration come from? Does it admit a universal property similar to the one given above for Hochschild homology itself?

This is the question answered by the papers of Moulinos–Robalo–Toën and Raksit.

# Weighing down the circle

Let $R$ be a smooth commutative $k$-algebra. The Postnikov filtration on $\HH(R/k)$ is a filtration by complexes with $S^1$-action. It follows that in general the HKR filtration $\F^\star\HH(R/k)$ is a filtered complex with $S^1$-action. The circle acts trivially on the graded pieces. However, we would like the circle to act in such a way as to recover the classical `B-operator’ on Hochschild homology, which gives the de Rham differential $\Omega^i_{R/k}\rightarrow\Omega^{i+1}_{R/k}$ in the smooth case.

Here, note that giving a complex with $S^1$-action amounts to giving a dg module over the chain algebra $\C_\bullet(S^1,k)$. If we let $B$ be a generator of $\H_1(S^1,k)$, then we get an operator $B\colon \HH(R/k)\rightarrow\HH(R/k)[-1]$. However, on the associated gradeds of the HKR filtration, this operator

$B\colon\gr^\star\HH(R/k)\rightarrow\gr^\star\HH(R/k)[-1]$

is necessarily nullhomotopic as each $\Lambda^i\L_{R/k}[i]\rightarrow\Lambda^i\L_{R/k}[i-1]$ is nullhomotopic.1 What we want instead is for $B$ to have weight $1$ so that it induces an operator

$B\colon\gr^\star\HH(R/k)\rightarrow\gr^{\star+1}\HH(R/k)[-1]$

which on graded pieces is the de Rham differential

$d\colon\Lambda^i\L_{R/k}[i]\rightarrow\Lambda^{i+1}\L_{R/k}[i].$

Moulinos–Robalo–Toën and Raksit attack this problem in different ways. The former is essentially geometric, and works over $\bZ_{(p)}$, while the latter is essentially algebraic and works over $\bZ_{(p)}$. The geometric approach crucially uses the perspective that filtered complexes are quasicoherent sheaves on $\bA^1/\Gm$. In this picture, the underlying object being filtered is the value of the quasicoherent sheaf at the generic point $\Gm/\Gm\simeq\ast$ while the associated graded is the restriction at $0/\Gm$.

# The MRT approach

Moulinos–Robalo–Toën construct in  a group scheme $\bH_{p^\infty}$ over $\bA^1/\Gm$ which correctly interpolates between the circle at the generic point and the homology of the circle at the special fiber. (See Definition 2.3.7.)

From now on, we assume that our base $k$ is a commutative $\bZ_{(p)}$-algebra. Let $\bW$ be the ring scheme of $p$-typical Witt vectors. As sheaf of sets, $\bW$ is equivalent to $\prod_{i=0}^\infty\bG_a$, but this ignores the additive and multiplicative structures on both sides. There is a Frobenius endomorphism $F\colon\bW\rightarrow\bW$. The authors define

$\mathbf{Ker}=\ker(F)$

and

$\mathbf{Fix}=\ker(1-F).$

These are both group schemes.

Theorem (MRT ). The group stack $B\bH_{p^\infty}$ over $\bA^1/\Gm$ has underlying group stack $B\mathbf{Fix}$ and associated graded $B\mathbf{Ker}$. The cohomology of $B\mathbf{Ker}$ is the cohomology of the circle $\H^\star(S^1,k)$ as a graded object. The classifying stack $B\mathbf{Fix}$ is the affinization of the circle; in particular, $\C^\bullet(B\mathbf{Fix},\Oscr)\simeq\C^\bullet(S^1,k)$ as $\bE_\infty$-rings.

One can summarize by saying that $\bH_{p^\infty}$ is a filtration on $\mathbf{Fix}$ with associated graded $\mathbf{Ker}$.

Definition. The filtered circle is defined to be the group stack $S^1_\fil=B\bH_{p^\infty}$ over $\bA^1/\Gm$.

To construct a filtered version of Hochschild homology, recall first that if $X=\Spec(R)$, then the (derived) loop stack $\Lscr X=X^{S^1}\simeq\bMap_k(S^1,X)$ has global sections $\C^\bullet(\Lscr X,\Oscr)\simeq\HH(R/k)$. The group $S^1$ acts via rotation of loops.

Now, given $X=\Spec(R)$ over $\Spec k$, we can form the filtered loop stack

$\Lscr_\fil X=\bMap_{\bA^1/\Gm}(S^1_\fil,X\times \bA^1/\Gm).$

Definition. The global sections $\C^\bullet(\Lscr_\fil X,\Oscr)$ is defined to be $\HH_\fil(R/k)$. It is a quasicoherent sheaf on $\bA^1/\Gm$ with $S^1_\fil$-action. By the previous theorem, the underlying object is $\HH(R/k)$.

Theorem (MRT). The associated graded of $\HH_\fil(R/k)$ is $\Lambda^\star\L_{R/k}[\star]$ with action by $\gr^\star S^1_\fil$ given by the de Rham differential. Moreover, upon taking $S^1_\fil$-fixed points $\HH_\fil(R/k)^{\h S^1_\fil}$, one obtains a filtration on $\HC^-(R/k)=\HH(R/k)^{\h S^1}$ with

$\gr^n\HC^-(R/k)\simeq\widehat{\dR}_{R/k}^{\geq n}[2n],$

where $\widehat{\dR}_{R/k}$ denotes Hodge-complete derived de Rham cohomology.

For more on this filtration, see our previous post.

# Raksit’s approach

The approach here is somewhat different. Raksit works directly with the circle and upgrades it to a filtered object, without affinization.

Writes $\bT$ for $\C_\bullet(S^1,k)$. This is a bicommutative bialgebra in $\D(k)$ and we have $\Mod_{\bT}(\D(k))\simeq\D(k)^{BS^1}$. In particular, the symmetric monoidal structure on $\D(k)^{BS^1}$ comes from the $\bE_\infty$-comultiplication on $\bT$. Raksit lets $\bT_\fil$ denote $\tau_{\geq\star}\bT$, the Whitehead tower of $\bT$ viewed as a filtered object. This turns out again to be a bicommutative bialgebra, this time in $\F\D(k)=\Fun(\bZ^\op,\D(k))$, the $\infty$-category of decreasing filtrations. By definition, a complex with filtered circle action is a $\bT_\fil$-module in $\F\D(k)$.

In order to study Hochschild homology, we must bring some notion of derived commutative rings into play. For this, Raksit finds it more appropriate to work with the dual object $\bT_\fil^\vee$, which is the Whitehead filtration of cochains on the circle. Note that $\bT_\fil$-modules in $\F\D(k)$ are equivalent to $\bT_\fil^\vee$-comodules.

The advantage of $\bT_\fil^\vee$ is that the algebra structure is derived commutative and moreover the comultiplication makes $\bT_\fil^\vee$ into an $\bE_\infty$-coalgebra in $\dCAlg(\F\D(k))$, the $\infty$-category of derived commutative rings in filtered complexes (a notion which Raksit defines). By definition a derived commutative algebra in filtered complexes with filtered $S^1$-action is a $\bT_\fil^\vee$-comodule in $\dCAlg(\F\D(k))$.

Now, there is a limit-preserving functor

$\coMod_{\bT_\fil^\vee}(\dCAlg(\F\D(k)))\rightarrow\dCAlg_k$

obtained by forgetting the $\bT_\fil$-action to obtain derived commutative algebra $\F^\star S$ and then taking $\F^0S$.

Raksit defines filtered Hochschild homology as the left adjoint

$\HH_\fil(-/k)\colon\dCAlg_k\rightarrow\coMod_{\bT_\fil^\vee}(\dCAlg(\F\D(k))).$

The universal property becomes the following: given a derived commutative filtered ring $\F^\star S$ with filtered circle action, there is a natural equivalence

$\{\text{maps R\rightarrow \F^0 S}\}\simeq\{\text{\bT_\fil-equivariant maps \HH_\fil(R/k)\rightarrow \F^\star S}\}.$

Theorem (Raksit ). If $R$ is an animated commutative $k$-algebra, $\HH_\fil(R/k)$ is equivalent to $\HH(R/k)$ with the HKR filtration (as animated commutative rings with $S^1$-action). Moreover, the associated graded $\gr^\star\HH_\fil(R/k)$ is Hodge-complete derived de Rham cohomology.

Let us explain the final sentence. Consider first the difference between a $\bT$-module $\F^\star X$ in $\F\D(k)$ versus a $\bT_\fil$-module $\F^\star Y$. In the first case, on associated gradeds we get a graded $\bT$-module $\gr^\star X$ where $\bT$ has weight $0$. For the second, we get a $\gr^\star\bT_\fil$-module $\gr^\star Y$. But, $\gr^\star\bT_\fil\simeq k\oplus k(1)$. In particular, the action map $\gr^\star\bT_\fil\otimes_k\gr^\star Y\rightarrow\gr^\star Y$ induces maps $\gr^{\star}Y\rightarrow\gr^{\star+1}Y[-1]$.

Now, $\gr^\star\HH_\fil(R/k)$ is a $\gr^\star\bT_\fil$-module. Let $\Gr\D(k)$ be the $\infty$-category of $\bZ$-graded complexes. There is a shear down equivalence $\Gr\D(k)\xrightarrow{[-2\star]}\Gr\D(k)$ which sends a graded object $X(\star)$ to $X(\star)[-2\star]$. The shear down equivalence is symmetric monoidal. In particular, if we shear down, we get an action of $\gr^\star\bT_\fil[-2\star]\simeq k\oplus k[-1](1)$ on $\gr^\star\HH_\fil(R/k)[-2\star]\simeq\Lambda^\star\L_{R/k}[-\star]$. The action of this sheared down graded circle, gives the derived de Rham complex as a homotopy coherent chain complex:

$R\rightarrow\L_{R/k}\rightarrow\Lambda^2\L_{R/k}\rightarrow\cdots.$

Raksit also establishes the de Rham filtrations on $\HC^-(R/k)$ and $\HP(R/k)$ by applying the homotopy fixed points $\HH_\fil(R/k)^{\h\bT_\fil}$ or Tate construction $\HH_\fil(R/k)^{\t\bT_\fil}$, respectively.

As a bonus, Raksit also proves a universal property for Hodge-complete derived de Rham cohomology which is obtained by shearing down the associated graded of the universal property for $\bT_\fil$-equivariant filtered Hochschild homology.

# Final remarks

Each of the papers discussed here contains a lot more material than I’ve written about. There is some nice material in MRT on the geometry of the Witt vectors and affinization, while Raksit has written on derived algebraic contexts, a notion to formalize some constructions of Bhatt and Mathew. Both of these will see a lot of use in the future.

There is one thing that remains to be done: show that the filtered circles of MRT and Raksit agree in some appropriate sense. Specifically, I expect that

$\Mod_{\bT_\fil}\F\D(k)\simeq\D(B^2\bH_{p^\infty})$

as symmetric monoidal $\infty$-categories (and even as commutative $\F\D(k)$-algebras). This is alluded to in Remark 4.2.5 of MRT without proof.

 Moulinos, Robalo, and Toën, A universal HKR theorem, arXiv:1906.00118.

 Raksit, Hochschild homology and the derived de Rham complex revisited, arXiv:2007.025760.

# Footnotes

1. Here we use that the circle acts trivially on any (shifted) discrete complex. Since the HKR filtration has (shifted) discrete graded pieces in the smooth case, the circle action is trivial on the graded pieces in general.