ScalingStacks

It is very useful to have versions of the Hochschild chain and cochain complexes which calculate the trace and center. In the setting of an associative algebra object $A$ in a monoidal $\infty$-category $\mathcal{S}$, they will take the form of a simplicial object $\mathbf{N}^{cyc}_{*}(A)$ and cosimplicial object $\mathbf{N}_{cyc}^{*}(A)$ such that the geometric realization $\colim\mathbf{N}^{cyc}_{*}(A)$ is the trace $\mathcal{T}r(A)$ and the totalization $\lim\mathbf{N}_{cyc}^{*}(A)$ is the center $\mathcal{Z}(A)$. (Note that simplicial and cosimplicial objects here are taken in the $\infty$-categorical sense, and so the diagram identities hold up to coherent homotopies as in, e.g., [A], where the cyclic bar construction for $A_{\infty}$ H-spaces is developed.)

We construct the simplicial object $\mathbf{N}^{cyc}_{*}(A)$ and cosimplicial object $\mathbf{N}_{cyc}^{*}(A)$ as follows. Consider the adjunction

where $F(-)=A\otimes-$ is the induction, and $G$ is the forgetful functor. It determines a comonad $S\simeq FG$ acting on $\mathrm{Mod}_{A}(\mathcal{S})$.

As in classical algebra, for any $A$-module $M$, the comonad $S$ provides a canonical augmented simplicial object $C_{*}(M)$ with terms

$$C_{n-1}(M)\simeq(FG)^{n}(M)\simeq A^{\otimes n}\otimes M$$

such that the geometric realization of $C_{*}(M)$ is naturally equivalent to $M$. We will say that $C_{*}(M)$ is a simplicial resolution of $M$.

We can apply this technique to produce the familiar cyclic bar construction. Consider the special case of the above adjunction

where $F(-)=A\otimes-$ again is the induction, and $G$ is the forgetful functor from $A$-bimodules to right $A$-modules. Using the comonad $S\simeq FG$, we obtain a simplicial resolution $C_{*}(A)$ of the $A$-bimodule $A$ whose terms $C_{n-1}(A)\simeq A^{\otimes n+1}$ are $A$-bimodules which are free as left $A$-modules.

Now recall that the trace $\mathcal{T}r(A)$ is defined by the self-pairing $A\otimes_{A\otimes A^{\rm op}}A$. Since the tensor product commutes with colimits, in particular geometric realizations, we calculate

$$\mathcal{T}r(A)=A\otimes_{A\otimes A^{\rm op}}A\simeq A\otimes_{A\otimes A^{\rm op}}|C_{*}(A)|\simeq|A\otimes_{A\otimes A^{\rm op}}C_{*}(A)|.$$

Thus the geometric realization of $A\otimes_{A\otimes A^{\rm op}}C_{*}(A)$ calculates the trace $\mathcal{T}r(A)$.

We write $\mathbf{N}^{cyc}_{*}(A)$ for the simplicial object $A\otimes_{A\otimes A^{\rm op}}C_{*}(A)$ and refer to it as the Hochschild chain complex. Since $A$ is free as a right $A$-module, the terms of the simplicial object $C_{*}(A)$ are free as $A$-bimodules. Thus we can evaluate the terms of the Hochschild chain complex

$$\mathbf{N}^{cyc}_{n}(A)=A\otimes_{A\otimes A^{\rm op}}C_{n}(A)\simeq A\otimes_{A\otimes A^{\rm op}}A^{\otimes n+2}\simeq A^{\otimes n+1}$$

In particular, there are equivalences $\mathbf{N}^{cyc}_{0}(A)\simeq A$ and $\mathbf{N}^{cyc}_{1}(A)\simeq A\otimes A$, and the two simplicial maps $A\otimes A\rightarrow A$ are the multiplication and the opposite multiplication of $A$.

Similarly, recall that the center $\mathcal{Z}(A)$ is defined by the endomorphisms ${\mathcal{E}nd}_{A\otimes A^{\rm op}}(A)$. Since morphisms take colimits in the domain to limits, in particular geometric realizations to totalizations, we calculate

$$\mathcal{Z}(A)={\mathcal{E}nd}_{A\otimes A^{\rm op}}(A)\simeq{\mathcal{H}om}_{A\otimes A^{op}}(|C_{*}(A)|,A)\simeq|{\mathcal{H}om}_{A\otimes A^{op}}(C_{*}(A),A)|$$

Thus the totalization of ${\mathcal{H}om}_{A\otimes A^{op}}(C_{*}(A),A)$ calculates the center $\mathcal{Z}(A)$.

We write $\mathbf{N}_{cyc}^{*}(A)$ for the cosimplicial object ${\mathcal{H}om}_{A\otimes A^{op}}(C_{*}(A),A)$ and refer to it as the Hochschild cochain complex. As before, we can evaluate the terms of the Hochschild cochain complex

$$\mathbf{N}_{cyc}^{n}(A)={\mathcal{H}om}_{A\otimes A^{op}}(C_{n}(A),A)\simeq{\mathcal{H}om}_{A\otimes A^{op}}(A^{\otimes n+2},A)\simeq{\mathcal{H}om}(A^{\otimes n},A).$$

In particular, there are equivalences $\mathbf{N}_{cyc}^{0}(A)\simeq A$ and $\mathbf{N}_{cyc}^{1}(A)\simeq{\mathcal{H}om}(A,A)$, and the two cosimplicial maps $A\to{\mathcal{H}om}(A,A)$ are induced by the left and right multiplication of $A$.

We will apply the above constructions in the following setting. We will always take $\mathcal{S}$ to be the $\infty$-category ${\mathcal{P}r}^{\rm L}$ of presentable $\infty$-categories (with morphisms left adjoints). Then a monoidal presentable $\infty$-category $\mathcal{C}$ is an associative algebra object in $\mathcal{P}r^{\rm L}$. Thus we have the notion of its center (or Hochschild cohomology category) $\mathcal{Z}(\mathcal{C})=\Fun_{\mathcal{C}\otimes\mathcal{C}^{\rm op}}(\mathcal{C},\mathcal{C})\in\mathcal{P}r^{\rm L}$, and its trace (or Hochschild homology category) $\mathcal{T}r(\mathcal{C})=\mathcal{C}\otimes_{\mathcal{C}\otimes\mathcal{C}^{\rm op}}\mathcal{C}\in\mathcal{P}r^{\rm L}$.

Now we will specialize further to a geometric setting. Let $X$ be a perfect stack, and take $\mathcal{C}$ to be the presentable stable $\infty$-category $\qc(X)$ equipped with its (symmetric) monoidal tensor product. To calculate the center and trace of $\qc(X)$, we introduce the loop space

$$\mathcal{L}X=\Map(S^{1},X)\simeq X\times_{X\times X}X$$

where the fiber product is along two copies of the diagonal map.

For the proof of Theorem 5.3, it will be useful to introduce relative versions of the Hochschild chain and cochain complexes. In general, the setup for the construction will be a map of associative algebra objects $B\to A$ in a monoidal $\infty$-category $\mathcal{S}$. The usual Hochschild chain and cochain complexes introduced in the previous section will correspond to the case when $B$ is the unit $1_{\mathcal{S}}$.

Consider the adjunction

where $F(-)=A\otimes_{B}-$ is the induction, and $G$ is the forgetful functor from $A$-bimodules to $B\otimes A^{\rm op}$-modules. Using the comonad $S\simeq FG$, we obtain a simplicial resolution $C^{B}_{*}(A)$ of the $A$-bimodule $A$ with terms

$$C^{B}_{n-1}(A)\simeq A\otimes_{B}\otimes\cdots\otimes_{B}A,\quad\mbox{ with $n+1$ terms.}$$

To reduce the notation, we will denote the above expression by $A_{B}^{\otimes n+1}$.

As before, the geometric realization of $A\otimes_{A\otimes A^{\rm op}}C^{B}_{*}(A)$ calculates the trace $\mathcal{T}r(A)$, and the totalization of ${\mathcal{H}om}_{A\otimes A^{op}}(C^{B}_{*}(A),A)$ calculates the center $\mathcal{Z}(A)$. We write $\mathbf{N}^{B}_{*}(A)$ for the simplicial object $A\otimes_{A\otimes A^{\rm op}}C^{B}_{*}(A)$ and refer to it as the relative Hochschild chain complex. Similarly, we write $\mathbf{N}_{B}^{*}(A)$ for the cosimplicial object ${\mathcal{H}om}_{A\otimes A^{op}}(C^{B}_{*}(A),A)$ and refer to it as the relative Hochschild cochain complex.

Continuing as before, we can evaluate the terms of the relative Hochschild chain complex

$$\mathbf{N}^{B}_{n}(A)=A\otimes_{A\otimes A^{\rm op}}C^{B}_{n}(A)\simeq A\otimes_{A\otimes A^{\rm op}}A_{B}^{\otimes n+2}\simeq B\otimes_{B\otimes B^{\rm op}}A_{B}^{\otimes n+1}$$

Similarly, we can evaluate the terms of the relative Hochschild cochain complex

$$\mathbf{N}_{B}^{n}(A)={\mathcal{H}om}_{A\otimes A^{op}}(C^{B}_{n}(A),A)\simeq{\mathcal{H}om}_{A\otimes A^{op}}(A_{B}^{\otimes n+2},A)\simeq{\mathcal{H}om}_{B\otimes B^{\rm op}}(A_{B}^{\otimes n+1},B).$$

Now we will apply the preceding formalism to calculate the center of the monoidal $\infty$-category $\qc(X\times_{Y}X)$. Recall that our aim is to show that there is a canonical equivalence

$$\mathcal{Z}(\qc(X\times_{Y}X))\simeq\qc(\mathcal{L}Y).$$

where $\mathcal{L}Y$ is the loop space of $Y$.

Consider the symmetric monoidal $\infty$-category $\qc(X)$, together with the monoidal functor

$$\Delta_{*}:\qc(X)\to\qc(X\times_{Y}X)$$

obtained via pushforward along the relative diagonal $\Delta:X\to X\times_{Y}X$.

Set $A=\qc(X\times_{Y}X)$, and $B=\qc(X)$. By the preceding discussion, the center $\mathcal{Z}(A)$ is the limit of the relative Hochschild cochain complex $\mathbf{N}^{*}_{B}(A)$. Furthermore, its terms can be calculated

$$\mathbf{N}_{B}^{n}(A)\simeq{\mathcal{H}om}_{B\otimes B^{\rm op}}(A_{B}^{\otimes n+1},B).$$

Applying the results of Section 4, we can rewrite each term in the form

$$\mathbf{N}_{B}^{n}(A)\simeq\qc(\mathcal{L}Y\times_{Y}X\times_{Y}\cdots\times_{Y}X),\quad\mbox{ with $n+1$ copies of $X$.}$$

Here we have used the elementary identification

$$\mathcal{L}Y\times_{Y}X\times_{Y}\cdots\times_{Y}X\simeq X\times_{X\times X}((X\times_{Y}X)\times_{X}\cdots\times_{X}(X\times_{Y}X))$$

where the left hand side has $n+1$ copies of $X$, and the right hand side has $n+1$ copies of $X\times_{Y}X$.

We conclude that the terms of $\mathbf{N}_{B}^{*}(A)$ are nothing more than the terms of the cosimplicial $\infty$-category obtained by applying $\qc(-)$ to the C̆ech simplicial stack induced by the map

$$\tilde{p}:\mathcal{L}Y\times_{Y}X\to\mathcal{L}Y$$

obtained by base change from the original map $p:X\to Y$. It is straightforward to check that under this identification the coboundary maps are given by the usual C̆ech pullbacks. By assumption, $p$ satisfies descent, hence $\tilde{p}$ satisfies descent, and thus the totalization $\lim\mathbf{N}_{B}^{*}(A)$ also calculates the $\infty$-category $\qc(\mathcal{L}Y)$.

Finally, note that under this identification, the composition $\delta_{*}\pi^{*}:\qc(\mathcal{L}Y)\to\qc(X\times_{Y}X)$ corresponds to the composition

$$\qc(\mathcal{L}Y)\simeq\lim\mathbf{N}_{B}^{*}(A)\to\mathbf{N}_{B}^{0}(A)\to\mathbf{N}_{cyc}^{0}(A)\simeq\qc(X\times_{Y}X)$$

which is precisely the central functor $\mathfrak{z}$. This concludes the proof of Theorem 5.3.

Finally, we explain here an analogous description of traces, conditional on a still to be developed version of Grothendieck duality in the derived setting. Namely, assuming further that $p:X\to Y$ is proper with invertible dualizing sheaf and Grothendieck duality holds, we explain how to deduce an expected canonical equivalence

$$\mathcal{T}r(\qc(X\times_{Y}X))\simeq\qc(\mathcal{L}Y)$$

such that the trace $\mathfrak{tr}:\qc(X\times_{Y}X)\to\mathcal{T}r(\qc(X\times_{Y}X))$ is given by the correspondence $\pi_{*}\delta^{*}:\qc(X\times_{Y}X)\to\qc(\mathcal{L}Y)$.

We continue with the notation from the proof of Theorem 5.3 and the preceding sections. We will show that the geometric realization $\colim\mathbf{N}^{B}_{*}(A)$ of the relative Hochschild chain complex also calculates the $\infty$-category $\qc(\mathcal{L}Y)$. As before, applying the results of Section 4, we can rewrite the terms of $\mathbf{N}^{B}_{*}(A)$ in the form

$$\mathbf{N}^{B}_{n}(A)\simeq\qc(\mathcal{L}Y\times_{Y}X\times_{Y}\cdots\times_{Y}X)\quad\mbox{ with $n+1$ copies of $X$.}$$

Furthermore, it is straightforward to check that under this identification the boundary maps of $\mathbf{N}^{B}_{*}(A)$ are given by the pushforwards $\mathfrak{p}_{n*}$ which are right adjoints to the usual C̆ech pullbacks $\mathfrak{p}_{n}^{*}$.

To reduce notation, set $X_{n}=X\times_{Y}\cdots\times_{Y}X$ with $n+1$ copies of $X$.

Now suppose that $p:X\to Y$ is proper and has an invertible dualizing complex (Gorenstein). Then we expect Grothendieck duality to hold in the following form: the pushforwards $\mathfrak{p}_{n*}$ are also left adjoints to the pullbacks

$$\mathfrak{p}_{n}^{!}(-)\simeq\mathfrak{p}_{n}^{*}(-\otimes\omega^{-1}_{\mathcal{L}Y\times_{Y}X_{n-1}/Y})\otimes\omega_{\mathcal{L}Y\times_{Y}X_{n}/Y},$$

where $\omega_{\mathcal{L}Y\times_{Y}X_{n}/Y}$ denotes the relative dualizing sheaf of $\mathcal{L}Y\times_{Y}X_{n}\to Y$. Under this assumption, we find that the limit $\qc(\mathcal{L}Y)$ of the cosimplicial $\infty$-category $(\qc(\mathcal{L}Y\times_{Y}X_{n}),\mathfrak{p}_{n}^{*})$ admits the following alternative description.

First, we can identify the cosimplicial $\infty$-category $(\qc(\mathcal{L}Y\times_{Y}X_{n}),\mathfrak{p}_{n}^{*})$ with the cosimplicial $\infty$-category $(\qc(\mathcal{L}Y\times_{Y}X_{n},\mathfrak{p}_{n}^{!})$ via tensoring by the inverse of the relative dualizing sheaf $\omega_{\mathcal{L}Y\times_{Y}X_{n}/Y}$ on each simplex. In particular, we obtain an identification of their limits.

Second, we can consider the cosimplicial $\infty$-category $(\qc(\mathcal{L}Y\times_{Y}X_{n}),\mathfrak{p}_{n}^{!})$ as a diagram in the $\infty$-category $\mathcal{P}r^{\rm R}$ of presentable $\infty$-categories (with morphisms right adjoints). By [L2, Theorem 5.5.3.18], the calculation of the limit of the diagram does not depend on this choice of context. Then we can pass to the opposite $\infty$-category $\mathcal{P}r^{\rm L}$ of presentable $\infty$-categories (with morphisms left adjoints). To calculate a limit in $\mathcal{P}r^{\rm R}$ is the same as to calculate a colimit in $\mathcal{P}r^{\rm L}$. But we have seen that the trace $\mathcal{T}r(\mathcal{H})$ is precisely the colimit of the dual simplicial diagram $(\qc(\mathcal{L}Y\times_{Y}X_{n}),\mathfrak{p}_{n*})$. Thus we conclude that the geometric realization $\colim\mathbf{N}_{*}^{B}(A)$ also calculates the $\infty$-category $\qc(\mathcal{L}Y)$.

Finally, note that under this identification, the composition $\pi_{*}\delta^{*}:\qc(X\times_{Y}X)\to\qc(\mathcal{L}Y)$ corresponds to the composition

$$\qc(X\times_{Y}X)\simeq\mathbf{N}_{0}^{cyc}(A)\to\mathbf{N}_{0}^{B}(A)\to\colim\mathbf{N}_{*}^{cyc}(A)\simeq\qc(\mathcal{L}Y)$$

which is precisely the trace $\mathfrak{tr}$.