Subsection 5.2. (0NZX) · ScalingStacks

5.2. Centers of convolution categories

Let $p:X\to Y$ be a map of perfect stacks satisfying descent. Consider the convolution diagram

Equip $\qc(X\times_{Y}X)$ with the monoidal product defined by convolution

M\star N=p_{13*}(p_{12}^{*}(M)\otimes p_{23}^{*}(N)).

By Theorem 4.14, we have a monoidal equivalence

\qc(X\times_{Y}X)\simeq\Fun_{Y}(\qc(X),\qc(X)).

Consider the fundamental correspondence

where the maps are defined by the formulas

\pi={\rm id}_{\mathcal{L}Y}\times_{{\rm id}_{Y}}p=p\times_{p\times{\rm id}_{Y}}p\qquad\delta={\rm id}_{X}\times_{\pi_{Y}}{\rm id}_{X}.

where $\pi_{Y}:X\times Y\to Y$ is the obvious projection.

Passing to sheaves, we obtain a diagram

which by Theorem 4.14 admits the interpretation

where $\pi^{*}$ is the $\qc(X)$ -linear induction, and $\delta_{*}$ forgets the $\qc(X)$ -linear structure.

The aim of this section is to prove the following.

0NYE

Theorem 5.3. Suppose $p:X\to Y$ is a map of perfect stacks satisfying descent. Then there is a canonical equivalence

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

such that the forgetful functor $\mathfrak{z}:\mathcal{Z}(\qc(X\times_{Y}X))\to\qc(X\times_{Y}X)$ is given by the correspondence $\delta_{*}\pi^{*}:\qc(\mathcal{L}Y)\to\qc(X\times_{Y}X)$ .

The proof occupies the remainder of this section. We break up the argument into a general discussion and then its specific application.

At the end of the section, we also explain 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 $\pi_{*}\delta^{*}:\qc(X\times_{Y}X)\to\qc(\mathcal{L}Y)$ .

5.2.1. Relative cyclic bar construction

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).

5.2.2. Proof of Theorem 5.3

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.

5.2.3. Traces and Grothendieck duality

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 $\mathfrak{tr}$ .

David Ben-Zvi, John Francis, David Nadler

Original source: arXiv:0805.0157v5