ScalingStacks

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

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

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

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

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