ScalingStacks

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

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