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

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

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

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

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

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 $A$.