Termwise Hochschild cyclicity for bounded projective bimodule complexes

Given: AC, a field $k$, unital associative $k$-algebras $A$ and $B$, a bounded cochain complex $M$ of graded $(A,B)$-bimodules with termwise finite projective right $B$-terms, and a bounded cochain complex $N$ of graded $(B,A)$-bimodules with termwise finite projective right $A$-terms, all differentials of internal degree zero.

For each $j\ge 0$ the termwise Hochschild complex $H{H}_j(A,-)$ is obtained by applying the Hochschild complex functor $C_{\bullet }(A,-)$ degreewise to the coefficient complex and passing to homology; a bimodule map $u$ induces a chain map $C_{\bullet }(A,u)$ commuting with every Hochschild face, and the induced maps on homology assemble into the cochain differential of the termwise complex, whose cohomology is $H^i(H{H}_j(A,F^{\bullet }))$ (Hochschild chains and Hochschild homology with coefficients, Termwise Hochschild homology and iterated homology).

Assume AC. For a fixed $(A,B)$-bimodule $M^{\mathrm{\prime }}$ finite projective as a right $B$-module and a fixed $(B,A)$-bimodule $N^{\mathrm{\prime }}$ finite projective as a right $A$-module, there is a natural zigzag of chain-homotopy equivalences $C_{\bullet }(A,M^{\mathrm{\prime }}{\otimes }_B{N}^{\mathrm{\prime }})\simeq C_{\bullet }(B,N^{\mathrm{\prime }}{\otimes }_A{M}^{\mathrm{\prime }})$; it is realized through the double-bar resolutions and their cyclic rotation, and it is natural and involutive up to homotopy (Double bar comparison for cyclic bimodule tensor products).

The signed tensor totalization has $\mathrm{Tot}(M{\otimes }_{B}N{)}^r={⨁}_{i+l=r}{M}^i{\otimes }_B{N}^l$ with differential $d(m\otimes n)=d_{M}m\otimes n+(-1{)}^{i}m\otimes d_{N}n$ on the summand of cochain degree $i$; each total degree is a finite direct sum because $M$ and $N$ are bounded, and the differentials and the Koszul signs are as in Bounded graded bimodule complexes and signed tensor totalization.

Tensoring with a bimodule complex is additive and preserves chain maps, composition and chain homotopies: on a summand of cochain degree $i$ the second-factor homotopy enters with sign $(-1{)}^i$ and the first-factor homotopy enters with no extra sign (Bimodule tensor totalization respects differentials and homotopies).

A chain map induces the unique map on homology represented by its restriction to cycles followed by the homology quotient; the uniqueness clause makes this assignment preserve identities and compositions, and the cycle-quotient description preserves sums of chain maps (A chain map induces a well-defined map on homology).

Since $C_j(A,M)=M{\otimes }_k{A}^{{\otimes }_{k}j}$ is built from the tensor product, which is linear in its coefficient variable, the Hochschild complex functor $C_{\bullet }(A,-)$ is additive: a finite direct sum of coefficient bimodules satisfies $C_j(A,F\oplus G)\cong C_j(A,F)\oplus C_j(A,G)$ naturally, and $C_j(A,u+v)=C_j(A,u)+C_j(A,v)$. Thus the induced maps on homology are additive by [F5]. For a finite direct sum of coefficient complexes, the termwise complexes and their cohomology are the direct sums of the summands; kernels and images in the definition of cohomology commute with finite direct sums (Hochschild chains and Hochschild homology with coefficients, Cohomology object of a cochain complex).

Chain-homotopic maps induce the same map on homology; applying this in each Hochschild degree gives invariance of $H{H}_j$ under the chain-homotopy equivalences in [F2] (Chain-homotopic maps induce the same map on homology).