Termwise Hochschild homology and iterated homology

Let $k$ be a field, let $A$ be a unital associative $k$-algebra, and let $F=(F^i,d_F^i{)}_{i\in \mathbb{Z}}$ be a bounded cochain complex of $k$-central $A$-bimodules with differentials of internal degree zero (Bounded graded bimodule complexes and signed tensor totalization). Fix $j\ge 0$. Each differential $d_F^i:F^i\to F^{i+1}$ is a map of $A$-bimodules, so it commutes with every Hochschild face of the chains $C_j(A,-)=(-){\otimes }_k{A}^{{\otimes }_{k}j}$ and therefore induces a chain map $C_{\bullet }(A,d_F^i):(C_{\bullet }(A,F^i),b)⟶(C_{\bullet }(A,F^{i+1}),b)$ (Hochschild chains and Hochschild homology with coefficients). Writing $H{H}_j(A,F^i)=H_j(C_{\bullet }(A,F^i))$, the induced map on homology is the $k$-linear map $H{H}_j(A,d_F^i):H{H}_j(A,F^i)⟶H{H}_j(A,F^{i+1})$ provided by A chain map induces a well-defined map on homology. Because $d_F^{i+1}{d}_F^i=0$, the composite chain map $C_{\bullet }(A,d_F^{i+1})C_{\bullet }(A,d_F^i)$ is induced by the zero bimodule map and is therefore the zero chain map, so the composite $H{H}_j(A,d_F^{i+1})H{H}_j(A,d_F^i)$ vanishes; the functoriality in the same cited theorem also gives $H{H}_j(A,d_F^i)=\mathrm{i}\mathrm{d}$ when $d_F^i$ is an identity and $H{H}_j(C_2)H{H}_j(C_1)=H{H}_j(C_2{C}_1)$ for composable bimodule maps. Hence $(H{H}_j(A,F^{\bullet }),H{H}_j(A,d_F^{\bullet }))$ is a cochain complex of $k$-modules, the termwise Hochschild complex of $F$ in Hochschild degree $j$, and its cohomology is defined (Cohomology object of a cochain complex): $H^i(H{H}_j(A,F^{\bullet }))=\frac{\ker (H{H}_j(A,d_F^i))}{\mathrm{im}(H{H}_j(A,d_F^{i-1}))}\mathrm{.}$ The differentials $H{H}_j(A,d_F^i)$ are induced by maps of bimodules, so they are $k$-linear and preserve whatever outer structure the construction carries; in particular, if $F$ is a complex of graded bimodules and every $d_F^i$ has internal degree zero, then each internal-degree piece of $H{H}_j(A,F^i)$ is mapped to the same internal degree of $H{H}_j(A,F^{i+1})$.

In the internally graded case the three indices are kept separately: the Hochschild degree $j\ge 0$, the cochain index $i$ of $F$, and the internal degree, together with the cohomological degree $i$ of $H^i(H{H}_j(A,F^{\bullet }))$. No identification between these indices is asserted by this definition. In particular this construction is not defined to coincide with the Hochschild hyperhomology ${\mathrm{H}\mathrm{H}}^{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r},i-j}(A,F)$ of Hochschild hyperhomology of a bounded bimodule complex: the latter is the cohomology of the total complex in which the Hochschild boundary $b$ and the cochain differential $d_F$ are combined into one differential, whereas here $b$ is used first, separately for each $i$, and only the induced maps $H{H}_j(A,d_F^i)$ are totalized. The two constructions agree in general only through the spectral sequence of Termwise Hochschild spectral sequence of a bounded bimodule complex, whose second page is exactly the iterated group $H^i(H{H}_j(A,F^{\bullet }))$ and which may carry higher differentials.

Two degenerate readings fix the conventions. If $F$ is concentrated in a single cochain degree $r$, then the termwise complex has one term in degree $r$ and $H^r(H{H}_j(A,F^{\bullet }))=H{H}_j(A,F^r)$, with all other iterated groups zero. If $F=0$ then the termwise complex is zero, so every iterated group vanishes. If the differential of $F$ vanishes then all maps $H{H}_j(A,d_F^i)$ vanish, so $H^i(H{H}_j(A,F^{\bullet }))=H{H}_j(A,F^i)$ in the sense that the termwise complex is the direct sum of its terms with zero differential; this is the situation of the matrix and two-term examples on the companion page.