Hochschild hyperhomology of a bounded bimodule complex

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): $F^i=0$ outside a finite interval of integers, each $d_F^i:F^i\to F^{i+1}$ is a map of $A$-bimodules, $d_F^{i+1}{d}_F^i=0$, and each $d_F^i$ preserves the internal degree when the bimodules carry one (Associative graded algebras, bimodules, and internal shifts).

For $j\ge 0$ let $C_j(A,F^i)$ denote the Hochschild chains of the coefficient bimodule $F^i$, that is $C_j(A,F^i)=F^i{\otimes }_k{A}^{{\otimes }_{k}j}$ with $C_0(A,F^i)=F^i$, with the Hochschild boundary $b_j:C_j(A,F^i)\to C_{j-1}(A,F^i)$ given by the alternating sum of the faces (Hochschild chains and Hochschild homology with coefficients); recall $C_{-1}(A,F^i)=0$ and $b_0=0$. The Hochschild hyperhomology total complex has cochain degree $n$ term $T^n(A,F):={⨁}_{\begin{array}{c}i-j=n\\ j\ge 0\end{array}}{C}_j(A,F^i),$ with differential $D:T^n(A,F)\to T^{n+1}(A,F)$ acting on the $(i,j)$ summand as $D:=d_F+(-1{)}^{i}b,x⟼d_F(x)+(-1{)}^{i}b(x)\in C_j(A,F^{i+1})\oplus C_{j-1}(A,F^i)\mathrm{.}$

Each total degree is a finite direct sum: if $F$ is supported in the interval $[a,b]$, then for fixed $n$ the pair $(i,j)$ satisfies $i-j=n$ and $j\ge 0$, hence $i=n+j$ with $a\le i\le b$ and only the finitely many indices $i\in [a,b]\cap [n,\mathrm{\infty })$ contribute, each by the single summand $C_{i-n}(A,F^i)$. The differential is well defined and $D^2=0$: a bimodule map commutes with every Hochschild face and therefore with the boundary $b$, so $d_{F}b=b{d}_F$, while $d_F^2=0$ and $b^2=0$; on the $(i,j)$ summand the coefficient of $b$ in $D$ is $(-1{)}^i$ and the coefficient of $b$ in $D$ on the $(i+1,j)$ summand is $(-1{)}^{i+1}$, so the two mixed composites $(-1{)}^{i+1}b{d}_F$ and $(-1{)}^i{d}_{F}b$ cancel. Thus $(T^{\bullet }(A,F),D)$ is a cochain complex of $k$-modules and its cohomology is defined (Cohomology object of a cochain complex): ${\mathrm{H}\mathrm{H}}^{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r},n}(A,F):=H^n(T^{\bullet }(A,F))=\frac{\ker (D:T^n(A,F)\to T^{n+1}(A,F))}{\mathrm{im}(D:T^{n-1}(A,F)\to T^n(A,F))}\mathrm{.}$

Both structure maps preserve internal degree, so the internal grading descends to $T^{\bullet }(A,F)$ and to ${\mathrm{H}\mathrm{H}}^{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r},n}(A,F)$. When $A$ and $F$ are internally graded, the ordinary tensor grading on $C_j(A,F^i)=F^i{\otimes }_k{A}^{{\otimes }_{k}j}$ is the sum grading: for homogeneous $f\in F^i$ and $a_1,\dots ,a_j\in A$, the tensor $f\otimes a_1\otimes \cdots \otimes a_j$ has internal degree ${\mathrm{deg}}_{\mathrm{i}\mathrm{n}\mathrm{t}}(f)+{\sum }_{t=1}^j{\mathrm{deg}}_{\mathrm{i}\mathrm{n}\mathrm{t}}(a_t)$. If $A$ is concentrated in internal degree zero, this reduces to ${\mathrm{deg}}_{\mathrm{i}\mathrm{n}\mathrm{t}}(f)$. The maps $d_F$ and $b$ are homogeneous of internal degree zero. When $A$ is graded, the Hochschild boundary used here is the ordinary boundary of Hochschild chains and Hochschild homology with coefficients: no Koszul sign is inserted into a face merely because the entries have nonzero internal degree.

For each integer $p$ the subspaces $F^p{T}^n(A,F):={⨁}_{\begin{array}{c}i\ge p\\ i-j=n\\ j\ge 0\end{array}}{C}_j(A,F^i)$ form a decreasing filtration of $T^n(A,F)$ by subcomplexes (each summand of $F^p{T}^{\bullet }$ has its $D$-image again in $F^p{T}^{\bullet +1}$, because $d_F$ raises $i$ and $b$ fixes $i$). This filtration is finite at each total degree, exhaustive and separated, since $F$ is bounded. The separate indices $i$ and $j$ therefore enter only through this filtration: the decomposition of $T^n(A,F)$ into its $(i,j)$ summands is not a direct sum decomposition compatible with $D$, and one may not read ${\mathrm{H}\mathrm{H}}^{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r},n}(A,F)$ off as a direct sum of the homologies of the individual summands $C_j(A,F^i)$. What descends automatically is the internal grading and the total cohomological degree $n$; the pair $(i,j)$ becomes a filtered piece, exactly as used by the termwise spectral sequence of Termwise Hochschild spectral sequence of a bounded bimodule complex.

Three specializations record the conventions used throughout this page. If $F$ is concentrated in cochain degree $0$, then $T^{-j}(A,F)=C_j(A,F)$ for $j\ge 0$, with differential $b$ raising the total cochain degree by one; thus ${\mathrm{H}\mathrm{H}}^{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r},-j}(A,F)=H{H}_j(A,F)$, and the hyperhomology vanishes in positive total degrees. If $F=0$, then $T^{\bullet }(A,F)=0$ and all hyperhomology groups vanish. If the differential of $F$ vanishes, then $D$ is, on each fixed-$i$ column, the boundary $b$ up to the sign $(-1{)}^i$. The total complex is the direct sum of these reindexed Hochschild complexes, so ${\mathrm{H}\mathrm{H}}^{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r},n}(A,F)\cong {⨁}_{i-j=n}H{H}_j(A,F^i)\mathrm{.}$ This special decomposition does not extend to a general nonzero $d_F$; in that case the $(i,j)$ pieces give the filtration, and the termwise groups $H^i(H{H}_j(A,F))$ occur on the second page of Termwise Hochschild spectral sequence of a bounded bimodule complex.

This definition is the complex-level Hochschild construction of the source: for a complex of bimodules the Hochschild complex is formed degreewise and then totalized, and the hyperhomology is the homology of that total complex (BPW §3.8.6, printed p.38; Khovanov, printed pp.6–7). No projective resolution is fixed here; the identification of $T^{\bullet }(A,F)$ with the total complex of the reindexed two-sided bar resolution tensored over $A^e$ with $F$, and the consequent resolution independence of ${\mathrm{H}\mathrm{H}}^{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r},n}(A,F)$, is the content of Hochschild hyperhomology is independent of a projective resolution.