Hochschild Hyperhomology and Cyclic Tensor Invariance

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.

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.

Assume the Axiom of Choice (AC). Let $k$ be a field, let $A$ and $B$ be unital associative $k$-algebras, let $M$ be an $(A,B)$-bimodule that is finite projective as a right $B$-module, and let $N$ be a $(B,A)$-bimodule that is finite projective as a right $A$-module. Put $P_A:=\mathrm{Bar}(A){\otimes }_A(M{\otimes }_{B}N),Q_A:=\mathrm{Tot}(\mathrm{Bar}(A){\otimes }_{A}M{\otimes }_B\mathrm{Bar}(B){\otimes }_{B}N),$ the second total complex being the signed total complex of the double complex whose homological bidegree $(p,q)$ term is ${\mathrm{Bar}}_p(A){\otimes }_{A}M{\otimes }_B{\mathrm{Bar}}_q(B){\otimes }_{B}N$ and whose standard homological total differential is $d_A+(-1{)}^p{d}_B$, where $d_A$ and $d_B$ lower the $A$-bar degree $p$ and the $B$-bar degree $q$, respectively. This sign makes the two cross terms cancel, so the total differential squares to zero. Then $P_A$ and $Q_A$ are projective resolutions of $M{\otimes }_{B}N$ in the abelian category of right $A^e$-modules, and the exchanged constructions $P_B:=\mathrm{Bar}(B){\otimes }_B(N{\otimes }_{A}M),Q_B:=\mathrm{Tot}(\mathrm{Bar}(B){\otimes }_{B}N{\otimes }_A\mathrm{Bar}(A){\otimes }_{A}M)$ are projective resolutions of $N{\otimes }_{A}M$ in right $B^e$-modules. After enveloping coinvariants the two middle double-bar complexes are isomorphic by the cyclic rotation $\rho :\mathrm{coInv}(Q_A)⟶\mathrm{coInv}(Q_B),\rho ([x\otimes m\otimes y\otimes n])=(-1{)}^{pq}[y\otimes n\otimes x\otimes m],$ on a block in bar degrees $p$ and $q$, with the homological Koszul sign. This map is defined on enveloping coinvariant classes; the raw balanced tensors are not claimed to rotate before quotienting. The outer comparisons provide a natural zigzag of chain-homotopy equivalences $C_{\bullet }(A,M{\otimes }_{B}N)\simeq C_{\bullet }(B,N{\otimes }_{A}M),$ natural up to homotopy and involutive up to homotopy. Consequently there is a natural isomorphism $H{H}_j(A,M{\otimes }_{B}N)\cong H{H}_j(B,N{\otimes }_{A}M)$ for every $j\ge 0$. No left-projectivity of $M$ or $N$ is asserted or used.

Assume the Axiom of Choice (AC). Let $k$ be a field, let $A$ be a unital associative $k$-algebra, and let $F=(F^i,d_F^i)$ be a bounded cochain complex of $k$-central $A$-bimodules with differentials of internal degree zero, in the sense of Bounded graded bimodule complexes and signed tensor totalization. Regard the two-sided bar complex $\mathrm{Bar}(A)$ as a complex of right $A^e$-modules and reindex it by $\stackrel{}{\mathrm{Bar}}{}^i:={\mathrm{Bar}}_{-i}(A)$, $\stackrel{}{\mathrm{Bar}}{}^i=0$ for $i>0$. Then the tensor total complex $\mathrm{Tot}(\stackrel{}{\mathrm{Bar}}{\otimes }_{A^e}F{)}^n={⨁}_{i-p=n}{\mathrm{Bar}}_p(A){\otimes }_{A^e}{F}^i$ is identified with the Hochschild hyperhomology complex $T^{\bullet }(A,F)$ of Hochschild hyperhomology of a bounded bimodule complex by the bar-to-Hochschild map on each summand, multiplied by $(-1{)}^{ip}$. Indeed, the source tensor differential is $b+(-1{)}^p{d}_F$, while the target differential is $d_F+(-1{)}^{i}b$; the factor $(-1{)}^{ip}$ intertwines both components. The direct-sum index is $i-p=n$, since the reindexed bar degree is $-p$.

Consequently the hyperhomology ${\mathrm{H}\mathrm{H}}^{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r},n}(A,F)$ can be computed from any supplied bounded-above projective resolution $P\to A$ of $A$ in right $A^e$-modules by $H^n\mathrm{Tot}\left(P{\otimes }_{A^e}F\right)$, and any two such resolutions give canonically isomorphic hyperhomology, the isomorphism being natural in $F$ up to chain homotopy. Moreover every quasi-isomorphism $F\to G$ of bounded cochain complexes of $k$-central $A$-bimodules with internal-degree-zero differentials induces an isomorphism ${\mathrm{H}\mathrm{H}}^{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r},n}(A,F)\to {\mathrm{H}\mathrm{H}}^{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r},n}(A,G)$ for every $n$; when the quasi-isomorphism has internal degree zero this isomorphism is compatible with the internal gradings, so it maps the internal-degree-$r$ part of ${\mathrm{H}\mathrm{H}}^{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r},n}(A,F)$ isomorphically onto the internal-degree-$r$ part of ${\mathrm{H}\mathrm{H}}^{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r},n}(A,G)$. The Axiom of Choice enters only through the basis of $A$ used to make each bar term projective and through the comparison choices between projective resolutions; the sign and exactness computations are choice-free.

Assume the Axiom of Choice (AC). Let $k$ be a field, let $A$ and $B$ be unital associative $k$-algebras, let $M$ be a bounded cochain complex of graded $(A,B)$-bimodules with termwise finite projective right $B$-terms, and let $N$ be a bounded cochain complex of graded $(B,A)$-bimodules with termwise finite projective right $A$-terms; all differentials have internal degree zero (Bounded graded bimodule complexes and signed tensor totalization). Then for every Hochschild degree $j\ge 0$, every cochain degree $r$ and every internal degree there is a natural isomorphism $H^r(H{H}_j(A,\mathrm{Tot}(M{\otimes }_{B}N)))\cong H^r(H{H}_j(B,\mathrm{Tot}(N{\otimes }_{A}M))),$ where the termwise Hochschild complexes are those of Termwise Hochschild homology and iterated homology. The isomorphism is induced, termwise over the pairs $(M^i,N^l)$, by the double-bar rotation of Double bar comparison for cyclic bimodule tensor products, multiplied on the $(i,l)$-summand by the Koszul sign $(-1{)}^{il}$; the twisted map is a cochain isomorphism before cohomology is taken, and the reverse rotation gives its inverse. This is an iterated-homology statement in its own right and is not inferred from an abutment of any hyperhomology spectral sequence.