Derived cyclicity of Hochschild hyperhomology

Every $M^i$ is flat as a right $B$-module and every $N^l$ is flat as a right $A$-module because the terms are projective. The bounded signed totals therefore compute the derived tensor products under [F5]; [F3] gives their induced outer bimodule structures, internal gradings, and finite diagonals.

Put $X=\mathrm{Tot}(M{\otimes }_{B}N)$ and form the standard total complexes $Q_A=\mathrm{Tot}(\mathrm{Bar}(A){\otimes }_{A}M{\otimes }_B\mathrm{Bar}(B){\otimes }_{B}N)$ and $Q_B$ with $A,B$ and $M,N$ exchanged, using the reindexed bar degrees $-p,-q$. The associator and signed-totalization rules [F3, F7] group these as $U_A{\otimes }_B{U}_B$ and $U_B{\otimes }_A{U}_A$, and [F6] ensures the augmentation maps tensor to cochain maps. The map $Q_A\to P_A:=\mathrm{Bar}(A){\otimes }_{A}X$ induced by the $B$-bar augmentation is a quasi-isomorphism: for each $N^l$, [F9] is a left $B$-bar resolution, tensoring it with the bounded complex $M$ preserves that quasi-isomorphism by [F4] and termwise right $B$-flatness, and the outer bounded-above right $A$-flat bar complex preserves it again by [F4]. The augmentation $P_A\to X$ is a quasi-isomorphism by the outer bar contraction. Each total degree of $Q_A$ is a finite sum, since $i,l$ range over bounded intervals and $p+q=i+l-n$. Each term of $Q_A$ is projective as a right $A^e$-module by applying the explicit module-level projectivity proof of [F2] to each pair $(M^i,N^l)$ and taking the finite direct sum on that diagonal; each term of $P_A$ is projective by the same proof and the finite projectivity of each $M^i{\otimes }_B{N}^l$. Thus $Q_A,P_A$ are bounded-above complexes of projective right $A^e$-modules, both quasi-isomorphic to $X$. The symmetric statements hold for $Q_B$ and $P_B$.

The quasi-isomorphism $Q_A\to P_A$ is a chain-homotopy equivalence. Its cone is bounded above, acyclic, and termwise projective. Starting at the highest nonzero degree, acyclicity makes the preceding differential onto the top term; that surjection splits because the top term is projective. Its kernel is a direct summand of the preceding projective term and is projective by [F10]. Repeating downward gives splittings of all cycle epimorphisms; [F12] supplies the recursive choices and [F11] makes the cone contractible. Since the differentials and cycle epimorphisms preserve internal degree, each splitting may be taken to have degree zero: extract the degree-zero homogeneous component of an underlying module splitting, which remains a splitting because the epimorphism has degree zero. Thus the contraction and induced comparison after coinvariants preserve internal degree. Hence after applying the additive coinvariant functor, ${\mathrm{coInv}}_A(Q_A)\to {\mathrm{coInv}}_A(P_A)$ remains a chain-homotopy equivalence; likewise on the $B$-side. To identify these models with hyperhomology, use the common triple total $\mathrm{Bar}(A{)}_{\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}{A}^e}{\otimes }_{A^e}{P}_A$, viewing the bimodule complex $P_A$ as a left $A^e$-complex. Its maps to $\mathrm{Bar}(A){\otimes }_{A^e}X$ and to $A{\otimes }_{A^e}{P}_A={\mathrm{coInv}}_A(P_A)$ are quasi-isomorphisms by [F4]: the bar resolution is bounded above and flat as a right $A^e$-complex, while $P_A$ is bounded above and termwise flat as a left $A^e$-complex by [F14]. Each cone is a bounded-above acyclic complex of $k$-vector spaces. Under AC, every term is projective and every epimorphism onto a cycle splits; after reindexing, [F11] therefore makes both cones contractible. These contractions can be taken internally degree zero by the homogeneous-splitting argument above. Thus both triple-model maps are chain-homotopy equivalences, including after coinvariants, rather than only quasi-isomorphisms. The first model is the reindexed-bar model of [F1], hence computes ${\mathrm{H}\mathrm{H}}^{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r}}(A,X)$. The same argument identifies ${\mathrm{coInv}}_B(Q_B)$ with ${\mathrm{H}\mathrm{H}}^{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r}}(B,\mathrm{Tot}(N{\otimes }_{A}M))$.

Group $U_A=\mathrm{Tot}(\mathrm{Bar}(A){\otimes }_{A}M)$ and $U_B=\mathrm{Tot}(\mathrm{Bar}(B){\otimes }_{B}N)$, so $Q_A=\mathrm{Tot}(U_A{\otimes }_B{U}_B)$ and $Q_B=\mathrm{Tot}(U_B{\otimes }_A{U}_A)$. For homogeneous blocks of cochain degrees $r=i-p$ and $s=l-q$, the map $[u\otimes v]\mapsto (-1{)}^{rs}[v\otimes u]$ is well defined on coinvariants: a $B$-balance relation maps to the two classes identified by $B$-coinvariants, and an $A$-coinvariant relation maps to the two classes identified by $A$-balance. It is invertible by the reverse switch. For the standard tensor total differential, the first-block component commutes because $(-1{)}^{(r+1)s}=(-1{)}^{rs}(-1{)}^s$, and the second-block component commutes because $(-1{)}^r(-1{)}^{r(s+1)}=(-1{)}^{rs}$. This grouped calculation covers the $A$-bar and $M$ differentials in the first block and the $B$-bar and $N$ differentials in the second block. The reverse switch has the same sign, so the composite multiplies by $(-1{)}^{rs+sr}=1$.

By 2.2 the middle rotation is a cochain isomorphism ${\mathrm{coInv}}_A(Q_A)\to {\mathrm{coInv}}_B(Q_B)$ whose reverse is its inverse. The map $Q_A\to P_A$ induces a quasi-isomorphism after coinvariants by 2.1, and the two maps from the triple total in 2.1 are quasi-isomorphisms; therefore these maps identify the cohomology of each coinvariant model with the bar hyperhomology model. The corresponding maps on cohomology are isomorphisms, and composing them with the isomorphism induced by the middle rotation gives the claimed cyclic isomorphism. Using the inverse cohomology identifications and the reverse rotation gives its inverse; equivalently, chosen chain-homotopy inverse comparisons induce those inverse cohomology maps by [F13]. All comparisons and homotopies preserve internal degree by 2.1, and the constructions are natural up to homotopy in maps of the bounded bimodule complexes because bar augmentation, tensor totalization, coinvariants, and rotation are natural. The termwise projectivity assumptions are used only on the indicated right sides, and no arbitrary-projective-target derived-functor assertion enters.