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Termwise Hochschild homology respects bimodule chain homotopies

Statement

Let $k$ be a field, let $A$ be a unital associative $k$ -algebra, and let $F = (F^{i}, d_{F}^{i})$ and $G = (G^{i}, d_{G}^{i})$ be bounded cochain complexes of $k$ -central $A$ -bimodules with differentials of internal degree zero (Bounded graded bimodule complexes and signed tensor totalization). Let $f, g : F \to G$ be bimodule-linear cochain maps of cochain degree zero and let $h : F \to G$ be a bimodule-linear cochain homotopy of cochain degree $- 1$ with $f - g = d_{G} h + h d_{F}$ on $F^{∙}$ (A chain homotopy).

Then for every $j \geq 0$ the induced maps $H H_{j} (A, f), H H_{j} (A, g) : H H_{j} (A, F^{∙}) \to H H_{j} (A, G^{∙})$ on the termwise Hochschild complexes of Termwise Hochschild homology and iterated homology are cochain-homotopic; the homotopy is induced by $H H_{j} (A, h) : H H_{j} (A, F^{i}) \to H H_{j} (A, G^{i - 1})$ , and hence $H H_{j} (A, f)$ and $H H_{j} (A, g)$ induce the same map $H^{i} (H H_{j} (A, F)) \to H^{i} (H H_{j} (A, G))$ for every $i$ (Cohomology object of a cochain complex). Consequently a bimodule chain-homotopy equivalence $F \to G$ induces isomorphisms $H^{i} (H H_{j} (A, F)) \to H^{i} (H H_{j} (A, G))$ for all $i, j$ . Everything here is choice-free, and no invariance of the termwise groups under arbitrary quasi-isomorphisms is asserted.

Facts & Assumptions

Given: a field $k$ , a unital associative $k$ -algebra $A$ , bounded cochain complexes $F, G$ of $k$ -central $A$ -bimodules with internal-degree-zero differentials, bimodule-linear cochain maps $f, g : F \to G$ of cochain degree zero, and a bimodule-linear cochain homotopy $h : f ≃ g$ of cochain degree $- 1$ with $f - g = d_{G} h + h d_{F}$ .

[F1]

The Hochschild chain complex of a $k$ -central bimodule $M$ has $C_{j} (A, M) = M \otimes_{k} A^{\otimes_{k} j}$ with boundary $b_{j}$ the alternating sum of faces; the termwise complex of a bounded complex $F$ in Hochschild degree $j$ is $H H_{j} (A, F^{∙})$ with differentials $H H_{j} (A, d_{F}^{i})$ induced by the bimodule maps $d_{F}^{i}$ , and its cohomology is $H^{i} (H H_{j} (A, F^{∙}))$ (Hochschild chains and Hochschild homology with coefficients, Termwise Hochschild homology and iterated homology).

[F2]

$H H_{j} (A, F^{i}) = H_{j} (C_{∙} (A, F^{i}))$ is the homology of the Hochschild chain complex, and a chain map $u : C_{∙} \to D_{∙}$ induces a well-defined map $H_{j} (u)$ (A chain map induces a well-defined map on homology).

[F3]

A chain homotopy $s$ between chain maps of chain complexes satisfies $f_{n} - g_{n} = d_{n + 1} s_{n} + s_{n - 1} d_{n}$ in each degree; homotopic chain maps induce the same map on homology (A chain homotopy).

[F4]

A map of $k$ -central $A$ -bimodules commutes with every Hochschild face, since the faces multiply the coefficient by algebra elements on either side; hence a bimodule map $u : M \to N$ induces a chain map $C_{∙} (A, u) : C_{∙} (A, M) \to C_{∙} (A, N)$ natural in the bimodule (Hochschild chains and Hochschild homology with coefficients, Enveloping algebra and the bimodule–module dictionary).

[F5]

For fixed $j$ and composable bimodule maps the assignment $u \mapsto C_{j} (A, u)$ is additive: $C_{j} (A, u + v) = C_{j} (A, u) + C_{j} (A, v)$ , because the tensor product of a map with an identity is linear in the map; it also preserves identities and composition, so $H H_{j} (A, -)$ is additive on maps (Hochschild chains and Hochschild homology with coefficients).

[F6]

Chain-homotopic maps induce the same map on homology; after reindexing cochain degree $i$ as homological degree $- i$ , homotopic cochain maps induce the same map on cohomology (Chain-homotopic maps induce the same map on homology).

Proof

technique · direct

1.1F1F2F4givenalgebra

Fix $j \geq 0$ . For each $i$ the bimodule map $d_{F}^{i} : F^{i} \to F^{i + 1}$ commutes with every Hochschild face by [F4], so it induces a chain map $C_{∙} (A, d_{F}^{i}) : C_{∙} (A, F^{i}) \to C_{∙} (A, F^{i + 1})$ , and hence a map $H H_{j} (A, d_{F}^{i})$ on homology by [F2]. The same applies to $d_{G}$ , $f$ , $g$ and $h$ ; the homotopy $h$ has cochain degree $- 1$ , so $C_{∙} (A, h)$ is a degree- $(- 1)$ family of maps of Hochschild complexes.

1.2F1F2F3F5givenalgebra

Because $C_{∙} (A, -)$ is additive on maps by [F5], applying it to the homotopy identity $f - g = d_{G} h + h d_{F}$ gives exactly $C_{∙} (A, f) - C_{∙} (A, g) = C_{∙} (A, d_{G}) C_{∙} (A, h) + C_{∙} (A, h) C_{∙} (A, d_{F})$ . Passing to homology with [F2], this is the displayed homotopy identity $H H_{j} (A, f) - H H_{j} (A, g) = H H_{j} (A, d_{G}) H H_{j} (A, h) + H H_{j} (A, h) H H_{j} (A, d_{F})$ in Hochschild degree $j$ , valid in every cochain degree; the family $H H_{j} (A, h)$ has cochain degree $- 1$ and is a cochain homotopy of the termwise complexes by [F3].

2.1F2F6step 1.1step 1.2givenalgebra

Applying [F6] in each cochain degree $i$ , the cochain-homotopic maps $H H_{j} (A, f)$ and $H H_{j} (A, g)$ induce the same map $H^{i} (H H_{j} (A, F)) \to H^{i} (H H_{j} (A, G))$ on the cohomology of the termwise complex, and the induced map depends only on the cochain-homotopy class of the map of coefficient complexes. Everything in the argument is a computation of maps of $k$ -vector spaces, so no choice is used, and no statement about arbitrary quasi-isomorphisms is made. In the graded case the statement is ungraded unless $f, g, h$ are also internal-degree-zero; under that extra condition the induced homotopy and maps preserve internal degree.

3.1F2F3step 2.1givenalgebra∎

Now let $f : F \to G$ be a bimodule chain-homotopy equivalence, with bimodule-linear homotopy inverse $g : G \to F$ of cochain degree zero and two bimodule-linear homotopies $f g ≃ {i d}_{G}$ and $g f ≃ {i d}_{F}$ of cochain degree $- 1$ . By 2.1 and functoriality, the induced maps on $H^{i} (H H_{j} (A, -))$ compose to the identity in both orders, so they are inverse isomorphisms for every $i, j$ .

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Sources

Beliakova–Putyra–Wehrli, Quantum Link Homology via Trace Functor I, §3.8.6, printed p.38 (standard reference, not scraped)
Charles A. Weibel, An Introduction to Homological Algebra, Chapter 9, §9.1, printed pp.300–304 (standard reference, not scraped)
Charles A. Weibel, An Introduction to Homological Algebra, Chapter 1, §1.4, Lemma 1.4.5, printed p.17 (standard reference, not scraped)