TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedaudited 2026-10-02

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Termwise Hochschild spectral sequence of a bounded bimodule complex

Statement

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 (Bounded graded bimodule complexes and signed tensor totalization). Write $T^{∙} (A, F)$ for the Hochschild hyperhomology complex of Hochschild hyperhomology of a bounded bimodule complex and, for every $p$ , $F^{p} T^{n} (A, F) : = ⨁_{\begin{array}{c} i \geq p \\ i - j = n \\ j \geq 0 \end{array}} C_{j} (A, F^{i}) .$ Then the decreasing filtration $F^{∙} T^{∙}$ by subcomplexes is finite, exhaustive and separated in every total degree, and its spectral sequence is a natural cohomological spectral sequence $E_{1}^{i, - j} = H H_{j} (A, F^{i}), E_{2}^{i, - j} = H^{i} (H H_{j} (A, F^{∙})),$ with differentials $d_{r} : E_{r}^{i, - j} \to E_{r}^{i + r, - j - r + 1}$ (Cohomological spectral sequence), abutting to the finite image filtration on the hyperhomology ${H H}^{h y p e r, i - j} (A, F)$ (Abutment to a filtered object): $E_{\infty}^{i, - j} ≅ {gr}^{i} {H H}^{h y p e r, i - j} (A, F) .$ The second page is the iterated homology of Termwise Hochschild homology and iterated homology. When $A$ and $F$ carry internal gradings and the differentials have internal degree zero, the spectral sequence is compatible with the internal grading: each term $E_{r}^{i, - j}$ splits as a direct sum over internal degrees, all differentials preserve the internal degree, and the abutment isomorphism is internal-degree preserving. The result asserts nothing about the vanishing of higher differentials: it is not claimed that the spectral sequence degenerates at any page, and extension problems in passing from $E_{\infty}$ to ${H H}^{h y p e r}$ are not excluded.

Facts & Assumptions

Given: a field $k$ , a unital associative $k$ -algebra $A$ , and a bounded cochain complex $F$ of $k$ -central $A$ -bimodules with internal-degree-zero differentials.

[F1]

The hyperhomology complex has $T^{n} (A, F) = ⨁_{i - j = n, j \geq 0} C_{j} (A, F^{i})$ with $D = d_{F} + (- 1)^{i} b$ on the $(i, j)$ summand; the subspaces $F^{p} T^{n} : = ⨁_{i \geq p, i - j = n} C_{j} (A, F^{i})$ form a decreasing filtration by subcomplexes that is finite at each total degree because $F$ is bounded, and ${H H}^{h y p e r, n} (A, F) = H^{n} (T^{∙} (A, F))$ (Hochschild hyperhomology of a bounded bimodule complex).

[F2]

For fixed $j$ the maps $H H_{j} (A, d_{F}^{i}) : H H_{j} (A, F^{i}) \to H H_{j} (A, F^{i + 1})$ induced by the coefficient differentials make $(H H_{j} (A, F^{∙}), H H_{j} (A, d_{F}^{∙}))$ a cochain complex, and its cohomology $H^{i} (H H_{j} (A, F^{∙}))$ is the iterated Hochschild homology (Termwise Hochschild homology and iterated homology).

[F3]

The cochain complex $K = K^{∙}$ with a decreasing filtration by subcomplexes produces a cohomological spectral sequence with $E_{0}^{p, q} = {gr}^{p} K^{p + q}$ , $E_{1}^{p, q} ≅ H^{p + q} ({gr}^{p} K)$ , differentials $d_{r} : E_{r}^{p, q} \to E_{r}^{p + r, q - r + 1}$ , and, when the filtration is degreewise finite, abutment $E_{\infty}^{p, q} ≅ {gr}^{p} H^{p + q} (K)$ with the decreasing image filtration; the construction is natural in the filtered complex (The cohomological filtered complex construction).

[F4]

If the filtration on every $C_{n}$ of a chain complex is finite, the spectral sequence of the filtered complex stabilizes pointwise and naturally abuts to $H_{n} (C)$ with the image filtration, which is finite, exhaustive and separated; no uniform filtration bound in $n$ is needed (Bounded filtered complex spectral sequence abuts to filtered homology).

[F5]

A filtered chain map induces a morphism of the associated spectral sequences, respecting identities, composition and the differentials (A filtered chain map induces a morphism of spectral sequences).

[F6]

A cochain complex over an abelian category is a graded family of objects with degree-raising differentials squaring to zero; a filtration by subcomplexes restricts the differential to each filtration level (Cochain complex in an abelian category).

[F7]

A cohomological spectral sequence is a family of bigraded objects $E_{r}^{p, q}$ with differentials $d_{r}$ of bidegree $(r, 1 - r)$ satisfying $d_{r}^{2} = 0$ and $E_{r + 1} ≅ H (E_{r}, d_{r})$ (Cohomological spectral sequence).

Proof

technique · direct

1.1F1givenalgebra

By [F1] the filtration $F^{p} T^{∙}$ is a decreasing filtration by subcomplexes: $D$ preserves $F^{p} T^{∙}$ , since $d_{F}$ raises $i$ and $b$ preserves $i$ . It is exhaustive and separated in each total degree, and finite there because for $i - j = n$ with $0 \leq j$ and $F$ supported on a finite interval $[a, b]$ only the indices $i \in [a, b] \cap [n, \infty)$ contribute, a finite set; the quotient $F^{p} T^{n} / F^{p + 1} T^{n}$ retains exactly the summand with $i = p$ . Thus ${gr}^{p} T^{n} = C_{p - n} (A, F^{p})$ for $p \geq n$ , and it is zero for $p < n$ (equivalently, $C_{j} = 0$ for $j < 0$ ). In particular $E_{0}^{p, q} = C_{- q} (A, F^{p})$ , zero for $q > 0$ .

1.2F1F3F6F7givenalgebra

Because the filtration is decreasing and degreewise finite, [F3] (with [F4] for the chain-level abutment statement, read in the degreewise-finite form supplied by [F3]) gives a cohomological spectral sequence in the sense of [F7] whose $E_{0}$ -page is the associated graded of $T^{∙} (A, F)$ and whose $E_{1}$ -page is the cohomology of the associated graded. On ${gr}^{p} T^{n}$ the differential induced by $D = d_{F} + (- 1)^{i} b$ is the summand- $b$ term $(- 1)^{p} b$ ; the $d_{F}$ -part raises the filtration index and therefore contributes zero to the associated-graded differential. Hence $H^{p + q} ({gr}^{p} T^{∙}) = H^{n} (C_{∙} (A, F^{p}), (- 1)^{p} b) = H H_{j} (A, F^{p})$ with $j = - q$ , because multiplying a complex by the invertible constant $(- 1)^{p}$ does not change homology. Thus $E_{1}^{i, - j} = H H_{j} (A, F^{i})$ .

1.3F1F2F3givenalgebra

The $E_{1}$ -differential is induced by the part of $D$ that raises the filtration index, namely $d_{F}$ ; on the summand ${gr}^{i} T^{∙}$ it is the map on homology $H H_{j} (A, d_{F}^{i}) : H H_{j} (A, F^{i}) \to H H_{j} (A, F^{i + 1})$ induced by the coefficient differential, with the standard sign convention of $D$ . By [F2] these maps make the iterated complex, and $E_{2}^{i, - j} = H^{i} (H H_{j} (A, F^{∙}))$ is its cohomology.

1.4F1F3F4givenalgebra

Convergence is the degreewise-finite abutment of [F3]: the filtration on $T^{n} (A, F)$ is finite, so the spectral sequence stabilizes pointwise and $E_{\infty}^{i, - j} ≅ {gr}^{i} {H H}^{h y p e r, i - j} (A, F)$ with the decreasing image filtration; [F4] supplies the corresponding chain-level statement, both being the same theorem transported across the reindexing $C_{n} = T^{- n}$ , $F_{p} C_{n} = F^{- p} T^{- n}$ stated in [F3]. The filtration is finite, exhaustive and separated, so no convergence condition beyond degreewise finiteness is used.

2.1F1F3F5step 1.1step 1.4givenalgebra∎

Naturality: a degree-zero bimodule map of bounded coefficient complexes $u : F \to F^{'}$ commutes with both $d_{F}$ and the Hochschild boundary, so it preserves the filtrations of 1.1 and is a filtered chain map; by [F5] it induces a morphism of the associated spectral sequences, compatible with all pages and with the abutment. If $A$ and the coefficient complexes are internally graded and all maps have internal degree zero, then $T^{∙} (A, F)$ , the filtration levels $F^{p} T^{n}$ , and every differential are internal-degree homogeneous; hence each $E_{r}^{i, - j}$ inherits a direct-sum decomposition by internal degree, all $d_{r}$ preserve it, and the abutment isomorphism of 1.4 is internal-degree preserving. No claim of degeneration or of vanishing of higher differentials is made, and extension problems in reconstructing ${H H}^{h y p e r}$ from $E_{\infty}$ are not excluded.

Depends on

Used by

Total and separate Hochschild degrees for a two-term complex Example

Dependency tree · two levels

32 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

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 5, §5.4–5.6, printed pp.133–143 (standard reference, not scraped)
Mikhail Khovanov, Triply-graded link homology and Hochschild homology of Soergel bimodules, printed pp.5–7 (standard reference, not scraped)