LemmaStatement: AI-generatedProof: AI-generated

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.

Functorial finite coefficients for derived complexes

PDF p. 4: Definition 2.2

Statement

For $C \in D (Z)$ and a positive integer $n$ , put $C / n = C \otimes_{Z}^{L} Z / n$ . This construction and its cohomology groups are functorial in $C$ . For every integer $i$ there is a natural short exact sequence $0 \to H^{i} (C) / n H^{i} (C) \to H^{i} (C / n) \to H^{i + 1} (C) [n] \to 0,$ where $H^{i + 1} (C) [n]$ is the subgroup annihilated by $n$ .

Facts & Assumptions

[F1]

Derived finite coefficients and the Bockstein sequence identifies derived tensoring with $Z / n$ with the cone of multiplication by $n$ and supplies the displayed natural exact sequence for complexes.

[F2]

Cohomology is a functor on $D (Z)$ (Cohomology descends to the derived category).

Proof

Given: $C \in D (Z)$ and a positive integer $n$ .

Goal: Establish functoriality and the exact sequence.

1.1F1F2

Choose a complex representing $C$ . By [F1], its multiplication-by- $n$ cone represents $C \otimes^{L} Z / n$ . Derived tensor product is a functor on the derived category; applying [F2] gives functorial cohomology groups, independent of the representative.

2.1F1step 1.1∎

Apply the natural exact sequence of [F1] to the representative. Step 1.1 identifies its terms with the displayed derived-category terms and transports naturality to maps in $D (Z)$ . This proves the assertion.

Depends on

Used by · 4

Dependency tree · 3 levels

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