LemmaStatement: AI-generatedProof: AI-generated

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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.

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Derived finite coefficients preserve mapping cones

PDF p. 6: Finite-coefficient comparison

Statement

For a morphism $f : A \to B$ in $D (Z)$ and a positive integer $n$ , there is an isomorphism $cofib (f \otimes_{Z}^{L} Z / n) ≅ cofib (f) \otimes_{Z}^{L} Z / n .$ Here cofibers in the triangulated category $D (Z)$ are considered up to isomorphism. More precisely, fix the complex $P = [Z \overset{n}{\to} Z]$ in degrees $- 1$ and $0$ . For every cochain map $g : A^{'} \to B^{'}$ representing a derived morphism, there is a chain isomorphism $Cone (g) \otimes P ≅ Cone (g \otimes 1_{P}),$ natural for strictly commutative squares of cochain maps. Naturality refers to this chain-level construction with fixed $n$ and $P$ .

Facts & Assumptions

Given: A morphism $f$ in $D (Z)$ and $n > 0$ .

Goal: Identify the two indicated cofibers.

[F1]

The two-term complex $P = [Z \overset{n}{\to} Z]$ in degrees $- 1$ and $0$ is K-flat and resolves $Z / n$ . Thus tensoring with $P$ computes derived finite coefficients. Derived finite coefficients and the Bockstein sequence

Proof

1.1F1givenconstruct

Represent $f$ by a cochain map $g : A^{'} \to B^{'}$ after quasi-isomorphic replacements if necessary. Use the convention $Cone (g)^{k} = B^{' k} \oplus A^{' k + 1}$ , with differential $(b, a) \mapsto (d_{B^{'}} b + g (a), - d_{A^{'}} a)$ . Since $P$ has only two nonzero terms, each degree of the tensor total complex is a finite direct sum. Splitting its summands gives a degreewise isomorphism $Cone (g) \otimes P \to Cone (g \otimes 1_{P})$ , sending $b \otimes p$ to the target summand and $a \otimes p$ to the shifted source summand without changing either tensor.

2.1F1step 1.1algebra∎

For homogeneous $a \in A^{' i + 1}$ and $p \in P^{j}$ , the differential on the shifted source tensor has components $g (a) \otimes p$ , $- d_{A^{'}} a \otimes p$ and $(- 1)^{i} a \otimes d_{P} p$ on both sides. On $b \otimes p$ , both differentials are the ordinary tensor differential. Thus the degreewise identification is a chain isomorphism. For any strictly commutative square from $g$ to another cochain map, the induced cone maps act separately on the source and target summands; hence they commute with this identification. This proves the claimed chain-level naturality. By K-flatness of $P$ , tensoring with $P$ computes derived finite coefficients and preserves the quasi-isomorphisms used for the representatives. The two chain complexes therefore give the asserted isomorphism of derived objects, with cofibers considered up to isomorphism.

Depends on

Derived finite coefficients and the Bockstein sequence

Used by · 1

Rational cofiber implies finite-coefficient equivalence

Dependency tree · 3 levels

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