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Cohomology of an integer shift

PDF p. 4: Shift [2j] in equation (2.1)

Statement

For a cochain complex $C$ of abelian groups and integers $q, k$ , the convention $C [k]^{q} = C^{q + k}$ and $d_{C [k]}^{q} = (- 1)^{k} d_{C}^{q + k}$ gives a natural isomorphism $H^{q} (C [k]) ≅ H^{q + k} (C)$ .

Facts & Assumptions

Given: $C$ , $q$ , and $k$ as stated.

Goal: Compare the indicated cohomology groups.

[F1]

Derived category of an abelian category specifies the cochain shift and its sign.

Proof

1.1F1given

Multiplication of a homomorphism by $(- 1)^{k}$ changes neither its kernel nor its image. Hence the cycles and boundaries of $C [k]$ in degree $q$ are respectively the cycles and boundaries of $C$ in degree $q + k$ .

2.1step 1.1∎

The identity on representatives induces the displayed quotient isomorphism, and it commutes with maps of complexes because those maps commute with the differentials.

Depends on

Derived category of an abelian category

Used by · 1

Complex shifts and spectral homotopy indices

Dependency tree · 3 levels

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