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.

Cohomology after an even cochain shift

PDF p. 4: Property (4): shift by twice the weight

Statement

For a cochain complex $A$ , integers $i, j$ , and the standard shift convention $A [m]^{q} = A^{q + m}$ with differential multiplied by $(- 1)^{m}$ , there is a canonical isomorphism $H^{i} (A [- 2 j]) ≅ H^{i - 2 j} (A)$ .

Facts & Assumptions

Given: A cochain complex $A$ and integers $i, j$ .

Goal: Compute the cohomology of its indicated shift.

Proof

1.1givenalgebra

In degree $i$ the shifted group is $A^{i - 2 j}$ , and both adjacent differentials are the corresponding differentials of $A$ because $(- 1)^{- 2 j} = 1$ .

2.1step 1.1∎

The kernels and images defining the two displayed cohomology groups are consequently identical, giving the canonical isomorphism.

Used by · 1

Cycle-complex degree translation