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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Cycle-complex degree translation

PDF p. 4: Property (4): higher Chow indexing

Statement

Let $k$ be a field, $X$ a smooth finite-type $k$ -scheme, and $j \geq 0$ . Regard Bloch's homological cycle complex $z^{j} (X, ∙)$ as a cochain complex by placing its degree- $n$ group in cochain degree $- n$ . Then $H^{i} (z^{j} (X, ∙) [- 2 j]) ≅ {C H}^{j} (X, 2 j - i)$ for every integer $i$ , where the right side is higher Chow homology and is zero in negative homological degrees.

Facts & Assumptions

Given: $k, X, j, i$ as stated.

Goal: Translate the shift into higher Chow indexing.

[F1]

Bloch's complex and higher Chow groups are the specified chain complex and its homology; the complex is zero in negative degrees. Bloch cycle complexes and higher Chow groups

[F2]

Chain-to-cochain reindexing identifies degree- $q$ cohomology with degree- $(- q)$ homology. Reindexing a chain complex cohomologically

[F3]

The even shift changes cohomological degree $i$ to $i - 2 j$ . Cohomology after an even cochain shift

Proof

1.1F3given

By [F3], degree- $i$ cohomology of the shifted complex is degree- $(i - 2 j)$ cohomology of the reindexed cycle complex.

2.1F1F2step 1.1∎

By [F2] this is homology in degree $2 j - i$ , which is the higher Chow group by [F1]. If that degree is negative it is zero by the same definition.

Depends on

Used by · 1

Smooth motivic comparison

Dependency tree · 3 levels

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