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.

Restriction modulo its kernel

PDF p. 13: Supported Chow group

Statement

With $j^{*} : z^{r} (U, ∙) \to z^{r} (U ∖ T, ∙)$ as above and $K = \ker j^{*}$ , the induced chain map $z^{r} (U, ∙) / K \to z^{r} (U ∖ T, ∙)$ is degreewise injective and has the same cokernel complex as $j^{*}$ .

Statement references

The kernel of restriction is the supported subcomplex

Facts & Assumptions

Given: A field $k$ , a smooth finite-type $k$ -scheme $U$ , closed $T \subset U$ , and $r \geq 0$ .

Goal: Identify the quotient map and its cokernel.

[F1]

$K$ is the kernel subcomplex of the restriction chain map. The kernel of restriction is the supported subcomplex

Proof

1.1F1algebra

In each degree, the induced map from the quotient by the kernel is injective and has image equal to the image of the original homomorphism.

2.1F1step 1.1∎

Because the original map and the quotient projection commute with differentials, the induced map is a chain map. Its cokernel is the target modulo the same image, with the same induced differential.

Depends on

The kernel of restriction is the supported subcomplex

Used by · 1

Cycle complexes with fixed support

Dependency tree · 3 levels

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