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.

Constant integer coefficients and exact derived evaluation

PDF p. 4: Weight zero

Statement

Assume the Axiom of Choice and fixed compatible site and universe size bounds for the constructions below. On the preceding site let ${\underline{Z}}_{c d h} = a_{c d h} (U \mapsto Z)$ be the ordinary abelian sheafification of the constant presheaf, placed in degree zero. Define $R Γ_{c d h}^{F} (X, Z)$ to be derived evaluation of this sheaf at $X$ on that specified site. Its cohomology is denoted $H_{c d h, F}^{i} (X, Z)$ . It is not the constant complex $Z$ evaluated without sheafification. All negative cohomology groups of this derived value are zero.

Facts & Assumptions

[A1]

The Axiom of Choice and fixed compatible site and universe bounds are retained when applying Derived global sections on a scheme site: step 1.1 uses its K-injective derived-evaluation construction, and step 2.1 uses its nonnegative injective-resolution computation. The Axiom of Choice

Given: The hypotheses in the statement.

Goal: Establish the displayed construction or conclusion with its actual maps.

[F1]

Under [A1], Derived global sections on a scheme site applies to the explicitly chosen small site; The source qcqs cdh site supplies this site.

Proof

1.1A1F1construct

The preceding small site permits ordinary abelian sheafification, so the displayed sheaf and its degree-zero placement are defined. Apply the general-site construction of Derived global sections on a scheme site with this site and coefficient sheaf. Its completed supplier gives a K-injective derived evaluation, independent of choices.

2.1A1F1step 1.1∎

The same supplier computes the value of an ordinary sheaf by a nonnegative injective resolution. Evaluating that resolution has zero terms in negative degrees, hence zero cohomology there. This proves the negative-degree statement. The use of a resolved sheaf rather than the raw constant presheaf is essential to the definition and does not require a site comparison.

Depends on

Used by · 2

Dependency tree · 3 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.