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.

Relative objects for empty and split pairs

PDF p. 7: Motivic purity at a smooth closed point

Statement

For a cochain complex $B$ , the relative cone of ${id}_{B}$ is contractible and that of $0 \to B$ is $B$ . For the split inclusion $A \to A \oplus B$ , $a \mapsto (a, 0)$ , its cone is quasi-isomorphic to $B$ . Dually the support fibers of ${id}_{C} : C \to C$ , $C \to 0$ and $C \oplus D \to C$ are respectively zero, $C$ and $D$ in the derived category.

Thus whenever a motive-valued or coefficient-complex-valued functor is separately known to take the empty scheme to zero, the pair with empty support has zero relative/support object and the pair with full support has the whole object. If finite disjoint unions are additionally known to become direct sums, the relative/support object of the split pair $X = U ∐ Z$ is the $Z$ summand. These are formal boundary cases and do not prove purity for a positive-codimension smooth pair.

Facts & Assumptions

Given: The displayed complexes and maps; the functorial applications assume exactly the stated empty-value and, for the split case, direct-sum property.

Goal: Compute these special cases without invoking any Gysin statement.

[F1]

The relative cone is $Cone (f)^{n} = B^{n} \oplus A^{n + 1}$ with the differential already established. Relative cofibers and quotient complexes

[F2]

A contravariant enhanced Hom sends such a cofiber to the corresponding restriction fiber whenever that enhancement is supplied. Contravariant Hom reverses relative cofiber triangles

Proof

1.1F1algebra

For $f = {id}_{B}$ , the contraction $h (b_{0}, b_{1}) = (0, b_{0})$ satisfies $d h + h d = id$ by the calculation in [F1]. For $f : 0 \to B$ , its cone formula has no second summand and is exactly $B$ . For the split inclusion, $f$ is termwise injective and its quotient is $B$ , so the quotient conclusion of [F1] gives the claimed quasi-isomorphism.

2.1F1F2step 1.1algebra

The explicit fiber of a map is its cone shifted by $- 1$ , up to the standard sign identification. Thus the fiber of an identity is zero and that of $C \to 0$ is $C$ . For the projection $C \oplus D \to C$ , its fiber is $D$ plus the contractible fiber of the identity of $C$ , as is seen directly by separating the three coordinates of its cone. Hence it is isomorphic to $D$ in the derived category. These results also agree with [F2] in its specified enhanced setting.

3.1givenstep 1.1step 2.1∎

Substitute the restriction or inclusion maps of the stated pairs only when the functor's empty-value and disjoint-union properties have been supplied. They become exactly the maps computed in steps 1.1 through 2.1. This proves the conditional applications and shows that none uses the positive-codimension purity theorem it is intended to accompany.

Depends on

Used by · 1

Gysin purity for a smooth closed pair

Dependency tree · 3 levels

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