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Additivity of the supported coherent Euler class

PDF p. 15: Supported degree-zero dévissage

Statement

Assume the Axiom of Choice and the fixed suitable universes of the supplied supported perfect category. Let $U, T, C_{T}, A_{T}$ be as in the preceding construction. The formula $χ_{T} (E) = \sum_{n \in Z} (- 1)^{n} [H^{n} (E)]$ defines a homomorphism from the triangle Grothendieck group of $h C_{T}$ to $G_{0} (A_{T})$ . The sum is finite. It sends a sheaf in degree zero to its sheaf class, and is compatible with forgetting support into the corresponding unsuppported groups on $U$ .

Facts & Assumptions

[A1]

The assumed The Axiom of Choice and fixed universes discharge the supported-heart construction in [F1] when using its actual bounded t-structure in steps 1.1–2.1. The finite alternating-sum argument requires no additional choice.

Given: The Axiom of Choice, the fixed suitable universes, and the supported perfect category with its bounded t-structure and coherent supported heart.

Goal: Verify the Euler assignment on every relation, not merely on terms of a locally free resolution.

[F1]

Inherited bounded t structure on supported perfect complexes supplies finite coherent supported cohomology and the usual long exact cohomology sequence of a triangle.

[F2]

Cofiber and triangle presentations of K zero identifies the triangle group with the cofiber presentation. The abelian Grothendieck group is defined by short exact relations.

Proof

1.1A1F1F2

Cohomology is invariant under equivalence, and boundedness in [F1] makes the displayed assignment a finite sum on every object. For a triangle $E \to F \to G \to E [1]$ , the long exact sequence of cohomology is finite after deleting zero terms and consists of objects in $A_{T}$ . In any finite exact sequence of an abelian category, breaking at images gives short exact sequences; summing their Grothendieck relations makes its alternating sum zero, since each image occurs twice with opposite signs. Applying this to the cohomology sequence gives $χ_{T} (F) = χ_{T} (E) + χ_{T} (G)$ .

2.1A1F1F2step 1.1∎

The assignment therefore descends to the triangle group. For $M [0]$ it is $[M]$ , because the only nonzero cohomology is $H^{0} = M$ . Forgetting support changes neither the underlying sheaves nor their cohomology; it sends each short exact relation to the same relation. Thus the Euler homomorphisms commute with forgetting support.

Depends on

Used by · 2

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