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The hereditary Euler form is additive on the Grothendieck group

Statement

Let $A$ be a finite-field hereditary Hall category. For objects $M, N$ , define $⟨ [M], [N] ⟩ = \dim_{F_{q}} Hom (M, N) - \dim_{F_{q}} {YExt}^{1} (M, N) .$ This form is additive in short exact sequences in either variable, so it descends to a well-defined bilinear form on $K_{0} (A)$ .

Facts & Assumptions

Given: A finite-field hereditary Hall category and a short exact sequence $0 \to A \to B \to C \to 0$ .

[F1]

For every pair of objects, Hom and Yoneda ${Ext}^{1}$ are finite-dimensional $F_{q}$ -vector spaces and Yoneda ${Ext}^{2}$ vanishes (Finite-field hereditary Hall categories).

[F2]

The covariant and contravariant low-degree Yoneda exact sequences are exact through the ${Ext}^{2}$ boundary (Low-degree exact sequences for Yoneda Ext in both variables).

[F3]

$K_{0}$ imposes $[B] = [A] + [C]$ for every short exact sequence (Grothendieck group of an essentially small abelian category).

Proof

technique · direct

1.1F1F2given

Fix $X$ . The covariant sequence from [F2] ends $Hom (X, A) \to Hom (X, B) \to Hom (X, C) \to {YExt}^{1} (X, A) \to {YExt}^{1} (X, B) \to {YExt}^{1} (X, C) \to 0,$ because its next term is ${YExt}^{2} (X, A) = 0$ by [F1]. It is a finite exact sequence of finite-dimensional vector spaces.

1.2F1F2algebra

Apply the contravariant sequence in [F2]. Its finite exact segment is $Hom (C, X) \to Hom (B, X) \to Hom (A, X) \to {YExt}^{1} (C, X) \to {YExt}^{1} (B, X) \to {YExt}^{1} (A, X) \to 0.$ Alternating dimensions give $⟨ [B], [X] ⟩ = ⟨ [A], [X] ⟩ + ⟨ [C], [X] ⟩$ .

2.1step 1.1algebra

The alternating-dimension identity for the finite exact sequence of the previous step gives $\dim Hom (X, B) - \dim {YExt}^{1} (X, B) = (\dim Hom (X, A) - \dim {YExt}^{1} (X, A)) + (\dim Hom (X, C) - \dim {YExt}^{1} (X, C)) .$ Thus $⟨ [X], [B] ⟩ = ⟨ [X], [A] ⟩ + ⟨ [X], [C] ⟩$ .

3.1F3step 2.1step 1.2algebra∎

The two additivity identities in steps 2.1 and 1.2 respect every defining relation of $K_{0}$ in [F3]. The universal property of the quotient therefore extends the formula uniquely and additively in both variables to $K_{0} (A) \times K_{0} (A)$ . All entries are integers, proving the statement.

Depends on

Used by

Dependency tree · two levels

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Sources

Olivier Schiffmann, Lectures on Hall Algebras, arXiv:math/0611617v2 (standard reference, not scraped)
Barry Mitchell, Theory of Categories, Chapter VII, §5 (standard reference, not scraped)