Lemma 72.17.2 (0DN2)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 72: Algebraic Spaces over Fields
Section 72.17: Euler characteristics
Lemma 72.17.2 (cite)

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Lemma 72.17.2. Let $k$ be a field. Let $X$ be a proper algebraic space over $k$ . Let $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ be a short exact sequence of coherent modules on $X$ . Then

$\chi (X, \mathcal{F}_2) = \chi (X, \mathcal{F}_1) + \chi (X, \mathcal{F}_3)$

Proof. Consider the long exact sequence of cohomology

$0 \to H^0(X, \mathcal{F}_1) \to H^0(X, \mathcal{F}_2) \to H^0(X, \mathcal{F}_3) \to H^1(X, \mathcal{F}_1) \to \ldots$

associated to the short exact sequence of the lemma. The rank-nullity theorem in linear algebra shows that

$0 = \dim H^0(X, \mathcal{F}_1) - \dim H^0(X, \mathcal{F}_2) + \dim H^0(X, \mathcal{F}_3) - \dim H^1(X, \mathcal{F}_1) + \ldots$

This immediately implies the lemma. $\square$

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