Lemma 33.33.2. Let $k$ be a field. Let $X$ be a proper scheme 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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