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. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).