The Stacks project

70.17 Euler characteristics

In this section we prove some elementary properties of Euler characteristics of coherent sheaves on algebraic spaces proper over fields.

Definition 70.17.1. Let $k$ be a field. Let $X$ be a proper algebraic over $k$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. In this situation the Euler characteristic of $\mathcal{F}$ is the integer

\[ \chi (X, \mathcal{F}) = \sum \nolimits _ i (-1)^ i \dim _ k H^ i(X, \mathcal{F}). \]

For justification of the formula see below.

In the situation of the definition only a finite number of the vector spaces $H^ i(X, \mathcal{F})$ are nonzero (Cohomology of Spaces, Lemma 67.7.3) and each of these spaces is finite dimensional (Cohomology of Spaces, Lemma 67.20.3). Thus $\chi (X, \mathcal{F}) \in \mathbf{Z}$ is well defined. Observe that this definition depends on the field $k$ and not just on the pair $(X, \mathcal{F})$.

Lemma 70.17.2. 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. $\square$

Lemma 70.17.3. Let $k$ be a field. Let $f : Y \to X$ be a morphism of algebraic spaces proper over $k$. Let $\mathcal{G}$ be a coherent $\mathcal{O}_ Y$-module. Then

\[ \chi (Y, \mathcal{G}) = \sum (-1)^ i \chi (X, R^ if_*\mathcal{G}) \]

Proof. The formula makes sense: the sheaves $R^ if_*\mathcal{G}$ are coherent and only a finite number of them are nonzero, see Cohomology of Spaces, Lemmas 67.20.2 and 67.8.1. By Cohomology on Sites, Lemma 21.14.5 there is a spectral sequence with

\[ E_2^{p, q} = H^ p(X, R^ qf_*\mathcal{G}) \]

converging to $H^{p + q}(Y, \mathcal{G})$. By finiteness of cohomology on $X$ we see that only a finite number of $E_2^{p, q}$ are nonzero and each $E_2^{p, q}$ is a finite dimensional vector space. It follows that the same is true for $E_ r^{p, q}$ for $r \geq 2$ and that

\[ \sum (-1)^{p + q} \dim _ k E_ r^{p, q} \]

is independent of $r$. Since for $r$ large enough we have $E_ r^{p, q} = E_\infty ^{p, q}$ and since convergence means there is a filtration on $H^ n(Y, \mathcal{G})$ whose graded pieces are $E_\infty ^{p, q}$ with $p + 1 = n$ (this is the meaning of convergence of the spectral sequence), we conclude. $\square$


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