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33.33 Euler characteristics

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

Definition 33.33.1. Let $k$ be a field. Let $X$ be a proper scheme 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 Schemes, Lemma 30.4.5) and each of these spaces is finite dimensional (Cohomology of Schemes, Lemma 30.19.2). 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 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. $\square$

Lemma 33.33.3. Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{F}$ be a coherent sheaf with $\dim (\text{Supp}(\mathcal{F})) \leq 0$. Then

  1. $\mathcal{F}$ is generated by global sections,

  2. $H^0(X, \mathcal{F}) = \bigoplus _{x \in \text{Supp}(\mathcal{F})} \mathcal{F}_ x$,

  3. $H^ i(X, \mathcal{F}) = 0$ for $i > 0$,

  4. $\chi (X, \mathcal{F}) = \dim _ k H^0(X, \mathcal{F})$, and

  5. $\chi (X, \mathcal{F} \otimes \mathcal{E}) = n\chi (X, \mathcal{F})$ for every locally free module $\mathcal{E}$ of rank $n$.

Proof. By Cohomology of Schemes, Lemma 30.9.7 we see that $\mathcal{F} = i_*\mathcal{G}$ where $i : Z \to X$ is the inclusion of the scheme theoretic support of $\mathcal{F}$ and where $\mathcal{G}$ is a coherent $\mathcal{O}_ Z$-module. By definition of the scheme theoretic support the underlying topological space of $Z$ is $\text{Supp}(\mathcal{F})$. Since the dimension of $Z$ is $0$, we see $Z$ is affine (Properties, Lemma 28.10.5). Hence $\mathcal{G}$ is globally generated and the higher cohomology groups of $\mathcal{G}$ are zero (Cohomology of Schemes, Lemma 30.2.2). In fact, by Lemma 33.20.2 the scheme $Z$ is a finite disjoint union of spectra of local Artinian rings. Thus correspondingly $H^0(Z, \mathcal{G}) = \bigoplus _{z \in Z} \mathcal{G}_ z$. The cohomologies of $\mathcal{F}$ and $\mathcal{G}$ agree by Cohomology of Schemes, Lemma 30.2.4. Thus $H^ i(X, \mathcal{F}) = 0$ for $i > 0$ and $H^0(X, \mathcal{F}) = H^0(Z, \mathcal{G})$. In particular we have (3) is true. For $z \in Z$ corresponding to $x \in \text{Supp}(\mathcal{F})$ we have $\mathcal{G}_ z = (i_*\mathcal{G})_ x = \mathcal{F}_ x$. We conclude that (2) holds. Of course (2) implies (1). We have (4) by definition of the Euler characteristic $\chi (X, \mathcal{F})$ and (3). By the projection formula (Cohomology, Lemma 20.52.2) we have

\[ i_*(\mathcal{G} \otimes i^*\mathcal{E}) = \mathcal{F} \otimes \mathcal{E}. \]

Since $Z$ has dimension $0$ the locally free sheaf $i^*\mathcal{E}$ is isomorphic to $\mathcal{O}_ Z^{\oplus n}$ and arguing as above we see that (5) holds. $\square$

Lemma 33.33.4. Let $k'/k$ be an extension of fields. Let $X$ be a proper scheme over $k$. Let $\mathcal{F}$ be a coherent sheaf on $X$. Let $\mathcal{F}'$ be the pullback of $\mathcal{F}$ to $X_{k'}$. Then $\chi (X, \mathcal{F}) = \chi (X', \mathcal{F}')$.

Proof. This is true because

\[ H^ i(X_{k'}, \mathcal{F}') = H^ i(X, \mathcal{F}) \otimes _ k k' \]

by flat base change, see Cohomology of Schemes, Lemma 30.5.2. $\square$

Lemma 33.33.5. Let $k$ be a field. Let $f : Y \to X$ be a morphism of proper schemes 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 Schemes, Proposition 30.19.1 and Lemma 30.4.5. By Cohomology, Lemma 20.13.4 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 $p + q = n$ (this is the meaning of convergence of the spectral sequence), we conclude. Compare also with the more general Homology, Lemma 12.24.12. $\square$

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