The Stacks project

Lemma 70.18.2. Let $k$ be a field. Let $X$ be a proper algebraic space over $k$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Let $\mathcal{L}_1, \ldots , \mathcal{L}_ r$ be invertible $\mathcal{O}_ X$-modules. Let $d = \dim (\text{Supp}(\mathcal{F}))$. Let $Z_ i \subset X$ be the irreducible components of $\text{Supp}(\mathcal{F})$ of dimension $d$. Let $\overline{x}_ i$ be a geometric generic point of $Z_ i$ and set $m_ i = \text{length}_{\mathcal{O}_{X, \overline{x}_ i}} (\mathcal{F}_{\overline{x}_ i})$. Then

\[ \chi (X, \mathcal{F} \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}) - \sum \nolimits _ i m_ i\ \chi (Z_ i, \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}|_{Z_ i}) \]

is a numerical polynomial in $n_1, \ldots , n_ r$ of total degree $< d$.

Proof. We first prove a slightly weaker statement. Namely, say $\dim (X) = N$ and let $X_ i \subset X$ be the irreducible components of dimension $N$. Let $\overline{x}_ i$ be a geometric generic point of $X_ i$. The ├ętale local ring $\mathcal{O}_{X, \overline{x}_ i}$ is Noetherian of dimension $0$, hence for every coherent $\mathcal{O}_ X$-module $\mathcal{F}$ the length

\[ m_ i(\mathcal{F}) = \text{length}_{\mathcal{O}_{X, \overline{x}_ i}} (\mathcal{F}_{\overline{x}_ i}) \]

is an integer $\geq 0$. We claim that

\[ E(\mathcal{F}) = \chi (X, \mathcal{F} \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}) - \sum \nolimits _ i m_ i(\mathcal{F})\ \chi (Z_ i, \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}|_{Z_ i}) \]

is a numerical polynomial in $n_1, \ldots , n_ r$ of total degree $< N$. We will prove this using Cohomology of Spaces, Lemma 67.14.6. For any short exact sequence $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0$ we have $E(\mathcal{F}) = E(\mathcal{F}') + E(\mathcal{F}'')$. This follows from additivity of Euler characteristics (Lemma 70.17.2) and additivity of lengths (Algebra, Lemma 10.51.3). This immediately implies properties (1) and (2) of Cohomology of Spaces, Lemma 67.14.6. Finally, property (3) holds because for $\mathcal{G} = \mathcal{O}_ Z$ for any $Z \subset X$ irreducible reduced closed subspace. Namely, if $Z = Z_{i_0}$ for some $i_0$, then $m_ i(\mathcal{G}) = \delta _{i_0i}$ and we conclude $E(\mathcal{G}) = 0$. If $Z \not= Z_ i$ for any $i$, then $m_ i(\mathcal{G}) = 0$ for all $i$, $\dim (Z) < N$ and we get the result from Lemma 70.18.1.

Proof of the statement as in the lemma. Let $Z \subset X$ be the scheme theoretic support of $\mathcal{F}$. Then $\mathcal{F} = i_*\mathcal{G}$ for some coherent $\mathcal{O}_ Z$-module $\mathcal{G}$ (Cohomology of Spaces, Lemma 67.12.7) and we have

\[ \chi (X, \mathcal{F} \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}) = \chi (Z, \mathcal{G} \otimes i^*\mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes i^*\mathcal{L}_ r^{\otimes n_ r}) \]

by the projection formula (Cohomology on Sites, Lemma 21.48.1) and Cohomology of Spaces, Lemma 67.8.3. Since $|Z| = \text{Supp}(\mathcal{F})$ we see that $Z_ i \subset Z$ for all $i$ and we see that these are the irreducible components of $Z$ of dimension $d$. We may and do think of $\overline{x}_ i$ as a geometric point of $Z$. The map $i^\sharp : \mathcal{O}_ X \to i_*\mathcal{O}_ Z$ determines a surjection

\[ \mathcal{O}_{X, \overline{x}_ i} \to \mathcal{O}_{Z, \overline{x}_ i} \]

Via this map we have an isomorphism of modules $\mathcal{G}_{\overline{x}_ i} = \mathcal{F}_{\overline{x}_ i}$ as $\mathcal{F} = i_*\mathcal{G}$. This implies that

\[ m_ i = \text{length}_{\mathcal{O}_{X, \overline{x}_ i}} (\mathcal{F}_{\overline{x}_ i}) = \text{length}_{\mathcal{O}_{Z, \overline{x}_ i}} (\mathcal{G}_{\overline{x}_ i}) \]

Thus we see that the expression in the lemma is equal to

\[ \chi (Z, \mathcal{G} \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}) - \sum \nolimits _ i m_ i\ \chi (Z_ i, \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}|_{Z_ i}) \]

and the result follows from the discussion in the first paragraph (applied with $Z$ in stead of $X$). $\square$


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