Lemma 72.18.2. 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 étale 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 69.14.6. 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 72.17.2) and additivity of lengths (Algebra, Lemma 10.52.3). This immediately implies properties (1) and (2) of Cohomology of Spaces, Lemma 69.14.6. 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 72.18.1.
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 69.12.7) 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 21.50.1) and Cohomology of Spaces, Lemma 69.8.3. 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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