Lemma 33.45.2. Let $k$ be a field. Let $X$ be a proper scheme 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 $\xi _ i \in Z_ i$ be the generic point and set $m_ i = \text{length}_{\mathcal{O}_{X, \xi _ i}}(\mathcal{F}_{\xi _ i})$. 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.**
Consider pairs $(\xi , Z)$ where $Z \subset X$ is an integral closed subscheme of dimension $d$ and $\xi $ is its generic point. Then the finite $\mathcal{O}_{X, \xi }$-module $\mathcal{F}_\xi $ has support contained in $\{ \xi \} $ hence the length $m_ Z = \text{length}_{\mathcal{O}_{X, \xi }}(\mathcal{F}_\xi )$ is finite (Algebra, Lemma 10.62.3) and zero unless $Z = Z_ i$ for some $i$. Thus the expression of the lemma can be written as

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

where the sum is over integral closed subschemes $Z \subset X$ of dimension $d$. The assignment $\mathcal{F} \mapsto E(\mathcal{F})$ is additive in short exact sequences $0 \to \mathcal{F} \to \mathcal{F}' \to \mathcal{F}'' \to 0$ of coherent $\mathcal{O}_ X$-modules whose support has dimension $\leq d$. This follows from additivity of Euler characteristics (Lemma 33.33.2) and additivity of lengths (Algebra, Lemma 10.52.3). Let us apply Cohomology of Schemes, Lemma 30.12.3 to find a filtration

\[ 0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_ m = \mathcal{F} \]

by coherent subsheaves such that for each $j = 1, \ldots , m$ there exists an integral closed subscheme $V_ j \subset X$ and a nonzero sheaf of ideals $\mathcal{I}_ j \subset \mathcal{O}_{V_ j}$ such that

\[ \mathcal{F}_ j/\mathcal{F}_{j - 1} \cong (V_ j \to X)_* \mathcal{I}_ j \]

It follows that $V_ j \subset \text{Supp}(\mathcal{F})$ and hence $\dim (V_ j) \leq d$. By the additivity we remarked upon above it suffices to prove the result for each of the subquotients $\mathcal{F}_ j/\mathcal{F}_{j - 1}$. Thus it suffices to prove the result when $\mathcal{F} = (V \to X)_*\mathcal{I}$ where $V \subset X$ is an integral closed subscheme of dimension $\leq d$ and $\mathcal{I} \subset \mathcal{O}_ V$ is a nonzero coherent sheaf of ideals. If $\dim (V) < d$ and more generally for $\mathcal{F}$ whose support has dimension $< d$, then the first term in $E(\mathcal{F})$ has total degree $< d$ by Lemma 33.45.1 and the second term is zero. If $\dim (V) = d$, then we can use the short exact sequence

\[ 0 \to (V \to X)_*\mathcal{I} \to (V \to X)_*\mathcal{O}_ V \to (V \to X)_*(\mathcal{O}_ V/\mathcal{I}) \to 0 \]

The result holds for the middle sheaf because the only $Z$ occurring in the sum is $Z = V$ with $m_ Z = 1$ and because

\[ H^ i(X, ((V \to X)_*\mathcal{O}_ V) \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}) = H^ i(V, \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}|_ V) \]

by the projection formula (Cohomology, Section 20.52) and Cohomology of Schemes, Lemma 30.2.4; so in this case we actually have $E(\mathcal{F}) = 0$. The result holds for the sheaf on the right because its support has dimension $< d$. Thus the result holds for the sheaf on the left and the lemma is proved.
$\square$

## Comments (2)

Comment #6816 by Yuto Masamura on

Comment #6957 by Johan on