The Stacks project

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

In the filtration with for some integral closed subschemes and sheaves of ideals in the proof, we may assume that each is contained in , as I wrote in Lemma 30.12.3. Then we can immediately see that (although it is not mentioned in the proof). Moreover we do not have to think about all the integral closed subschemes in the summation at the beginning of the proof.

Comment #6957 by on

OK, thanks and fixed here. The structure of the proof is the write down some universal formula at the beginning of the proof and then work with that. So I didn't change it.

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