The Stacks project

Lemma 33.45.1. 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. The map

\[ (n_1, \ldots , n_ r) \longmapsto \chi (X, \mathcal{F} \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}) \]

is a numerical polynomial in $n_1, \ldots , n_ r$ of total degree at most the dimension of the support of $\mathcal{F}$.

Proof. We prove this by induction on $\dim (\text{Supp}(\mathcal{F}))$. If this number is zero, then the function is constant with value $\dim _ k \Gamma (X, \mathcal{F})$ by Lemma 33.33.3. Assume $\dim (\text{Supp}(\mathcal{F})) > 0$.

If $\mathcal{F}$ has embedded associated points, then we can consider the short exact sequence $0 \to \mathcal{K} \to \mathcal{F} \to \mathcal{F}' \to 0$ constructed in Divisors, Lemma 31.4.6. Since the dimension of the support of $\mathcal{K}$ is strictly less, the result holds for $\mathcal{K}$ by induction hypothesis and with strictly smaller total degree. By additivity of the Euler characteristic (Lemma 33.33.2) it suffices to prove the result for $\mathcal{F}'$. Thus we may assume $\mathcal{F}$ does not have embedded associated points.

If $i : Z \to X$ is a closed immersion and $\mathcal{F} = i_*\mathcal{G}$, then we see that the result for $X$, $\mathcal{F}$, $\mathcal{L}_1, \ldots , \mathcal{L}_ r$ is equivalent to the result for $Z$, $\mathcal{G}$, $i^*\mathcal{L}_1, \ldots , i^*\mathcal{L}_ r$ (since the cohomologies agree, see Cohomology of Schemes, Lemma 30.2.4). Applying Divisors, Lemma 31.4.7 we may assume that $X$ has no embedded components and $X = \text{Supp}(\mathcal{F})$.

Pick a regular meromorphic section $s$ of $\mathcal{L}_1$, see Divisors, Lemma 31.25.4. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the ideal of denominators of $s$ and consider the maps

\[ \mathcal{I}\mathcal{F} \to \mathcal{F},\quad \mathcal{I}\mathcal{F} \to \mathcal{F} \otimes \mathcal{L}_1 \]

of Divisors, Lemma 31.24.5. These are injective and have cokernels $\mathcal{Q}$, $\mathcal{Q}'$ supported on nowhere dense closed subschemes of $X = \text{Supp}(\mathcal{F})$. Tensoring with the invertible module $\mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}$ is exact, hence using additivity again we see that

\begin{align*} & \chi (X, \mathcal{F} \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}) - \chi (X, \mathcal{F} \otimes \mathcal{L}_1^{\otimes n_1 + 1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}) \\ & = \chi (\mathcal{Q} \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}) - \chi (\mathcal{Q}' \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}) \end{align*}

Thus we see that the function $P(n_1, \ldots , n_ r)$ of the lemma has the property that

\[ P(n_1 + 1, n_2, \ldots , n_ r) - P(n_1, \ldots , n_ r) \]

is a numerical polynomial of total degree $<$ the dimension of the support of $\mathcal{F}$. Of course by symmetry the same thing is true for

\[ P(n_1, \ldots , n_{i - 1}, n_ i + 1, n_{i + 1}, \ldots , n_ r) - P(n_1, \ldots , n_ r) \]

for any $i \in \{ 1, \ldots , r\} $. A simple arithmetic argument shows that $P$ is a numerical polynomial of total degree at most $\dim (\text{Supp}(\mathcal{F}))$. $\square$

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