The Stacks project

Lemma 70.18.1. 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. 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 scheme theoretic support of $\mathcal{F}$.

Proof. 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 it suffices to show

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

is a numerical polynomial in $n_1, \ldots , n_ r$ of total degree at most $\dim (X)$. Let us say property $\mathcal{P}$ holds for the coherent $\mathcal{O}_ X$-module $\mathcal{F}$ if the above is true.

We will prove this statement by devissage, more precisely we will check conditions (1), (2), and (3) of Cohomology of Spaces, Lemma 67.14.6 are satisfied.

Verification of condition (1). Let

\[ 0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0 \]

be a short exact sequence of coherent sheaves on $X$. By Lemma 70.17.2 we have

\[ P_{\mathcal{F}_2}(n_1, \ldots , n_ r) = P_{\mathcal{F}_1}(n_1, \ldots , n_ r) + P_{\mathcal{F}_3}(n_1, \ldots , n_ r) \]

Then it is clear that if 2-out-of-3 of the sheaves $\mathcal{F}_ i$ have property $\mathcal{P}$, then so does the third.

Condition (2) follows because $P_{\mathcal{F}^{\oplus m}}(n_1, \ldots , n_ r) = mP_\mathcal {F}(n_1, \ldots , n_ r)$.

Proof of (3). Let $i : Z \to X$ be a reduced closed subspace with $|Z|$ irreducible. We have to find a coherent module $\mathcal{G}$ on $X$ whose support is $Z$ such that $\mathcal{P}$ holds for $\mathcal{G}$. We will give two constructions: one using Chow's lemma and one using a finite cover by a scheme.

Proof existence $\mathcal{G}$ using a finite cover by a scheme. Choose $\pi : Z' \to Z$ finite surjective where $Z'$ is a scheme, see Limits of Spaces, Proposition 68.16.1. Set $\mathcal{G} = i_*\pi _*\mathcal{O}_{Z'} = (i \circ \pi )_*\mathcal{O}_{Z'}$. Note that $Z'$ is proper over $k$ and that the support of $\mathcal{G}$ is $Y$ (details omitted). We have

\[ R(\pi \circ i)_*(\mathcal{O}_{Z'}) = \mathcal{G} \quad \text{and}\quad R(\pi \circ i)_*(\pi ^*i^*(\mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}) ) = \mathcal{G} \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r} \]

The first equality holds because $i \circ \pi $ is affine (Cohomology of Spaces, Lemma 67.8.2) and the second equality follows from the first and the projection formula (Cohomology on Sites, Lemma 21.48.1). Using Leray (Cohomology on Sites, Lemma 21.14.6) we obtain

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

By the case of schemes (Varieties, Lemma 33.44.1) this is a numerical polynomial in $n_1, \ldots , n_ r$ of degree at most $\dim (Z')$. We conclude because $\dim (Z') \leq \dim (Z) \leq \dim (X)$. The first inequality follows from Decent Spaces, Lemma 66.12.7.

Proof existence $\mathcal{G}$ using Chow's lemma. We apply Cohomology of Spaces, Lemma 67.18.1 to the morphism $Z \to \mathop{\mathrm{Spec}}(k)$. Thus we get a surjective proper morphism $f : Y \to Z$ over $\mathop{\mathrm{Spec}}(k)$ where $Y$ is a closed subscheme of $\mathbf{P}^ m_ k$ for some $m$. After replacing $Y$ by a closed subscheme we may assume that $Y$ is integral and $f : Y \to Z$ is an alteration, see Lemma 70.8.5. Denote $\mathcal{O}_ Y(n)$ the pullback of $\mathcal{O}_{\mathbf{P}^ m_ k}(n)$. Pick $n > 0$ such that $R^ pf_*\mathcal{O}_ Y(n) = 0$ for $p > 0$, see Cohomology of Spaces, Lemma 67.20.1. We claim that $\mathcal{G} = i_*f_*\mathcal{O}_ Y(n)$ satisfies $\mathcal{P}$. Namely, by the case of schemes (Varieties, Lemma 33.44.1) we know that

\[ (n_1, \ldots , n_ r) \longmapsto \chi (Y, \mathcal{O}_ Y(n) \otimes f^*i^*(\mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r})) \]

is a numerical polynomial in $n_1, \ldots , n_ r$ of total degree at most $\dim (Y)$. On the other hand, by the projection formula (Cohomology on Sites, Lemma 21.48.1)

\begin{align*} i_*Rf_*\left( \mathcal{O}_ Y(n) \otimes f^*i^*(\mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r})\right) & = i_*Rf_*\mathcal{O}_ Y(n) \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r} \\ & = \mathcal{G} \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r} \end{align*}

the last equality by our choice of $n$. By Leray (Cohomology on Sites, Lemma 21.14.6) we get

\[ \chi (Y, \mathcal{O}_ Y(n) \otimes f^*i^*(\mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r})) = P_\mathcal {G}(n_1, \ldots , n_ r) \]

and we conclude because $\dim (Y) \leq \dim (Z) \leq \dim (X)$. The first inequality holds by Morphisms of Spaces, Lemma 65.35.2 and the fact that $Y \to Z$ is an alteration (and hence the induced extension of residue fields in generic points is finite). $\square$


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