The Stacks project

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 69.12.7) and we have

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 it suffices to show

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 69.14.6 are satisfied.

Verification of condition (1). Let

be a short exact sequence of coherent sheaves on $X$. By Lemma 72.17.2 we have

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 70.16.1. 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

The first equality holds because $i \circ \pi $ is affine (Cohomology of Spaces, Lemma 69.8.2) and the second equality follows from the first and the projection formula (Cohomology on Sites, Lemma 21.50.1). Using Leray (Cohomology on Sites, Lemma 21.14.6) we obtain

By the case of schemes (Varieties, Lemma 33.45.1) 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 68.12.7.

Proof existence $\mathcal{G}$ using Chow's lemma. We apply Cohomology of Spaces, Lemma 69.18.1 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 72.8.5. 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 69.20.1. We claim that $\mathcal{G} = i_*f_*\mathcal{O}_ Y(n)$ satisfies $\mathcal{P}$. Namely, by the case of schemes (Varieties, Lemma 33.45.1) we know that

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 21.50.1)

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

and we conclude because $\dim (Y) \leq \dim (Z) \leq \dim (X)$. The first inequality holds by Morphisms of Spaces, Lemma 67.35.2 and the fact that $Y \to Z$ is an alteration (and hence the induced extension of residue fields in generic points is finite). $\square$

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

is an integer $\geq 0$. We claim that

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

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

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

Thus we see that the expression in the lemma is equal to

and the result follows from the discussion in the first paragraph (applied with $Z$ in stead of $X$). $\square$

Proof. Any numerical polynomial of degree $e$ in $n_1, \ldots , n_ d$ can be written uniquely as a $\mathbf{Z}$-linear combination of the functions ${n_1 \choose k_1}{n_2 \choose k_2} \ldots {n_ d \choose k_ d}$ with $k_1 + \ldots + k_ d \leq e$. Apply this with $e = d$. Left as an exercise. $\square$

Proof. This is true because by Lemma 72.18.1 the function

is a numerical polynomial of total degree at most $d$ in $d + 1$ variables. $\square$

Proof. Immediate from Lemma 72.18.2 and the definitions. $\square$

Proof. In the statement $f(Z) \subset X$ is the scheme theoretic image of $f$ and it is also the reduced induced algebraic space structure on the closed subset $f(|Z|) \subset X$, see Morphisms of Spaces, Lemma 67.16.4. Then $Z$ and $f(Z)$ are reduced, proper (hence decent) algebraic spaces over $k$, whence integral (Definition 72.4.1). The left hand side is computed using the coefficient of $n_1 \ldots n_ d$ in the function

The equality follows from Lemma 72.17.3 and the projection formula (Cohomology, Lemma 20.54.2). If $f(Z)$ has dimension $< d$, then the right hand side is a polynomial of total degree $< d$ by Lemma 72.18.1 and the result is true. Assume $\dim (f(Z)) = d$. Then by dimension theory (Lemma 72.15.2) we find that the equivalent conditions (1) – (5) of Lemma 72.5.1 hold. Thus $\deg (Z \to f(Z))$ is well defined. By the already used Lemma 72.5.1 we find $f : Z \to f(Z)$ is finite over a nonempty open $V$ of $f(Z)$; after possibly shrinking $V$ we may assume $V$ is a scheme. Let $\xi \in V$ be the generic point. Thus $\deg (f : Z \to f(Z))$ the length of the stalk of $f_*\mathcal{O}_ Z$ at $\xi $ over $\mathcal{O}_{X, \xi }$ and the stalk of $R^ if_*\mathcal{O}_ X$ at $\xi $ is zero for $i > 0$ (for example by Cohomology of Spaces, Lemma 69.4.1). Thus the terms $\chi (X, R^ if_*\mathcal{O}_ Z \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ d^{\otimes n_ d})$ with $i > 0$ have total degree $< d$ and

modulo a polynomial of total degree $< d$ by Lemma 72.18.2. The desired result follows. $\square$

Proof. We may replace $X$ by $Z$ and $\mathcal{L}_ i$ by $\mathcal{L}_ i|_ Z$. Thus we may assume $X = Z$ and $\mathcal{L}_1 = \mathcal{O}_ X(D)$. Then $\mathcal{L}_1^{-1}$ is the ideal sheaf of $D$ and we can consider the short exact sequence

Set $P(n_1, \ldots , n_ d) = \chi (X, \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ d^{\otimes n_ d})$ and $Q(n_1, \ldots , n_ d) = \chi (D, \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ d^{\otimes n_ d}|_ D)$. We conclude from additivity (Lemma 72.17.2) that

Because the total degree of $P$ is at most $d$, we see that the coefficient of $n_1 \ldots n_ d$ in $P$ is equal to the coefficient of $n_2 \ldots n_ d$ in $Q$. $\square$