The Stacks project

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

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

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

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

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

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

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

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

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

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

by the projection formula (Cohomology, Section 20.54) 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. $\square$

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 33.45.1 the function

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

Proof. Immediate from Lemma 33.45.2 and the definitions. $\square$

Proof. The left hand side is computed using the coefficient of $n_1 \ldots n_ d$ in the function

The equality follows from Lemma 33.33.5 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 33.45.1 and the result is true. Assume $\dim (f(Z)) = d$. Let $\xi \in f(Z)$ be the generic point. By dimension theory (see Lemmas 33.20.3 and 33.20.4) the generic point of $Z$ is the unique point of $Z$ mapping to $\xi $. Then $f : Z \to f(Z)$ is finite over a nonempty open of $f(Z)$, see Morphisms, Lemma 29.51.1. Thus $\deg (f : Z \to f(Z))$ is defined and in fact it is equal to the length of the stalk of $f_*\mathcal{O}_ Z$ at $\xi $ over $\mathcal{O}_{X, \xi }$. Moreover, the stalk of $R^ if_*\mathcal{O}_ X$ at $\xi $ is zero for $i > 0$ because we just saw that $f|_ Z$ is finite in a neighbourhood of $\xi $ (so that Cohomology of Schemes, Lemma 30.9.9 gives the vanishing). 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 33.45.2. 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 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 $Q$. $\square$

Proof. We will prove this by induction on $d$. The case $d = 0$ follows from Lemma 33.33.3. Assume $d > 0$. By Lemma 33.45.6 we may assume that $Z$ is an integral closed subscheme. In fact, we may replace $X$ by $Z$ and $\mathcal{L}_ i$ by $\mathcal{L}_ i|_ Z$ to reduce to the case $Z = X$ is a proper variety of dimension $d$. By Lemma 33.45.5 we may replace $\mathcal{L}_1$ by a positive tensor power. Thus we may assume there exists a nonzero section $s \in \Gamma (X, \mathcal{L}_1)$ such that $X_ s$ is affine (here we use the definition of ample invertible sheaf, see Properties, Definition 28.26.1). Observe that $X$ is not affine because proper and affine implies finite (Morphisms, Lemma 29.44.11) which contradicts $d > 0$. It follows that $s$ has a nonempty vanishing scheme $Z(s) \subset X$. Since $X$ is a variety, $s$ is a regular section of $\mathcal{L}_1$, so $Z(s)$ is an effective Cartier divisor, thus $Z(s)$ has codimension $1$ in $X$, and hence $Z(s)$ has dimension $d - 1$ (here we use material from Divisors, Sections 31.13, 31.14, and 31.15 and from dimension theory as in Lemma 33.20.3). By Lemma 33.45.8 we have

By induction the right hand side is positive and the proof is complete. $\square$

Proof. The statement makes sense because $f^*\mathcal{L}$ is ample by Morphisms, Lemma 29.37.7. Having said this the result is a special case of Lemma 33.45.7. $\square$

Proof. This follows from the fact that the function $n \mapsto \chi (Z, \mathcal{L}|_ Z^{\otimes n})$ has degree $1$ and hence the leading coefficient is the difference of consecutive values. $\square$

Proof. This follows from the definitions, Lemma 33.45.9, and the vanishing of higher cohomology in Cohomology of Schemes, Lemma 30.17.1. $\square$