The Stacks project

70.18 Numerical intersections

In this section we play around with the Euler characteristic of coherent sheaves on proper algebraic spaces to obtain numerical intersection numbers for invertible modules. Our main tool will be the following lemma.

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). $\square$

The following lemma roughly shows that the leading coefficient only depends on the length of the coherent module in the generic points of its support.

Lemma 70.18.2. 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. 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 $\overline{x}_ i$ be a geometric generic point of $Z_ i$ and set $m_ i = \text{length}_{\mathcal{O}_{X, \overline{x}_ i}} (\mathcal{F}_{\overline{x}_ i})$. 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. 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

\[ m_ i(\mathcal{F}) = \text{length}_{\mathcal{O}_{X, \overline{x}_ i}} (\mathcal{F}_{\overline{x}_ i}) \]

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

\[ E(\mathcal{F}) = \chi (X, \mathcal{F} \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}) - \sum \nolimits _ i m_ i(\mathcal{F})\ \chi (Z_ i, \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}|_{Z_ i}) \]

is a numerical polynomial in $n_1, \ldots , n_ r$ of total degree $< N$. We will prove this using Cohomology of Spaces, Lemma 67.14.6. 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 70.17.2) and additivity of lengths (Algebra, Lemma 10.51.3). This immediately implies properties (1) and (2) of Cohomology of Spaces, Lemma 67.14.6. 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 70.18.1.

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 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 $Z_ i \subset Z$ 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

\[ \mathcal{O}_{X, \overline{x}_ i} \to \mathcal{O}_{Z, \overline{x}_ i} \]

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

\[ m_ i = \text{length}_{\mathcal{O}_{X, \overline{x}_ i}} (\mathcal{F}_{\overline{x}_ i}) = \text{length}_{\mathcal{O}_{Z, \overline{x}_ i}} (\mathcal{G}_{\overline{x}_ i}) \]

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

\[ \chi (Z, \mathcal{G} \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}) - \sum \nolimits _ i m_ i\ \chi (Z_ i, \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}|_{Z_ i}) \]

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

Definition 70.18.3. Let $k$ be a field. Let $X$ be a proper algebraic space over $k$. Let $i : Z \to X$ be a closed subspace of dimension $d$. Let $\mathcal{L}_1, \ldots , \mathcal{L}_ d$ be invertible $\mathcal{O}_ X$-modules. We define the intersection number $(\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot Z)$ as the coefficient of $n_1 \ldots n_ d$ in the numerical polynomial

\[ \chi (X, i_*\mathcal{O}_ Z \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ d^{\otimes n_ d}) = \chi (Z, \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ d^{\otimes n_ d}|_ Z) \]

In the special case that $\mathcal{L}_1 = \ldots = \mathcal{L}_ d = \mathcal{L}$ we write $(\mathcal{L}^ d \cdot Z)$.

The displayed equality in the definition follows from the projection formula (Cohomology, Section 20.49) and Cohomology of Schemes, Lemma 30.2.4. We prove a few lemmas for these intersection numbers.

Lemma 70.18.4. In the situation of Definition 70.18.3 the intersection number $(\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot Z)$ is an integer.

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. $\square$

Lemma 70.18.5. In the situation of Definition 70.18.3 the intersection number $(\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot Z)$ is additive: if $\mathcal{L}_ i = \mathcal{L}_ i' \otimes \mathcal{L}_ i''$, then we have

\[ (\mathcal{L}_1 \cdots \mathcal{L}_ i \cdots \mathcal{L}_ d \cdot Z) = (\mathcal{L}_1 \cdots \mathcal{L}_ i' \cdots \mathcal{L}_ d \cdot Z) + (\mathcal{L}_1 \cdots \mathcal{L}_ i'' \cdots \mathcal{L}_ d \cdot Z) \]

Proof. This is true because by Lemma 70.18.1 the function

\[ (n_1, \ldots , n_{i - 1}, n_ i', n_ i'', n_{i + 1}, \ldots , n_ d) \mapsto \chi (Z, \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes (\mathcal{L}_ i')^{\otimes n_ i'} \otimes (\mathcal{L}_ i'')^{\otimes n_ i''} \otimes \ldots \otimes \mathcal{L}_ d^{\otimes n_ d}|_ Z) \]

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

Lemma 70.18.6. In the situation of Definition 70.18.3 let $Z_ i \subset Z$ be the irreducible components of dimension $d$. Let $m_ i = \text{length}_{\mathcal{O}_{X, \overline{x}_ i}} (\mathcal{O}_{Z, \overline{x}_ i})$ where $\overline{x}_ i$ is a geometric generic point of $Z_ i$. Then

\[ (\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot Z) = \sum m_ i(\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot Z_ i) \]

Proof. Immediate from Lemma 70.18.2 and the definitions. $\square$

Lemma 70.18.7. Let $k$ be a field. Let $f : Y \to X$ be a morphism of algebraic spaces proper over $k$. Let $Z \subset Y$ be an integral closed subspace of dimension $d$ and let $\mathcal{L}_1, \ldots , \mathcal{L}_ d$ be invertible $\mathcal{O}_ X$-modules. Then

\[ (f^*\mathcal{L}_1 \cdots f^*\mathcal{L}_ d \cdot Z) = \deg (f|_ Z : Z \to f(Z)) (\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot f(Z)) \]

where $\deg (Z \to f(Z))$ is as in Definition 70.5.2 or $0$ if $\dim (f(Z)) < d$.

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 65.16.4. Then $Z$ and $f(Z)$ are reduced, proper (hence decent) algebraic spaces over $k$, whence integral (Definition 70.4.1). The left hand side is computed using the coefficient of $n_1 \ldots n_ d$ in the function

\[ \chi (Y, \mathcal{O}_ Z \otimes f^*\mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes f^*\mathcal{L}_ d^{\otimes n_ d}) = \sum (-1)^ i \chi (X, R^ if_*\mathcal{O}_ Z \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ d^{\otimes n_ d}) \]

The equality follows from Lemma 70.17.3 and the projection formula (Cohomology, Lemma 20.49.2). If $f(Z)$ has dimension $< d$, then the right hand side is a polynomial of total degree $< d$ by Lemma 70.18.1 and the result is true. Assume $\dim (f(Z)) = d$. Then by dimension theory (Lemma 70.15.2) we find that the equivalent conditions (1) – (5) of Lemma 70.5.1 hold. Thus $\deg (Z \to f(Z))$ is well defined. By the already used Lemma 70.5.1 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 67.4.1). 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

\[ \chi (X, f_*\mathcal{O}_ Z \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ d^{\otimes n_ d}) = \deg (f : Z \to f(Z)) \chi (f(Z), \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ d^{\otimes n_ d}|_{f(Z)}) \]

modulo a polynomial of total degree $< d$ by Lemma 70.18.2. The desired result follows. $\square$

Lemma 70.18.8. Let $k$ be a field. Let $X$ be a proper algebraic space over $k$. Let $Z \subset X$ be a closed subspace of dimension $d$. Let $\mathcal{L}_1, \ldots , \mathcal{L}_ d$ be invertible $\mathcal{O}_ X$-modules. Assume there exists an effective Cartier divisor $D \subset Z$ such that $\mathcal{L}_1|_ Z \cong \mathcal{O}_ Z(D)$. Then

\[ (\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot Z) = (\mathcal{L}_2 \cdots \mathcal{L}_ d \cdot D) \]

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

\[ 0 \to \mathcal{L}_1^{\otimes -1} \to \mathcal{O}_ X \to \mathcal{O}_ D \to 0 \]

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 70.17.2) that

\[ P(n_1, \ldots , n_ d) - P(n_1 - 1, n_2, \ldots , n_ d) = Q(n_1, \ldots , n_ d) \]

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 $Q$. $\square$


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