# stacks/stacks-project

Numerical intersection on proper algebraic spaces

We already had the basic lemma, but now we fleshed out the section a
bit more
 @@ -2148,8 +2148,51 @@ \section{Numerical intersections} mP_\mathcal{F}(n_1, \ldots, n_r)$. \medskip\noindent Proof of (3). Let$Z \subset X$be a reduced closed subspace with$|Z|$irreducible. We apply Proof of (3). Let$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}$. We will give two constructions: one using Chow's lemma and one using a finite cover by a scheme. \medskip\noindent Proof existence$\mathcal{G}$using a finite cover by a scheme. Choose$\pi : Z' \to Z$finite surjective where$Z'$is a scheme, see Limits of Spaces, Proposition \ref{spaces-limits-proposition-there-is-a-scheme-finite-over}. Set$\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 \ref{spaces-cohomology-lemma-affine-vanishing-higher-direct-images}) and the second equality follows from the first and the projection formula (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-projection-formula}). Using Leray (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-apply-Leray}) 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 \ref{varieties-lemma-numerical-polynomial-from-euler}) 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 \ref{decent-spaces-lemma-dimension-quasi-finite}. \medskip\noindent Proof existence$\mathcal{G}$using Chow's lemma. We apply Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-weak-chow} to the morphism$Z \to \Spec(k)$. Thus we get a surjective proper morphism$f : Y \to Z$over$\Spec(k)$@@ -2198,13 +2241,336 @@ \section{Numerical intersections} P_\mathcal{G}(n_1, \ldots, n_r) $$and we conclude because \dim(Y) \leq \dim(Z) \leq \dim(X). The first inequality hold by The first inequality holds by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-alteration-dimension-general} and the fact that Y \to Z is an alteration (and hence the induced extension of residue fields in generic points is finite). \end{proof} \noindent The following lemma roughly shows that the leading coefficient only depends on the length of the coherent module in the generic points of its support. \begin{lemma} \label{lemma-numerical-polynomial-leading-term} Let k be a field. Let X be a proper algebraic space over k. Let \mathcal{F} be a coherent \mathcal{O}_X-module. Let \mathcal{L}_1, \ldots, \mathcal{L}_r be invertible \mathcal{O}_X-modules. Let d = \dim(\text{Supp}(\mathcal{F})). Let Z_i \subset X be the irreducible components of \text{Supp}(\mathcal{F}) of dimension d. Let \overline{x}_i be a geometric generic point of Z_i and set m_i = \text{length}_{\mathcal{O}_{X, \overline{x}_i}} (\mathcal{F}_{\overline{x}_i}). Then$$ \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}) $$is a numerical polynomial in n_1, \ldots, n_r of total degree < d. \end{lemma} \begin{proof} We first prove a slightly weaker statement. Namely, say \dim(X) = N and let X_i \subset X be the irreducible components of dimension N. Let \overline{x}_i be a geometric generic point of X_i. The \'etale local ring \mathcal{O}_{X, \overline{x}_i} is Noetherian of dimension 0, hence for every coherent \mathcal{O}_X-module \mathcal{F} the length$$ 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}) $$is a numerical polynomial in n_1, \ldots, n_r of total degree < N. We will prove this using Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-property-higher-rank-cohomological-variant}. For any short exact sequence 0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0 we have E(\mathcal{F}) = E(\mathcal{F}') + E(\mathcal{F}''). This follows from additivity of Euler characteristics (Lemma \ref{lemma-euler-characteristic-additive}) and additivity of lengths (Algebra, Lemma \ref{algebra-lemma-length-additive}). This immediately implies properties (1) and (2) of Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-property-higher-rank-cohomological-variant}. Finally, property (3) holds because for \mathcal{G} = \mathcal{O}_Z for any Z \subset X irreducible reduced closed subspace. Namely, if Z = Z_{i_0} for some i_0, then m_i(\mathcal{G}) = \delta_{i_0i} and we conclude E(\mathcal{G}) = 0. If Z \not = Z_i for any i, then m_i(\mathcal{G}) = 0 for all i, \dim(Z) < N and we get the result from Lemma \ref{lemma-numerical-polynomial-from-euler}. \medskip\noindent Proof of the statement as in the lemma. Let Z \subset X be the scheme theoretic support of \mathcal{F}. Then \mathcal{F} = i_*\mathcal{G} for some coherent \mathcal{O}_Z-module \mathcal{G} (Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-coherent-support-closed}) and we have$$ \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}) $$by the projection formula (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-projection-formula}) and Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-relative-affine-cohomology}. Since |Z| = \text{Supp}(\mathcal{F}) we see that Z_i \subset Z for all i and we see that these are the irreducible components of Z of dimension d. We may and do think of \overline{x}_i as a geometric point of Z. The map i^\sharp : \mathcal{O}_X \to i_*\mathcal{O}_Z determines a surjection$$ \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}) $$and the result follows from the discussion in the first paragraph (applied with Z in stead of X). \end{proof} \begin{definition} \label{definition-intersection-number} Let k be a field. Let X be a proper algebraic space over k. Let i : Z \to X be a closed subspace of dimension d. Let \mathcal{L}_1, \ldots, \mathcal{L}_d be invertible \mathcal{O}_X-modules. We define the {\it intersection number} (\mathcal{L}_1 \cdots \mathcal{L}_d \cdot Z) as the coefficient of n_1 \ldots n_d in the numerical polynomial$$ \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) $$In the special case that \mathcal{L}_1 = \ldots = \mathcal{L}_d = \mathcal{L} we write (\mathcal{L}^d \cdot Z). \end{definition} \noindent The displayed equality in the definition follows from the projection formula (Cohomology, Section \ref{cohomology-section-projection-formula}) and Cohomology of Schemes, Lemma \ref{coherent-lemma-relative-affine-cohomology}. We prove a few lemmas for these intersection numbers. \begin{lemma} \label{lemma-intersection-number-integer} In the situation of Definition \ref{definition-intersection-number} the intersection number (\mathcal{L}_1 \cdots \mathcal{L}_d \cdot Z) is an integer. \end{lemma} \begin{proof} Any numerical polynomial of degree e in n_1, \ldots, n_d can be written uniquely as a \mathbf{Z}-linear combination of the functions {n_1 \choose k_1}{n_2 \choose k_2} \ldots {n_d \choose k_d} with k_1 + \ldots + k_d \leq e. Apply this with e = d. Left as an exercise. \end{proof} \begin{lemma} \label{lemma-intersection-number-additive} In the situation of Definition \ref{definition-intersection-number} the intersection number (\mathcal{L}_1 \cdots \mathcal{L}_d \cdot Z) is additive: if \mathcal{L}_i = \mathcal{L}_i' \otimes \mathcal{L}_i'', then we have$$ (\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) $$\end{lemma} \begin{proof} This is true because by Lemma \ref{lemma-numerical-polynomial-from-euler} the function$$ (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) $$is a numerical polynomial of total degree at most d in d + 1 variables. \end{proof} \begin{lemma} \label{lemma-intersection-number-in-terms-of-components} In the situation of Definition \ref{definition-intersection-number} let Z_i \subset Z be the irreducible components of dimension d. Let m_i = \text{length}_{\mathcal{O}_{X, \overline{x}_i}} (\mathcal{O}_{Z, \overline{x}_i}) where \overline{x}_i is a geometric generic point of Z_i. Then$$ (\mathcal{L}_1 \cdots \mathcal{L}_d \cdot Z) = \sum m_i(\mathcal{L}_1 \cdots \mathcal{L}_d \cdot Z_i) $$\end{lemma} \begin{proof} Immediate from Lemma \ref{lemma-numerical-polynomial-leading-term} and the definitions. \end{proof} \begin{lemma} \label{lemma-intersection-number-and-pullback} Let k be a field. Let f : Y \to X be a morphism of algebraic spaces proper over k. Let Z \subset Y be an integral closed subspace of dimension d and let \mathcal{L}_1, \ldots, \mathcal{L}_d be invertible \mathcal{O}_X-modules. Then$$ (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)) $$where \deg(Z \to f(Z)) is as in Definition \ref{definition-degree} or 0 if \dim(f(Z)) < d. \end{lemma} \begin{proof} In the statement f(Z) \subset X is the scheme theoretic image of f and it is also the reduced induced algebraic space structure on the closed subset f(|Z|) \subset X, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-scheme-theoretic-image-reduced}. Then Z and f(Z) are reduced, proper (hence decent) algebraic spaces over k, whence integral (Definition \ref{definition-integral-algebraic-space}). The left hand side is computed using the coefficient of n_1 \ldots n_d in the function$$ \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}) $$The equality follows from Lemma \ref{lemma-euler-characteristic-morphism} and the projection formula (Cohomology, Lemma \ref{cohomology-lemma-projection-formula}). If f(Z) has dimension < d, then the right hand side is a polynomial of total degree 0 (for example by Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-finite-higher-direct-image-zero}). Thus the terms \chi(X, R^if_*\mathcal{O}_Z \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_d^{\otimes n_d}) with i > 0 have total degree < d and$$ \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)}) $$modulo a polynomial of total degree < d by Lemma \ref{lemma-numerical-polynomial-leading-term}. The desired result follows. \end{proof} \begin{lemma} \label{lemma-numerical-intersection-effective-Cartier-divisor} Let k be a field. Let X be a proper algebraic space over k. Let Z \subset X be a closed subspace of dimension d. Let \mathcal{L}_1, \ldots, \mathcal{L}_d be invertible \mathcal{O}_X-modules. Assume there exists an effective Cartier divisor D \subset Z such that \mathcal{L}_1|_Z \cong \mathcal{O}_Z(D). Then$$ (\mathcal{L}_1 \cdots \mathcal{L}_d \cdot Z) = (\mathcal{L}_2 \cdots \mathcal{L}_d \cdot D) $$\end{lemma} \begin{proof} We may replace X by Z and \mathcal{L}_i by \mathcal{L}_i|_Z. Thus we may assume X = Z and \mathcal{L}_1 = \mathcal{O}_X(D). Then \mathcal{L}_1^{-1} is the ideal sheaf of D and we can consider the short exact sequence$$ 0 \to \mathcal{L}_1^{\otimes -1} \to \mathcal{O}_X \to \mathcal{O}_D \to 0 $$Set P(n_1, \ldots, n_d) = \chi(X, \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_d^{\otimes n_d}) and Q(n_1, \ldots, n_d) = \chi(D, \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_d^{\otimes n_d}|_D). We conclude from additivity (Lemma \ref{lemma-euler-characteristic-additive}) that$$ 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\$. \end{proof}