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Numerical intersection on proper algebraic spaces

We already had the basic lemma, but now we fleshed out the section a
bit more
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aisejohan committed Dec 6, 2017
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@@ -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})
$$
The equality follows from Lemma \ref{lemma-euler-characteristic-morphism}
and the projection formula
(Cohomology, Lemma \ref{cohomology-lemma-projection-formula}).
If $f(Z)$ has dimension $< d$, then the right hand side
is a polynomial of total degree $<d$ by
Lemma \ref{lemma-numerical-polynomial-from-euler}
and the result is true. Assume $\dim(f(Z)) = d$. Then
by dimension theory (Lemma \ref{lemma-alteration-dimension})
we find that the equivalent conditions (1) -- (5) of
Lemma \ref{lemma-finite-degree} hold. Thus
$\deg(Z \to f(Z))$ is well defined.
By the already used Lemma \ref{lemma-finite-degree}
we find $f : Z \to f(Z)$ is finite over a nonempty open
$V$ of $f(Z)$; after possibly shrinking $V$ we may assume
$V$ is a scheme. Let $\xi \in V$ be the generic point.
Thus $\deg(f : Z \to f(Z))$ the length of the stalk of
$f_*\mathcal{O}_Z$ at $\xi$ over $\mathcal{O}_{X, \xi}$
and the stalk of $R^if_*\mathcal{O}_X$ at $\xi$ is zero for $i > 0$
(for example by Cohomology of Spaces, Lemma
\ref{spaces-cohomology-lemma-finite-higher-direct-image-zero}).
Thus the terms $\chi(X, R^if_*\mathcal{O}_Z \otimes
\mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes
\mathcal{L}_d^{\otimes n_d})$ with $i > 0$ have total
degree $< d$ and
$$
\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}

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