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aisejohan committed Jul 22, 2019
1 parent dd3e808 commit f4540e2054a8b2c959dc9f616aa1895ef5a85afa
Showing with 17 additions and 56 deletions.
  1. +1 −1 chow.tex
  2. +16 −55 weil.tex
@@ -4572,7 +4572,7 @@ \section{Gysin homomorphisms}
sheaf and a global section $s \in \Gamma(X, \mathcal{L})$.
Let $D = Z(s)$ be the zero scheme of $s$, and
denote $i : D \to X$ the closed immersion.
We define, for every integer $k$, a (refined) {\it Gysin homomorphism}
We define, for every integer $k$, a {\it Gysin homomorphism}
$$
i^* : Z_{k + 1}(X) \to \CH_k(D).
$$
@@ -52,56 +52,17 @@ \section{Conventions and notation}
$R$-module structure on $A$.

\medskip\noindent
Let $k$ be a field and let $X$ be a scheme of finite type over $k$.
The Chow groups $\CH_k(X)$ of $X$ have been defined in
Let $k$ be a field. Let $X$ be a scheme of finite type over $k$.
The Chow groups $\CH_k(X)$ of $X$ of cycles of dimension $k$
modulo rational equivalence have been defined in
Chow Homology, Definition \ref{chow-definition-rational-equivalence}.
Given a proper morphism $f : X \to Y$ of schemes of finite
type over $k$ there is a pushforward map $f_* : \CH_k(X) \to \CH_k(Y)$,
see Chow Homology, Section \ref{chow-section-proper-pushforward} and
Lemma \ref{chow-lemma-proper-pushforward-rational-equivalence}.
If $X$ is smooth over $k$ and equidimensional of dimension $d$, then
we have
$$
\CH^i(X) = \CH_{d - i}(X)
$$
see Chow Homology, Section \ref{chow-lemma-identify-chow-for-smooth}
and recall that the isomorphism sends $c \in \CH^i(X)$ to
$c \cap [X]_d \in \CH_{d - i}(X)$.
If $X$ smooth over $k$ and quasi-compact, then we can write canonically
$$
X = X_0 \amalg X_1 \amalg X_2 \amalg \ldots \amalg X_n
$$
into open and closed subschemes $X_d$ which are
equidimensional of dimension $d$. Set $[X] = \sum [X_d]_d$
as an element of $\CH_*(X)$. Since
$\CH_k(X) = \prod \CH_k(X_d)$ and $\CH^i(X) = \prod \CH^i(X_d)$
the map $c \mapsto c \cap [X]$ still defines an isomorphism
$$
\CH^*(X) = \CH_*(X)
$$
but it is no longer compatible with gradings, namely,
$$
\CH^i(X) = \bigoplus\nolimits_d \CH_{d - i}(X_d)
$$
The reader may use this as an alternative definition of $\CH^i(X)$.
There is an intersection product
$(\alpha, \beta) \mapsto \alpha \cdot \beta$ on $\CH_*(X)$
with the property that it sends $\CH^i(X) \times \CH^j(X)$ into $\CH^{i + j}(X)$,
see Chow Homology, Section \ref{chow-section-intersection-product}.
If $f : Y \to X$ is a morphism of schemes smooth over $k$, then
there is a pullback map
$$
f^* : \CH^i(X) \to \CH^i(Y),\quad
\alpha \mapsto f^*\alpha
$$
which is compatible with intersection products, see
Chow Homology, Lemma \ref{}.
Moreover, if $f$ is also proper, then
we have $f_*(\alpha \cdot f^*\beta) = f_*\alpha \cdot \beta$, see
Chow Homology, Lemma \ref{}.
We have $\alpha \cdot \beta = \Delta^*(\alpha \times \beta)$
if $\alpha, \beta$ are cycles on $X$ smooth over $k$.

If $X$ is normal or Cohen-Macaulay, then we can also consider
the Chow groups $\CH^p(X)$ of cycles of codimension $p$, see
Chow Homology, Section \ref{chow-section-cycles-codimension}.
If $X$ is smooth and $\alpha$ and $\beta$ are cycles on $X$,
then $\alpha \cdot \beta$ denotes the intersection product of
$\alpha$ and $\beta$, see
Chow Homology, Section \ref{chow-section-intersection-product}.



@@ -527,25 +488,25 @@ \section{Correspondences}
Given $c \in \text{Corr}^r(X, Y)$ and $\beta \in \CH_k(Y) \otimes \mathbf{Q}$
we can define the {\it pullback} of $\beta$ by $c$ using the formula
$$
c^*(\beta) = \text{pr}_{1, *}(c \cap \text{pr}_2^*\beta)
c^*(\beta) = \text{pr}_{1, *}(c \cdot \text{pr}_2^*\beta)
\quad\text{in}\quad
\CH_{k - r}(X) \otimes \mathbf{Q}
$$
This makes sense because $\text{pr}_2$ is flat of relative dimension
$d$ on $X_d \times Y$, hence $\text{pr}_2^*\beta$ is a cycle of
dimension $d + k$ on $X_d \times Y$, hence $c \cap \text{pr}_2^*\alpha$
dimension $d + k$ on $X_d \times Y$, hence $c \cdot \text{pr}_2^*\alpha$
is a cycle of dimension $k - r$ on $X_d \times Y$ whose pushforward
by the proper morphism $\text{pr}_1$ is a cycle of the same dimension.
Similarly, switching to grading by codimension,
given $\alpha \in \CH^i(X) \otimes \mathbf{Q}$ we can define the
{\it pushforward} of $\alpha$ by $c$ using the formula
$$
c_*(\alpha) = \text{pr}_{2, *}(c \cap \text{pr}_1^*\alpha)
c_*(\alpha) = \text{pr}_{2, *}(c \cdot \text{pr}_1^*\alpha)
\quad\text{in}\quad
\CH^{i + r}(Y) \otimes \mathbf{Q}
$$
This makes sense because $\text{pr}_1^*\alpha$ is a cycle of codimension
$i$ on $X \times Y$, hence $c \cap \text{pr}_1^*\alpha$ is a cycle
$i$ on $X \times Y$, hence $c \cdot \text{pr}_1^*\alpha$ is a cycle
of codimension $i + d + r$ on $X_d \times Y$, which pushes forward
to a cycle of codimension $i + r$ on $Y$.

@@ -1403,7 +1364,7 @@ \section{Projective space bundle formula}
\begin{proof}
Denote $q_i : P \times_X P \to P$ the projections. Observe that we have
the transversal intersection
$\Delta_{P/X} = (p \times p)^{-1}\Delta_X \cap (P \times_X P)$
$\Delta_{P/X} = (p \times p)^{-1}\Delta_X \cdot (P \times_X P)$
in $P \times P$. Thus it suffices to show that the class of
$\Delta_{P/X} \subset P \times_X P$ is of the form
$$
@@ -4161,7 +4122,7 @@ \section{Weil cohomology theories, III}
have filtrations whose successive quotients are invertible
modules, this reduces us to the case where $\alpha$ is
of the form $\xi_1 \cap \ldots \cap \xi_t \cap [X]$
for some first chern classes $\xi_i$ if invertible modules $\mathcal{L}_i$.
for some first chern classes $\xi_i$ of invertible modules $\mathcal{L}_i$.
Since any invertible module is a difference of very ample
invertible modules, this reduces us to the case where
$\mathcal{L}_i$ is very ample.

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