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Fixed weil through the projection formula

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aisejohan committed Aug 19, 2019
1 parent 67d6b5f commit af523bf189c45a07c9ad88cd16bbbeb3fd4b80ef
Showing with 90 additions and 53 deletions.
  1. +24 −8 chow.tex
  2. +66 −45 weil.tex
@@ -4739,7 +4739,8 @@ \section{Gysin homomorphisms}
Definition \ref{definition-gysin-homomorphism} and assume
that $\mathcal{L}|_D \cong \mathcal{O}_D$. In this case we
can define a canonical map $i^* : Z_{k + 1}(X) \to Z_k(D)$
on cycles, by requiring that $i^*[W] = 0$ whenever $W \subset D$.
on cycles, by requiring that $i^*[W] = 0$ whenever $W \subset D$
is an integral closed subscheme.
The possibility to do this will be useful later on.
\end{remark}

@@ -7635,6 +7636,20 @@ \section{Cycles of given codimension}
Warning: the property for a morphism to have codimension $r$
is not preserved by base change.

\begin{remark}
\label{remark-fundamental-class}
Let $(S, \delta)$ be as in Situation \ref{situation-setup}.
Let $X$ be locally of finite type over $S$ satisfying the
equivalent conditions of Lemma \ref{lemma-locally-equidimensional}.
Let $X = \coprod X_n$ be the decomposition into open and closed
subschemes such that every irreducible component of $X_n$ has
$\delta$-dimension $n$. In this situation we sometimes set
$$
[X] = \sum\nolimits_n [X_n]_n \in \CH^0(X)
$$
This class is a kind of ``fundamental class'' of $X$ in Chow theory.
\end{remark}




@@ -11119,9 +11134,9 @@ \section{Higher codimension gysin homomorphisms}
\medskip\noindent
In this paragraph $Z$ and $Y$ are effective Cartier divisors on $X$
integral of dimension $n$, we have $\mathcal{N}'' = \mathcal{C}_{Y/X}$.
In this case $c(Y \to X, \mathcal{C}_{Y/X}^\vee) \cap [X] = [Y]$ by
In this case $c(Y \to X, \mathcal{C}_{Y/X}^\vee) \cap [X] = [Y]_{n - 1}$ by
Lemma \ref{lemma-gysin-fundamental}. Thus we have to prove that
$c(Z \to X, \mathcal{N}) \cap [X] = c(Z \to Y, \mathcal{N}') \cap [Y]$.
$c(Z \to X, \mathcal{N}) \cap [X] = c(Z \to Y, \mathcal{N}') \cap [Y]_{n - 1}$.
Denote $N$ and $N'$ the vector bundles over $Z$ associated to
$\mathcal{N}$ and $\mathcal{N}'$. Consider the commutative diagram
$$
@@ -11156,7 +11171,7 @@ \section{Higher codimension gysin homomorphisms}
$$
in $\CH_{n - 1}(N')$. Now observe that
$\gamma = c(Z \to X, \mathcal{N}) \cap [X]$ and
$\gamma' = c(Z \to Y, \mathcal{N}') \cap [Y]$
$\gamma' = c(Z \to Y, \mathcal{N}') \cap [Y]_{n - 1}$
are characterized by $p^*\gamma = [C_Z X]_n$ in $\CH_n(N)$
and by $(p')^*\gamma' = [C_Z Y]_{n - 1}$ in $\CH_{n - 1}(N')$.
Hence the proof is finished as $i^* \circ p^* = (p')^*$ by
@@ -11516,7 +11531,8 @@ \section{Chow groups and K-groups revisited}
[\mathcal{F}] \longmapsto ch(X \to Y, i_*\mathcal{F}) \cap [Y]
$$
where $[Y] = \sum [Y_i] \in \CH_*(Y)$ is the sum of the classes of the
irreducible components of $Y$.
irreducible components of $Y$ (compare with
Remark \ref{remark-fundamental-class}).
If $\mathcal{F}$ is (set theoretically) supported on a closed subscheme
$Z \subset X$, then we have
$$
@@ -11881,8 +11897,8 @@ \section{Gysin maps for local complete intersection morphisms}
\item for every generic point $\xi \in X'$ the local ring
$\mathcal{O}_{Y', f'(\xi)}$ is Cohen-Macaulay.
\end{enumerate}
Then $f^![Y'] = [X']$ where $[Y']$ and $[X']$ are defined in
the proof.
Then $f^![Y'] = [X']$ where $[Y']$ and $[X']$ are as in
Remark \ref{remark-fundamental-class}.
\end{lemma}
\begin{proof}
@@ -11892,7 +11908,7 @@ \section{Gysin maps for local complete intersection morphisms}
$x' \mapsto n_{x'}$, $x \mapsto n_x$, $y' \mapsto n_{y'}$, and
$y \mapsto n_y$ are locally constant. Let $X'_n$, $X_n$, $Y'_n$,
and $Y_n$ be the open and closed subscheme of $X'$, $X$, $Y'$, and
$Y$ where the function has value $n$. We set
$Y$ where the function has value $n$. Recall that
$[X'] = \sum [X'_n]_n$ and $[Y'] = \sum [Y'_n]_n$.
Having said this, it is clear that to prove the lemma we
may replace $X'$ by one of its connected components
111 weil.tex
@@ -57,8 +57,10 @@ \section{Conventions and notation}
modulo rational equivalence have been defined in
Chow Homology, Definition \ref{chow-definition-rational-equivalence}.
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}.
the Chow groups $\CH^p(X)$ of cycles of codimension $p$
(Chow Homology, Section \ref{chow-section-cycles-codimension})
and then $[X] \in \CH^0(X)$ denotes the ``fundamental class'' of $X$, see
Chow Homology, Remark \ref{chow-remark-fundamental-class}.
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
@@ -675,8 +677,9 @@ \section{Correspondences}
$$
[Z] = \text{pr}_{13, *}(\text{pr}_{12}^*[\Delta] \cdot \text{pr}_{23}^*[Z])
$$
in the chow group of $X \times Y$. After replacing $X$ and $Y$ by the
irreducible component containing the image of $X$ under the two projections
in the chow group of $X \times Y$ for any integral closed subscheme $Z$
of $X \times Y$. After replacing $X$ and $Y$ by the
irreducible component containing the image of $Z$ under the two projections
we may assume $X$ and $Y$ are integral as well. Then we have to show
$$
[Z] = \text{pr}_{13, *}([\Delta \times Y] \cdot [X \times Z])
@@ -1372,16 +1375,15 @@ \section{Chow groups of motives}
$$
c_* : \CH^i(M) \to \CH^i(N)
$$
for all $i \in \mathbf{Z}$. We omit the verification that this
is compatible with compositions of morphisms of motives (this follows
from Lemma \ref{lemma-composition-correspondences}).
for all $i \in \mathbf{Z}$. This is compatible with compositions of
morphisms of motives by Lemma \ref{lemma-composition-correspondences}.
This functoriality of Chow groups can also be deduced from the following
lemma.

\begin{lemma}
\label{lemma-chow-groups-representable}
Let $k$ be a base field. The functor $\CH^i(-)$ on the category
of motives $M_k$ is representable by $\mathbf{1}(i)$, i.e., we
of motives $M_k$ is representable by $\mathbf{1}(-i)$, i.e., we
have
$$
\CH^i(M) = \Hom_{M_k}(\mathbf{1}(-i), M)
@@ -1390,11 +1392,13 @@ \section{Chow groups of motives}
\end{lemma}

\begin{proof}
Immediate from the definitions.
Immediate from the definitions and
Lemma \ref{lemma-composition-correspondences}.
\end{proof}

\noindent
The reader can imagine that we can use this, the Yoneda lemma, and
The reader can imagine that we can use
Lemma \ref{lemma-chow-groups-representable}, the Yoneda lemma, and
the duality in Lemma \ref{lemma-dual} to obtain the following.

\begin{lemma}[Manin]
@@ -1411,7 +1415,7 @@ \section{Chow groups of motives}
of $M \otimes h(X)(m)$ are the same as the Chow groups of of $M \otimes h(X)$
up to a shift in degrees. Hence our assumption implies
that $c \otimes 1 : M \otimes L \to N \otimes L$ induces an isomorphism on
Chow gruops for every object $L$ of $M_k$. By
Chow groups for every object $L$ of $M_k$. By
Lemma \ref{lemma-chow-groups-representable}
we see that
$$
@@ -1448,12 +1452,15 @@ \section{Projective space bundle formula}
}
$$
over $X$ with $\mathcal{O}_P(1)$ normalized so that
$p_*(\mathcal{O}_P(1)) = \mathcal{E}$.
Denote $\xi = c_1(\mathcal{O}_P(1)) \cap [P] \in \CH^1(P)$ the first
chern class. For $i = 0, \ldots, r - 1$ consider the correspondences
$p_*(\mathcal{O}_P(1)) = \mathcal{E}$. Recall that
$$
[\Gamma_p] \in \text{Corr}^0(X, P) \subset \CH^*(X \times P) \otimes \mathbf{Q}
$$
c_i = [\Gamma_p] \cdot \text{pr}_2^*\xi^i \in
\CH^{d + i}(X \times P) = \text{Corr}^i(X, P)
See Remark \ref{example-graph-correspondence}.
For $i = 0, \ldots, r - 1$ consider the correspondences
$$
c_i = c_1(\text{pr}_2^*\mathcal{O}_P(1))^i \cap [\Gamma_p]
\in \text{Corr}^i(X, P)
$$
We may and do think of $c_i$ as a morphism $h(X)(i) \to h(P)$.

@@ -1475,36 +1482,42 @@ \section{Projective space bundle formula}
after taking the product with any smooth projective scheme $Z$.
Observe that $P \times Z \to X \times Z$ is the projective
bundle associated to the pullback of $\mathcal{E}$ to $X \times Z$.
Hence the statement on Chow groups is true
by the projective space bundle formula given
in Chow Homology, Lemma \ref{chow-lemma-chow-ring-projective-bundle}.
Hence the statement on Chow groups is true by the projective space bundle
formula given in
Chow Homology, Lemma \ref{chow-lemma-chow-ring-projective-bundle}.
Namely, pushforward of cycles along $[\Gamma_p]$ is given by pullback
of cycles by $p$ according to Lemma \ref{lemma-functor-and-cycles} and
Chow Homology, Lemma \ref{chow-lemma-lci-gysin-flat}. Hence pushforward
along $c_i$ sends $\alpha$ to $c_1(\mathcal{O}_P(1))^i \cap p^*\alpha$.
Some details omitted.
\end{proof}

\noindent
In the situation above, for $j = 0, \ldots, r - 1$ consider
the correspondences
$$
c'_j = \text{pr}_1^*\xi^{r - 1 - j} \cdot [\Gamma_p^t] \in
\CH^{d + r - 1 - j}(P \times X) = \text{Corr}^{-j}(P, X)
c'_j = c_1(\text{pr}_1^*\mathcal{O}_P(1))^{r - 1 - j} \cap [\Gamma_p^t] \in
\text{Corr}^{-j}(P, X)
$$
For $i, j \in \{0, \ldots, r - 1\}$ we have
$$
c'_j \circ c_i =
\text{pr}_{13, *}(
\text{pr}_{12}^*[\Gamma_p] \cdot \text{pr}_{23}^*[\Gamma_p^t]
\cdot \text{pr}_2^*\xi^{i + r - 1 - j})
\text{pr}_{13, *}\left(
c_1(\text{pr}_2^*\mathcal{O}_P(1))^{i + r - 1 - j} \cap
(\text{pr}_{12}^*[\Gamma_p] \cdot \text{pr}_{23}^*[\Gamma_p^t])
\right)
$$
The cycles $\text{pr}_{12}^{-1}\Gamma_p$ and
$\text{pr}_{23}^{-1}\Gamma_p^t$ intersect transversally and
with intersection equal to the image of
$(p, 1, p) : P \to X \times P \times X$.
Observe that the fibres of
$(p, p) : \text{pr}_{13} \circ (p, 1, p) P \to X \times X$
$(p, p) = \text{pr}_{13} \circ (p, 1, p) : P \to X \times X$
have dimension $r - 1$. We immediately conclude
$c'_j \circ c_i = 0$ for $i + r - 1 - j < r - 1 \Leftrightarrow i < j$.
$c'_j \circ c_i = 0$ for $i + r - 1 - j < r - 1$, in other words when $i < j$.
On the other hand, by the projective space bundle formula
(Chow Homology, Lemma \ref{chow-lemma-chow-ring-projective-bundle})
the cycle $\xi^{r - 1}$ maps
the cycle $c_1(\mathcal{O}_P(1))^{r - 1} \cap [P]$ maps
to $[X]$ in $X$. Hence for $i = j$ the pushforward above
gives the class of the diagonal and hence
we see that
@@ -1535,40 +1548,48 @@ \section{Projective space bundle formula}
$$
[\Delta_P] =
\left(\sum\nolimits_{i = 0, \ldots, r - 1}
\text{pr}_1^*\alpha_i \cdot \text{pr}_2^*\xi^i\right)
{r - 1 \choose i} c_{r - 1 - i}(\text{pr}_1^*\mathcal{S}^\vee) \cap
c_1(\text{pr}_2^*\mathcal{O}_P(1))^i\right)
\cap
(p \times p)^*[\Delta_X]
$$
for some $\alpha_i \in \CH^{r - 1 - i}(P)$ with $\alpha_{r - 1} = 1$.
where $\mathcal{S}$ is the kernel of the canonical surjection
$p^*\mathcal{E} \to \mathcal{O}_P(1)$.
\end{lemma}

\begin{proof}
Denote $q_i : P \times_X P \to P$ the projections. Observe that we have
the transversal intersection
$\Delta_{P/X} = (p \times p)^{-1}\Delta_X \cdot (P \times_X P)$
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
Observe that $(p \times p)^*[\Delta_X] = [P \times_X P]$.
Since $\Delta_P \subset P \times_X P \subset P \times P$
and since capping with chern classes commutes with proper pushforward
(Chow Homology, Lemma \ref{chow-lemma-pushforward-cap-cj})
it suffices to show that the class of
$\Delta_P \subset P \times_X P$ in $\CH^*(P \times_X P)$
is equal to
$$
\left(\sum\nolimits_{i = 0, \ldots, r - 1}
q_1^*\alpha_i \cdot q_2^*\xi^i\right)
{r - 1 \choose i} c_{r - 1 - i}(q_1^*\mathcal{S}^\vee) \cap
c_1(q_2^*\mathcal{O}_P(1))^i\right)
\cap
[P \times_X P]
$$
for some $\alpha_i \in \CH^{r - 1 - i}(P)$ with $\alpha_{r - 1} = 1$.
Denote $\mathcal{S}$ the kernel of the canonical surjective map
$p^*\mathcal{E} \to \mathcal{O}_P(1)$. Also set
$q = p \circ q_1 = p \circ q_2 : P \times_X P \to X$.
Next, consider the maps
where $q_i : P \times_X P \to P$, $i = 1, 2$ are the projections.
Set $q = p \circ q_1 = p \circ q_2 : P \times_X P \to X$.
Consider the maps
$$
q_1^*\mathcal{S} \otimes q_2^*\mathcal{O}_P(-1) \to
q^*\mathcal{E} \otimes q^*\mathcal{E}^\vee \to
\mathcal{O}_{P \times_X P}
$$
The source is a module locally free of rank $r - 1$ and a local calculation
shows that this map vanishes exactly along $\Delta_{P/X}$.
By Chow Homology, Lemma \ref{chow-lemma-top-chern-class}
the class $[\Delta_{P/X}]$ is the top chern class of the dual
where the final arrow is the pullback by $q$ of the evaluation map
$\mathcal{E} \otimes_{\mathcal{O}_X} \mathcal{E}^\vee \to \mathcal{O}_X$.
The source of the composition is a module locally free of rank $r - 1$
and a local calculation shows that this map vanishes exactly along
$\Delta_P$. By Chow Homology, Lemma \ref{chow-lemma-top-chern-class}
the class $[\Delta_P]$ is the top chern class of the dual
$$
q_1^*\mathcal{S}^\vee \otimes q_2^*\mathcal{O}_P(1)
$$
The lemma follows from Chow Homology, Lemma
The desired result follows from Chow Homology, Lemma
\ref{chow-lemma-chern-classes-E-tensor-L}.
\end{proof}

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