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A bit more in weil and in chow

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aisejohan committed Aug 19, 2019
1 parent eea7c43 commit 00062d0521e7929108ed7dfd49ed296eddace74e
Showing with 178 additions and 26 deletions.
  1. +108 −8 chow.tex
  2. +70 −18 weil.tex
116 chow.tex
@@ -11667,8 +11667,8 @@ \section{Gysin maps for local complete intersection morphisms}
as the assumptions are preserved by base change by $X' \to X$
locally of finite type. After replacing $P$ by an open neighbourhood
of $s(Z)$ we may assume $P \to X$ is smooth of fixed relative dimension $r$.
Say $\dim_\delta(Z) = n$. Then $p^{-1}(Z)$ is equidimensional of dimension
$r + n$ and $p^*[Z]$ is given by $[p^{-1}(Z)]_{n + r}$.
Say $\dim_\delta(Z) = n$. Then every irreducible component of $p^{-1}(Z)$
has dimension $r + n$ and $p^*[Z]$ is given by $[p^{-1}(Z)]_{n + r}$.
Observe that $s(X) \cap p^{-1}(Z) = s(Z)$ scheme theoretically. Hence by the
same reference as used above $s(X) \cap p^{-1}(Z)$ is a closed subscheme
regularly embedded in $\overline{p}^{-1}(Z)$ of codimension $r$.
@@ -11785,8 +11785,8 @@ \section{Gysin maps for local complete intersection morphisms}
$g$ is smooth of relative dimension $t$.
Then $f^*[Y] = [X]_{n + s}$ and $g^*[Y] = [P]_{n + t}$.
On the other hand $i$ is a regular immersion of codimension $t - s$.
Thus $i^![P]_{n + t} = [X]_{n + 2}$ follows from
Lemma \ref{lemma-gysin-fundamental} and the proof is complete.
Thus $i^![P]_{n + t} = [X]_{n + s}$ (Lemma \ref{lemma-lci-gysin-easy})
and the proof is complete.
\end{proof}
\begin{lemma}
@@ -11857,6 +11857,84 @@ \section{Gysin maps for local complete intersection morphisms}
immediately from Lemma \ref{lemma-gysin-commutes}.
\end{proof}
\begin{lemma}
\label{lemma-lci-gysin-easy}
Let $(S, \delta)$ be as in Situation \ref{situation-setup}.
Consider a cartesian diagram
$$
\xymatrix{
X' \ar[d]_{f'} \ar[r] &
X \ar[d]^f \\
Y' \ar[r] &
Y
}
$$
of schemes locally of finite type over $S$. Assume
\begin{enumerate}
\item $f$ is a local complete intersection morphism and
the gysin map exists for $f$,
\item $X$, $X'$, $Y$, $Y'$ satisfy the equivalent conditions of
Lemma \ref{lemma-locally-equidimensional},
\item for $x' \in X'$ with images $x$, $y'$, and $y$
in $X$, $Y'$, and $Y$ we have $n_{x'} - n_{y'} = n_x - n_y$
where $n_{x'}$, $n_x$, $n_{y'}$, and $n_y$ are as in the lemma, and
\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.
\end{lemma}
\begin{proof}
Recall that $n_{x'}$ is the common value of $\delta(\xi)$
where $\xi$ is the generic point of an irreducible component
passing through $x'$. Moreover, the functions
$x' \mapsto n_{x'}$, $x \mapsto n_x$, $y' \mapsto n_{y'}$, 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
$[X'] = \sum [X'_n]_n$ 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
and $X$, $Y'$, $Y$ by the connected component that
it maps into. Then we know that $X'$, $X$, $Y'$, and
$Y$ are $\delta$-equidimensional in the sense that
each irreducible component has the same $\delta$-dimension.
Say $n'$, $n$, $m'$, and $m$ is this common value
for $X'$, $X$, $Y'$, and $Y$. The last assumption
means that $n' - m' = n - m$.
\medskip\noindent
Choose a factorization $f = g \circ i$ where $i : X \to P$
is an immersion and $g : P \to Y$ is smooth. As $X$ is connected,
we see that the relative dimension of $P \to Y$ at points of $i(X)$
is constant. Hence after replacing $P$ by an open neighbourhood
of $i(X)$, we may assume that $P \to Y$ has constant relative dimension
and $i : X \to P$ is a closed immersion.
Denote $g' : Y' \times_Y P \to Y'$ the base change of $g$ and denote
$i' : X' \to Y' \times_Y P$ the base change of $i$.
It is clear that $g^*[Y] = [P]$ and $(g')^*[Y'] = [Y' \times_Y P]$.
Finally, if $\xi' \in X'$ is a generic point, then
$\mathcal{O}_{Y' \times_Y P, i'(\xi)}$ is Cohen-Macaulay.
Namely, the local ring map
$\mathcal{O}_{Y', f'(\xi)} \to \mathcal{O}_{Y' \times_Y P, i'(\xi)}$
is flat with regular fibre
(see Algebra, Section \ref{algebra-section-smooth-overview}),
a regular local ring is Cohen-Macaulay
(Algebra, Lemma \ref{algebra-lemma-regular-ring-CM}),
$\mathcal{O}_{Y', f'(\xi)}$ is Cohen-Macaulay by assumption
(4) and we get what we want from
Algebra, Lemma \ref{algebra-lemma-CM-goes-up}.
Thus we reduce to the case discussed in the next paragraph.
\medskip\noindent
Assume $f$ is a regular closed immersion and $X'$, $X$, $Y'$, and
$Y$ are $\delta$-equidimensional of $\delta$-dimensions
$n'$, $n$, $m'$, and $m$ and $m' - n' = m - n$.
In this case we obtain the result immediately from
Lemma \ref{lemma-gysin-easy}.
\end{proof}
@@ -12346,8 +12424,7 @@ \section{Intersection products}
connected components, hence we may assume $X$ is smooth over $k$
and equidimensional of dimension $d$ and $Y$ is smooth over $k$
and equidimensional of dimension $e$. Observe that
$f^![Y]_e = [X]_d$ (because this is true both if $f$ is smooth
and if $f$ is a regular immersion; small detail omitted).
$f^![Y]_e = [X]_d$ (see for example Lemma \ref{lemma-lci-gysin-easy}).
Write $\alpha = c \cap [Y]_e$ and $\beta = c' \cap [Y]_e$
and hence $\alpha \cdot \beta = c \cap c' \cap [Y]_e$,
see Lemma \ref{lemma-identify-chow-for-smooth}.
@@ -12392,8 +12469,7 @@ \section{Intersection products}
connected components, hence we may assume $X$ is smooth over $k$
and equidimensional of dimension $d$ and $Y$ is smooth over $k$
and equidimensional of dimension $e$. Observe that
$f^![Y]_e = [X]_d$ (because this is true both if $f$ is smooth
and if $f$ is a regular immersion; small detail omitted).
$f^![Y]_e = [X]_d$ (see for example Lemma \ref{lemma-lci-gysin-easy}).
Write $\alpha = c \cap [X]_d$ and $\beta = c' \cap [Y]_e$,
see Lemma \ref{lemma-identify-chow-for-smooth}. We have
\begin{align*}
@@ -12459,6 +12535,30 @@ \section{Intersection products}
Lemma \ref{lemma-gysin-easy} are satisfied and we conclude.
\end{proof}
\begin{lemma}
\label{lemma-intersect-regularly-embedded}
Let $k$ be a field. Let $X$ be a scheme smooth over $k$. Let $i : Y \to X$ be
a regular closed immersion. Let $\alpha \in \CH_*(X)$. If $Y$ is
equidimensional of dimension $e$, then
$\alpha \cdot [Y]_e = i_*(i^!(\alpha))$ in $\CH_*(X)$.
\end{lemma}
\begin{proof}
After decomposing $X$ into connected components we may and do assume $X$
is equidimensional of dimension $d$. Write $\alpha = c \cap [X]_n$
with $x \in A^*(X)$, see Lemma \ref{lemma-identify-chow-for-smooth}. Then
$$
i_*(i^!(\alpha)) = i_*(i^!(c \cap [X]_n)) =
i_*(c \cap i^![X]_n) = i_*(c \cap [Y]_e) =
c \cap i_*[Y]_e = \alpha \cdot [Y]_e
$$
The first equality by choice of $c$. Then second equality by
Lemma \ref{lemma-lci-gysin-commutes}. The third because
$i^![X]_d = [Y]_e$ in $\CH_*(Y)$ (Lemma \ref{lemma-lci-gysin-easy}).
The fourth because bivariant classes commute with proper pushforward.
The last equality by Lemma \ref{lemma-identify-chow-for-smooth}.
\end{proof}
@@ -758,9 +758,9 @@ \section{Correspondences}
Example \ref{example-graph-correspondence}. Then
\begin{enumerate}
\item pushforward of cycles by the correspondence $[\Gamma_f]$
corresponds to pullback of cycles by $f$,
agrees with the gysin map $f^! : \CH^*(Y) \to \CH^*(X)$,
\item pullback of cycles by the correspondence $[\Gamma_f]$
corresponds to pushforward of cycles by $f$,
agrees with the pushforward map $f_* : \CH_*(Y) \to \CH_*(X)$,
\item if $X$ and $Y$ are equidimensional of dimensions $d$ and $e$,
then
\begin{enumerate}
@@ -769,13 +769,43 @@ \section{Correspondences}
corresponds to pushforward of cycles by $f$, and
\item pullback of cycles by the correspondence
$[\Gamma_f^t]$ of Remark \ref{remark-transpose}
corresponds to pullback of cycles by $f$.
corresponds to the gysin map $f^!$.
\end{enumerate}
\end{enumerate}
\end{lemma}

\begin{proof}
Omitted.
Proof of (1). Recall that
$[\Gamma_f]_*(\alpha) =
\text{pr}_{2, *}([\Gamma_f] \cdot \text{pr}_1^*\alpha)$.
We have
$$
[\Gamma_f] \cdot \text{pr}_1^*\alpha =
(f, 1)_*((f, 1)^! \text{pr}_1^*\alpha) =
(f, 1)_*((f, 1)^! \text{pr}_1^!\alpha) =
(f, 1)_*(f^!\alpha)
$$
The first equality by Chow Homology, Lemma
\ref{chow-lemma-intersect-regularly-embedded}.
The second by
Chow Homology, Lemma \ref{chow-lemma-lci-gysin-flat}.
The third because $\text{pr}_1 \circ (f, 1) = f$ and
Chow Homology, Lemma \ref{chow-lemma-lci-gysin-composition}.
Then we coclude because
$\text{pr}_{2, *} \circ (f, 1)_* = 1_*$ by
Chow Homology, Lemma \ref{chow-lemma-compose-pushforward}.

\medskip\noindent
Proof of (2). Recall that $[\Gamma_f]_*(\beta) =
\text{pr}_{1, *}([\Gamma_f] \cdot \text{pr}_2^*\beta)$.
Arguing exactly as above we have
$$
[\Gamma_f] \cdot \text{pr}_2^*\beta = (f, 1)_*\beta
$$
Thus the result follows as before.

\medskip\noindent
Proof of (3). Proved in exactly the same manner as above.
\end{proof}

\begin{example}
@@ -803,15 +833,8 @@ \section{Correspondences}
\noindent
The category of correspondences is a symmetric monoidal category.
Given smooth projective schemes $X$ and $Y$ over $k$, we define
$X \otimes Y = X \times Y$. As associativity constraint
$$
X \otimes (Y \otimes Z) = (X \otimes Y) \otimes Z
$$
we use the usual associativity constraint on products of schemes.
The unit object will be $\Spec(k)$. The commutativity will be
given by the isomorphism $X \times Y \to Y \times X$ switching the factors.
Given four smooth projective schemes $X, X', Y, Y'$ over $k$
we define a tensor product
$X \otimes Y = X \times Y$. Given four smooth projective schemes
$X, X', Y, Y'$ over $k$ we define a tensor product
$$
\otimes :
\text{Corr}^r(X, Y) \times \text{Corr}^{r'}(X', Y')
@@ -825,7 +848,13 @@ \section{Correspondences}
$$
where $\text{pr}_{13} : X \times X' \times Y \times Y' \to X \times Y$
and $\text{pr}_{24} : X \times X' \times Y \times Y' \to X' \times Y'$
are the projections.
are the projections. As associativity constraint
$$
X \otimes (Y \otimes Z) = (X \otimes Y) \otimes Z
$$
we use the usual associativity constraint on products of schemes.
The commutativity constraint will be given by the isomorphism
$X \times Y \to Y \times X$ switching the factors.

\begin{lemma}
\label{lemma-tensor-product}
@@ -846,11 +875,11 @@ \section{Correspondences}
Set
$\eta_X = [\Gamma_{X \to X \times X}] \in \text{Corr}^0(X \times X, X)$,
$\eta_Y = [\Gamma_{Y \to Y \times Y}] \in \text{Corr}^0(Y \times Y, Y)$,
$[X] \in \text{Corr}^d(X, \Spec(k))$, and
$[Y] \in \text{Corr}^e(Y, \Spec(k))$. The diagram
$[X] \in \text{Corr}^{-d}(X, \Spec(k))$, and
$[Y] \in \text{Corr}^{-e}(Y, \Spec(k))$. The diagram
$$
\xymatrix{
X \otimes Y \ar[r]_{a^t \otimes \text{id}} \ar[d]_{\text{id} \otimes a} &
X \otimes Y \ar[r]_{a \otimes \text{id}} \ar[d]_{\text{id} \otimes a^t} &
Y \otimes Y \ar[r]_{\eta_Y} &
Y \ar[d]^{[Y]} \\
X \otimes X \ar[r]^{\eta_X} &
@@ -862,7 +891,30 @@ \section{Correspondences}
\end{lemma}

\begin{proof}
Going either way around the diagram a computation shows that we
Recall that $\text{Corr}^r(W, \Spec(k)) = \CH_{-r}(W)$ for any
smooth projective scheme $W$ over $k$
and given $c \in \text{Corr}^s(W', W)$ the composition
with $c$ agrees with pullback by $c$ as a map
$\CH_{-r}(W) \to \CH_{-r - s}(W')$
(Lemma \ref{lemma-composition-correspondences}).
Finally, we have Lemma \ref{lemma-functor-and-cycles}
which tells us how to convert this into usual
pushforward and pullback of cycles.
We have
$$
(a \otimes \text{id})^* \eta_Y^* [Y] =
(a \otimes \text{id})^* [\Delta_Y] =
(f \times \text{id})_*\Delta_Y = [\Gamma_f]
$$
and the other way around we get
$$
(\text{id} \otimes a^t)^* \eta_X^* [X] =
(\text{id} \otimes a^t)^* [\Delta_X] =
(\text{id} \times f)^![\Delta_X] = [\Gamma_f]
$$
The last equality follows from
Chow Homology, Lemma \ref{chow-lemma-lci-gysin-easy}.
In other words, going either way around the diagram we
obtain the element of $\text{Corr}^d(X \times Y, \Spec(k))$
corresponding to the cycle $\Gamma_f \subset X \times Y$.
\end{proof}

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