Skip to content
Permalink
Browse files

Improve exposition of a lemma in weil

It was just a trivial consequence of the blow up formula...
  • Loading branch information
aisejohan committed Nov 27, 2019
1 parent 4f25a75 commit e998b8ffb90e2fd8beb77b9a4e8a4648b0a87225
Showing with 117 additions and 110 deletions.
  1. +14 −0 chow.tex
  2. +62 −0 obsolete.tex
  3. +1 −1 tags/tags
  4. +40 −109 weil.tex
@@ -11761,6 +11761,20 @@ \section{An Adams operator}
complicated and another approach has to be used.
\end{remark}
\begin{lemma}
\label{lemma-minus-adams-operator}
Let $X$ be a scheme. There is a ring map
$\psi^{-1} : K_0(\textit{Vect}(X)) \to K_0(\textit{Vect}(X))$
which sends $[\mathcal{E}]$ to $[\mathcal{E}^\vee]$
when $\mathcal{E}$ is finite locally free
and is compatible with pullbacks.
\end{lemma}
\begin{proof}
The only thing to check is that taking duals is compatible with
short exact sequences and with pullbacks. This is clear.
\end{proof}
\begin{remark}
\label{remark-chern-classes-K}
Let $(S, \delta)$ be as in Situation \ref{situation-setup}.
@@ -2531,6 +2531,68 @@ \section{Modifications}
\section{Intersection theory}
\label{section-intersection-theory}

\begin{lemma}
\label{lemma-good-blowing-up}
Let $b : X' \to X$ be the blowing up of a smooth projective
scheme over a field $k$ in a smooth closed subscheme $Z \subset X$.
Picture
$$
\xymatrix{
E \ar[r]_j \ar[d]_\pi & X' \ar[d]^b \\
Z \ar[r]^i & X
}
$$
Assume there exists an element of $K_0(X)$ whose restriction to
$Z$ is equal to the class of $\mathcal{C}_{Z/X}$ in $K_0(Z)$.
Then $[Lb^*\mathcal{O}_Z] = [\mathcal{O}_E] \cdot \alpha''$
in $K_0(X')$ for some $\alpha'' \in K_0(X')$.
\end{lemma}

\begin{proof}
The schemes $X$, $X'$, $E$, $Z$ are smooth and projective over
$k$ and hence we have $K'_0(X) = K_0(X) = K_0(\textit{Vect}(X)) =
K_0(D^b_{\textit{Coh}}(X)))$
and similarly for the other $3$. See
Derived Categories of Schemes, Lemmas \ref{perfect-lemma-Noetherian-Kprime},
\ref{perfect-lemma-Kprime-K}, and \ref{perfect-lemma-K-is-old-K}.
We will switch between these versions at will in this proof.
Consider the short exact sequence
$$
0 \to \mathcal{F} \to \pi^*\mathcal{C}_{Z/X} \to \mathcal{C}_{E/X'} \to 0
$$
of finite locally free $\mathcal{O}_E$-modules defining $\mathcal{F}$.
Observe that $\mathcal{C}_{E/X'} = \mathcal{O}_{X'}(-E)|_E$
is the restriction of the invertible $\mathcal{O}_X$-module
$\mathcal{O}_{X'}(-E)$.
Let $\alpha \in K_0(X)$ be an element such that
$i^*\alpha = [\mathcal{C}_{Z/X}]$ in $K_0(Z)$.
Let $\alpha' = b^*\alpha - [\mathcal{O}_{X'}(-E)]$.
Then $j^*\alpha' = [\mathcal{F}]$. We deduce that
$j^*\lambda^i(\alpha') = [\wedge^i(\mathcal{F})]$ by
Weil Cohomology Theories, Lemma \ref{weil-lemma-lambda-operations}.
This means that $[\mathcal{O}_E] \cdot \alpha' = [\wedge^i\mathcal{F}]$
in $K_0(X)$, see
Derived Categories of Schemes, Lemma \ref{perfect-lemma-projection-formula}.
Let $r$ be the maximum codimension of an irreducible component of $Z$
in $X$. A computation which we omit shows that
$H^{-i}(Lb^*\mathcal{O}_Z) = \wedge^i\mathcal{F}$
for $i \geq 0, 1, \ldots, r - 1$ and zero in other degrees.
It follows that in $K_0(X)$ we have
\begin{align*}
[Lb^*\mathcal{O}_Z] & =
\sum\nolimits_{i = 0, \ldots, r - 1} (-1)^i[\wedge^i\mathcal{F}] \\
& =
\sum\nolimits_{i = 0, \ldots, r - 1} (-1)^i[\mathcal{O}_E] \lambda^i(\alpha') \\
& =
[\mathcal{O}_E] \left(\sum\nolimits_{i = 0, \ldots, r - 1}
(-1)^i \lambda^i(\alpha')\right)
\end{align*}
This proves the lemma with
$\alpha'' = \sum_{i = 0, \ldots, r - 1} (-1)^i \lambda^i(\alpha')$.
\end{proof}



\begin{lemma}
\label{lemma-gysin-factors-principal}
Let $(S, \delta)$ be as in Chow Homology, Situation \ref{chow-situation-setup}.
0FID,weil-section-c1
0FIE,weil-lemma-chern-classes
0FIF,weil-lemma-cycle-classes
0FIG,weil-lemma-good-blowing-up
0FIG,obsolete-lemma-good-blowing-up
0FIH,weil-lemma-divide-pullback-good-blowing-up
0FII,weil-lemma-A5-A6-imply
0FIJ,weil-lemma-poincare-duality
149 weil.tex
@@ -4649,11 +4649,8 @@ \section{Weil cohomology theories, III}
by Lemma \ref{lemma-adams-and-chern} as desired.
\end{proof}

\noindent
The following couple of lemmas can be significantly generalized.

\begin{lemma}
\label{lemma-good-blowing-up}
\label{lemma-divide-pullback-good-blowing-up}
Let $b : X' \to X$ be the blowing up of a smooth projective
scheme over $k$ in a smooth closed subscheme $Z \subset X$.
Picture
@@ -4665,119 +4662,53 @@ \section{Weil cohomology theories, III}
$$
Assume there exists an element of $K_0(X)$ whose restriction to
$Z$ is equal to the class of $\mathcal{C}_{Z/X}$ in $K_0(Z)$.
Then $[Lb^*\mathcal{O}_Z] = [\mathcal{O}_E] \cdot \alpha''$
in $K_0(X')$ for some $\alpha'' \in K_0(X')$.
\end{lemma}

\begin{proof}
The schemes $X$, $X'$, $E$, $Z$ are smooth and projective over
$k$ and hence we have $K'_0(X) = K_0(X) = K_0(\textit{Vect}(X)) =
K_0(D^b_{\textit{Coh}}(X)))$
and similarly for the other $3$. See
Derived Categories of Schemes, Lemmas \ref{perfect-lemma-Noetherian-Kprime},
\ref{perfect-lemma-Kprime-K}, and \ref{perfect-lemma-K-is-old-K}.
We will switch between these versions at will in this proof.
Consider the short exact sequence
$$
0 \to \mathcal{F} \to \pi^*\mathcal{C}_{Z/X} \to \mathcal{C}_{E/X'} \to 0
$$
of finite locally free $\mathcal{O}_E$-modules defining $\mathcal{F}$.
Observe that $\mathcal{C}_{E/X'} = \mathcal{O}_{X'}(-E)|_E$
is the restriction of the invertible $\mathcal{O}_X$-module
$\mathcal{O}_{X'}(-E)$.
Let $\alpha \in K_0(X)$ be an element such that
$i^*\alpha = [\mathcal{C}_{Z/X}]$ in $K_0(Z)$.
Let $\alpha' = b^*\alpha - [\mathcal{O}_{X'}(-E)]$.
Then $j^*\alpha' = [\mathcal{F}]$. We deduce that
$j^*\lambda^i(\alpha') = [\wedge^i(\mathcal{F})]$ by
Lemma \ref{lemma-lambda-operations}.
This means that $[\mathcal{O}_E] \cdot \alpha' = [\wedge^i\mathcal{F}]$
in $K_0(X)$, see
Derived Categories of Schemes, Lemma \ref{perfect-lemma-projection-formula}.
Let $r$ be the maximum codimension of an irreducible component of $Z$
in $X$. A computation which we omit shows that
$H^{-i}(Lb^*\mathcal{O}_Z) = \wedge^i\mathcal{F}$
for $i \geq 0, 1, \ldots, r - 1$ and zero in other degrees.
It follows that in $K_0(X)$ we have
\begin{align*}
[Lb^*\mathcal{O}_Z] & =
\sum\nolimits_{i = 0, \ldots, r - 1} (-1)^i[\wedge^i\mathcal{F}] \\
& =
\sum\nolimits_{i = 0, \ldots, r - 1} (-1)^i[\mathcal{O}_E] \lambda^i(\alpha') \\
& =
[\mathcal{O}_E] \left(\sum\nolimits_{i = 0, \ldots, r - 1}
(-1)^i \lambda^i(\alpha')\right)
\end{align*}
This proves the lemma with
$\alpha'' = \sum_{i = 0, \ldots, r - 1} (-1)^i \lambda^i(\alpha')$.
\end{proof}

\begin{lemma}
\label{lemma-divide-pullback-good-blowing-up}
In Lemma \ref{lemma-good-blowing-up} assume every irreducible component
of $Z$ has codimension $r$ in $X$.
Then there exists a cycle $\theta \in \CH^{r - 1}(X') \otimes \mathbf{Q}$
such that $b^![Z] = [E] \cdot \theta$ in $\CH^r(X') \otimes \mathbf{Q}$ and
$\pi_*j^!(\theta) = [Z]$ in $\CH^r(Z) \otimes \mathbf{Q}$.
Assume every irreducible component of $Z$ has codimension $r$ in $X$.
Then there exists a cycle $\theta \in \CH^{r - 1}(X')$
such that $b^![Z] = [E] \cdot \theta$ in $\CH^r(X')$ and
$\pi_*j^!(\theta) = [Z]$ in $\CH^r(Z)$.
\end{lemma}

\begin{proof}
We resume the notation of the proof of Lemma \ref{lemma-good-blowing-up}.
Recall that $[Z] = ch_r(\mathcal{O}_Z) \cap [X]$, see
Chow Homology, Lemma \ref{chow-lemma-actual-computation}.
Since $b^![X] = [X']$ and since $b^!$ commutes with chern classes
(Chow Homology, Remark \ref{chow-remark-gysin-chern-classes}) we obtain
\begin{align*}
b^![Z] & =
b^! (ch_r(\mathcal{O}_Z) \cap [X]) \\
& =
ch_r(\mathcal{O}_Z) \cap b^![X] \\
& =
ch_r(\mathcal{O}_Z) \cap [X'] \\
& =
ch_r(Lb^*\mathcal{O}_Z) \cap [X']
\end{align*}
The final equality because $ch$ commutes with pullback and
because pullback on $b^* : K_0(X) \to K_0(X')$ is given by derived
pullback if we make the identification
$K_0(X) = K_0(D_{perf}(\mathcal{O}_X))$.
Using the expression in the proof of Lemma \ref{lemma-good-blowing-up}
we see
$$
ch(Lb^*\mathcal{O}_Z) =
ch(\mathcal{O}_E) \circ ch\left(
\sum\nolimits_{i = 0, \ldots, r - 1} (-1)^i\lambda^i(\alpha')
\right)
The scheme $X$ is smooth and projective over $k$ and hence we have
$K_0(X) = K_0(\textit{Vect}(X))$. See
Derived Categories of Schemes, Lemmas
\ref{perfect-lemma-resolution-property-ample} and
\ref{perfect-lemma-K-is-old-K}.
Let $\alpha \in K_0(\text{Vect}(X))$ be an element
whose restriction to $Z$ is $[\mathcal{F}]$.
By Chow Homology, Lemma \ref{chow-lemma-minus-adams-operator}
there exists an element $\alpha^\vee$ which restricts to
$\mathcal{C}_{Z/X}^\vee$. By the blow up formula
(Chow Homology, Lemma \ref{chow-lemma-blow-up-formula})
we have
$$
We have
b^![Z] = b^!i_*[Z] = j_* res(b^!)([Z]) =
j_*(c_{r - 1}(\mathcal{F}^\vee) \cap \pi^*[Z]) =
j_*(c_{r - 1}(\mathcal{F}^\vee) \cap [E])
$$
ch(\mathcal{O}_E) \cap [X'] =
ch(\mathcal{O}_{X'}) \cap [X'] - ch(\mathcal{O}_{X'}(-E)) \cap [X'] =
[E] - (1/2)[E]^2 + (1/6)[E]^3 - \ldots
where $\mathcal{F}$ is the kernel of the surjection
$\pi^*\mathcal{C}_{Z/X} \to \mathcal{C}_{E/X'}$.
Observe that $b^*\alpha^\vee - [\mathcal{O}_{X'}(E)]$
is an element of $K_0(\text{Vect}(X'))$ which
restricts to $[\pi^*\mathcal{C}_{Z/X}^\vee] - [\mathcal{C}_{E/X'}^\vee] =
[\mathcal{F}^\vee]$ on $E$. Since capping with chern classes
commutes with $j_*$ we conclude that the above is equal to
$$
We can indeed formally ``divide'' the expression above by $[E]$.
Thus it makes sense to take
c_{r - 1}(b^*\alpha^\vee - [\mathcal{O}_{X'}(E)]) \cap [E]
$$
\theta = \text{degree }r - 1\text{ part of }
ch(\sum (-1)^i\lambda^i(\alpha')) \cap
([X'] - (1/2)[E] + (1/6)[E]^2 - \ldots)
in the chow group of $X'$. Hence we see that setting
$$
Then we have $b^![Z] = [E] \cdot \theta$ by construction.
Recall that the restriction of $\alpha'$ to $E$ is $\mathcal{F}$.
Thus $j^!\theta$ is equal to the degree $r - 1$ part of
\begin{align*}
&
ch(\sum (-1)^i\wedge^i(\mathcal{F})) \cap
j^!([X'] - (1/2)[E] + (1/6)[E]^2 - \ldots)
\\
& =
((-1)^{r - 1}c_{r - 1}(\mathcal{F}) + \ldots) \cap
([E] - (1/2)j^![E] + (1/6)j^![E]^2 - \ldots) \\
& = (-1)^{r - 1}c_{r - 1}(\mathcal{F}) \cap [E] + \ldots
\end{align*}
by a computation similar to the proof of
Chow Homology, Lemma \ref{chow-lemma-compute-koszul}.
To prove that $\pi_*$ of this is equal to $[Z]$ it
\theta = c_{r - 1}(b^*\alpha^\vee - [\mathcal{O}_{X'}(E)]) \cap [X']
$$
we get the first relation $\theta \cdot [E] = b^![Z]$
for example by Chow Homology, Lemma \ref{chow-lemma-identify-chow-for-smooth}.
For the second relation observe that
$$
j^!\theta = j^!(c_{r - 1}(b^*\alpha^\vee - [\mathcal{O}_{X'}(E)]) \cap [X'])
= c_{r - 1}(\mathcal{F}^\vee) \cap j^![X'] =
c_{r - 1}(\mathcal{F}^\vee) \cap [E]
$$
in the chow groups of $E$. To prove that $\pi_*$ of this is equal to $[Z]$ it
suffices to prove that the degree of the codimension $r - 1$ cycle
$(-1)^{r - 1}c_{r - 1}(\mathcal{F}) \cap [E]$ on the fibres of $\pi$ is $1$.
This is a computation we omit.

0 comments on commit e998b8f

Please sign in to comment.
You can’t perform that action at this time.