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Another sanity check for localized chern classes

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aisejohan committed May 11, 2019
1 parent 99046ca commit a9bae57217fbe29122d8441b68e11f2485f425dc
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  1. +96 −27 chow.tex
  2. +108 −8 flat.tex
  3. +28 −0 more-algebra.tex
123 chow.tex
@@ -5215,14 +5215,28 @@ \section{Bivariant intersection theory}
on $A^*(X)$ is commutative, but we will see that chern classes live
in its center.

\begin{remark}
\label{remark-restriction-bivariant}
Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X \to Y$
and $Y' \to Y$ be morphisms of schemes locally of finite type over $S$.
Let $X' = Y' \times_Y X$. Then there is an obvious restriction map
$$
A^p(X \to Y) \longrightarrow A^p(X' \to Y'),\quad
c \longmapsto c'
$$
obtained by viewing a scheme $Y''$ locally of finite type over $Y'$
as a scheme locally of finite type over $Y$ and settting
$c' \cap \alpha'' = c \cap \alpha''$ for any $\alpha'' \in A_k(Y'')$.
This restriction operation is compatible with compositions in an
obvious manner.
\end{remark}

\begin{remark}
\label{remark-pullback-cohomology}
Let $(S, \delta)$ be as in Situation \ref{situation-setup}.
Let $f : X \to Y$ be a morphism of schemes locally of finite type over $S$.
Then there is a canonical $\mathbf{Z}$-algebra map $A^*(Y) \to A^*(X)$.
Namely, given $c \in A^p(Y)$ and $X' \to X$, then we can let $f^*c$
be defined by the map $c \cap - : A_k(X') \to A_{k - p}(X')$ which is
given by thinking of $X'$ as a scheme over $Y$.
Let $f : Y' \to Y$ be a morphism of schemes locally of finite type over $S$.
As a special case of Remark \ref{remark-restriction-bivariant}
there is a canonical $\mathbf{Z}$-algebra map $f^* : A^*(Y) \to A^*(Y')$.
\end{remark}

\begin{lemma}
@@ -7433,28 +7447,28 @@ \section{Preparation for localized chern classes}
$c' = (E' \to Z)_* \circ c'_2 \circ c'_1$
with $c'_1 \in A^0(W'_\infty \to X)$ and $c'_2 \in A^p(E' \to W'_\infty)$.
By Lemma \ref{lemma-gysin-at-infty-independent} we have
$g_{\infty, *} \circ c'_1 = c_1$. Above $E' \subset W'_\infty$
denotes the inverse image of $Z$ in $W'_\infty$, hence $E'$ is also
$g_{\infty, *} \circ c'_1 = c_1$. Here $E' \subset W'_\infty$
denotes the inverse image of $Z$ in $W'_\infty$. Hence $E'$ is also
the inverse image of $E$ in $W'_\infty$ by $g_\infty$.
Since moreover $Q' = g^*Q$ we find that $c'_2$ is simply the
restriction of $c_2$ to schemes lying over $W'_\infty$.
Thus we obtain
restriction of $c_2$ to schemes lying over $W'_\infty$, see
Remark \ref{remark-restriction-bivariant}. Thus we obtain
\begin{align*}
c' \cap \alpha
& =
(E' \to Z)_*(c'_2 \cap c'_1 \cap \alpha) \\
& =
(E \to Z)_*(E' \to E)_*(c'_2 \cap c'_1 \cap \alpha) \\
(E \to Z)_*(E' \to E)_*(c_2 \cap c'_1 \cap \alpha) \\
& =
(E \to Z)_*(c_2 \cap g_{\infty, *}(c'_1 \cap \alpha)) \\
& =
(E \to Z)_*(c_2 \cap c_1 \cap \alpha) \\
& =
c \cap \alpha
\end{align*}
In the third equality we used the relationship between $c'_2$ and $c_2$
and the fact that $c_2$ commutes with proper pushforward as it is a
bivariant class. This proves the independence.
In the third equality we used that $c_2$
commutes with proper pushforward as it is a
bivariant class.
\end{proof}


@@ -7532,10 +7546,21 @@ \section{Localized chern classes}

\begin{lemma}
\label{lemma-independent-loc-chern}
The localized classes constructed above are independent of the choices.
The localized classes constructed above are independent of choices.
\end{lemma}

\begin{proof}
\begin{proof}[First proof]
The only choice we made above was the choice of the bounded complex
$\mathcal{E}^\bullet$ of finite locally free $\mathcal{O}_X$-modules
representing $E$. However, in
More on Flatness, Lemma \ref{flat-lemma-complex-and-divisor-derived}
we have seen that the blowing up $b : W \to \mathbf{P}^1_X$
and the isomorphism class $Q$ of $\mathcal{Q}^\bullet$ in $D(\mathcal{O}_W)$
only depends on the isomorphism class of $Lp^*E$ in
$D(\mathcal{O}_{\mathbf{P}^1_X})$.
\end{proof}

\begin{proof}[Second proof]
The only choice we made above was the choice of the bounded complex
$\mathcal{E}^\bullet$ of finite locally free $\mathcal{O}_X$-modules
representing $E$. Let us temporarily denote
@@ -7563,18 +7588,10 @@ \section{Localized chern classes}
}
$$
where all morphisms are blowing ups which are isomorphisms over
$\mathbf{A}^1_X$\footnote{In fact, this step isn't necessary as
the reader can show that although the ideals defining the blowups
$b$ and $b'$ are not the same, the blowing ups $W$ and $W'$ are
isomorphic over $\mathbf{P}^1_X$. Namely, a simple local calculation
shows the ideals defining $b$
and $b'$ will be off locally on $\mathbf{P}^1_X$ by a power of the
ideal defining the effective Cartier divisor $(\mathbf{P}^1_X)_\infty$. We
won't use this, if only to demonstrate that there is a lot
of flexibility in our construction.}. In fact, the lemma shows
that $W''$ is the strict transform of $W$ with respect to the
blowing up $b'$ and $W''$ is the strict transform of $W'$
with respect to the blowing up $b$.
$\mathbf{A}^1_X$. Note that $W''$ is the strict transform of $W$
with respect to the blowing up $b'$ and $W''$ is the strict transform of $W'$
with respect to the blowing up $b$
(see Divisors, Lemma \ref{divisors-lemma-strict-transform}).

\medskip\noindent
By Lemma \ref{lemma-localized-chern-pre-independent}
@@ -7604,6 +7621,58 @@ \section{Localized chern classes}
some bounded complex of finite locally free modules, we conclude.
\end{proof}

\noindent
Here is another sanity check.

\begin{lemma}
\label{lemma-base-change-loc-chern}
In the situation above let $f : X' \to X$ be a morphism of schemes
which is locally of finite type. Denote $E' = Lf^*E$ and $Z' = f^{-1}(Z)$.
Then the bivariant class
$$
P_p(Z' \to X', E') \in A^p(Z' \to X'),
\quad\text{resp.}\quad
c_p(Z' \to X', E') \in A^p(Z' \to X')
$$
constructed above using $X', Z', E'$ is the restriction
(Remark \ref{remark-restriction-bivariant}) of the
bivariant class $P_p(Z \to X, E) \in A^p(Z \to X)$,
resp.\ $c_p(Z \to X, E) \in A^p(Z \to X)$.
\end{lemma}

\begin{proof}
Choose a bounded complex $\mathcal{E}^\bullet$ of finite locally free
$\mathcal{O}_X$-modules representing $E$. Denote
$(\mathcal{E}')^\bullet = f^*\mathcal{E}^\bullet$.
Observe that $\mathbf{P}^1_{X'} \to \mathbf{P}^1_X$ is a morphism of
schemes such that the pullback of the effective Cartier divisor
$(\mathbf{P}^1_X)_\infty$ is the effective Cartier divisor
$(\mathbf{P}^1_{X'})_\infty$. By More on Flatness, Lemma
\ref{flat-lemma-complex-and-divisor-blowup-base-change}
we obtain a commutative diagram
$$
\xymatrix{
W' \ar[rd]_{b'} \ar[r]_-g &
\mathbf{P}^1_{X'} \times_{\mathbf{P}^1_X} W \ar[d]_r \ar[r]_-q &
W \ar[d]^b \\
&
\mathbf{P}^1_{X'} \ar[r] &
\mathbf{P}^1_X
}
$$
such that $W'$ is the strict transform of $\mathbf{P}^1_{X'}$
with respect to $b$ and such that
$(\mathcal{Q}')^\bullet = g^*q^*\mathcal{Q}^\bullet$.
The restriction of the bivariant class $P_p(Z \to X, E)$,
resp.\ $c_p(Z \to X, E)$ corresponds to the class constructed in
Lemma \ref{lemma-localized-chern-pre} using the
proper morphism $r$ and the complex $q^*\mathcal{Q}^\bullet$.
On the other hand, the bivariant class $P_p(Z' \to X', E')$,
resp.\ $c_p(Z' \to X', E')$ corresponds to the
proper morphism $b'$ and the complex $(\mathcal{Q}')^\bullet$.
Thus we conclude by Lemma \ref{lemma-localized-chern-pre-independent}.
\end{proof}

\begin{definition}
\label{definition-localized-chern}
Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be a scheme
116 flat.tex
@@ -13532,7 +13532,7 @@ \section{Blowing up complexes, II}

\begin{lemma}
\label{lemma-complex-and-divisor-blowup}
In Situation \ref{situation-complex-and-divisor} let $b : X' \longrightarrow X$
In Situation \ref{situation-complex-and-divisor} let $b : X' \to X$
be the blowing up of the product of the ideals $\mathcal{J}_i$ from
Remark \ref{remark-complex-and-divisor-ideal}.
Denote $D' = b^{-1}D$ with ideal sheaf
@@ -13622,9 +13622,8 @@ \section{Blowing up complexes, II}
$D' \subset X'$, and $\mathcal{Q}^\bullet$ be as in
Lemma \ref{lemma-complex-and-divisor-blowup}.
Let $U \subset X$ be as in Lemma \ref{lemma-complex-and-divisor-blowup-good}.
Then there exists a closed subscheme $T \subset D'$ cut out by a finite
type quasi-coherent sheaf of ideals which
scheme theoretically contains $D' \cap b^{-1}(U)$
Then there exists a closed immersion $T \to D'$ of finite presentation
with $D' \cap b^{-1}(U) \subset T$ scheme theoretically
such that $\mathcal{Q}^\bullet|_T$ has finite locally free cohomology sheaves.
\end{lemma}

@@ -13657,7 +13656,7 @@ \section{Blowing up complexes, II}

\medskip\noindent
To glue this affine local construction, we remark that in the proof of
More on Algebra, Lemma \ref{more-algebra-lemma-eta-vanishing-beta}
More on Algebra, Lemma \ref{more-algebra-lemma-eta-vanishing-beta-plus}
the ideal cutting out $T$ is constructed with a certain universal
property. Namely, the result of
More on Algebra, Lemma \ref{more-algebra-lemma-eta-locally-free}
@@ -13693,6 +13692,108 @@ \section{Blowing up complexes, II}
This is obvious.
\end{proof}

\begin{lemma}
\label{lemma-complex-and-divisor-derived}
Let $X$ be a scheme and let $D \subset X$ be an effective Cartier divisor. Let
$E \in D(\mathcal{O}_X)$ be a perfect object. Let $U \subset X$ be the maximal
open over which the cohomology sheaves $H^i(E)$ are locally free.
There exists a proper morphism
$b : X' \longrightarrow X$ and an object $Q \in D(\mathcal{O}_{X'})$
with the following properties
\begin{enumerate}
\item $D' = b^{-1}D$ is an effective Cartier divisor,
\item $Q = L\eta_{\mathcal{I}'}Lb^*E$ where $\mathcal{I}'$
is the ideal sheaf of $D'$,
\item $Q$ is a perfect object of $D(\mathcal{O}_{X'})$,
\item there exists a closed immersion $T \to D'$ of finite presentation
with $D' \cap b^{-1}(U) \subset T$ scheme theoretically such that
$Q|_T$ has finite locally free cohomology sheaves,
\item for any open subscheme $V \subset X$ such that
$E|_V$ can be represented by a bounded complex $\mathcal{E}^\bullet$
of finite locally free $\mathcal{O}_V$-modules, the base changes
of $X' \to X$, $Q$, $D'$, and $T$ to $V$ are given by the
constructions of Lemmas \ref{lemma-complex-and-divisor-blowup} and
\ref{lemma-complex-and-divisor-blowup-T}.
\end{enumerate}
\end{lemma}

\begin{proof}
We first construct the morphism $b : X' \to X$ by glueing the blowings up
constructed over opens $V \subset X$ as in (5). By
Constructions, Lemma \ref{constructions-lemma-relative-glueing}
to do this it suffices to show that given $V \subset X$ open
and two bounded complexes $\mathcal{E}^\bullet$
and $(\mathcal{E}')^\bullet$ of finite locally free $\mathcal{O}_V$-modules
representing $E|_V$ the resulting blowing ups are canonically isomorphic.
To do this, it suffices, by the universal property of blowing up
of Divisors, Lemma \ref{divisors-lemma-universal-property-blowing-up},
to show that the ideals $\mathcal{J}_i$ and $\mathcal{J}'_i$ from
Remark \ref{remark-complex-and-divisor-ideal}
constructed using $\mathcal{E}^\bullet$ and $(\mathcal{E}')^\bullet$
locally differ by multiplication by an invertible ideal.
We will in fact show that they differ locally by a power of the
ideal sheaf $\mathcal{I}$ of $D$. By More on Algebra, Lemma
\ref{more-algebra-lemma-compare-representatives-perfect}
working locally it suffices to prove the relationship when
$$
(\mathcal{E}')^\bullet =
\mathcal{E}^\bullet \oplus ( \ldots \to 0 \to
\mathcal{O}_V \xrightarrow{1} \mathcal{O}_V \to 0 \to \ldots)
$$
with the two summands $\mathcal{O}_V$ placed in degrees $i$ and $i + 1$ say.
Computing minors explicitly one finds that
$\mathcal{J}'_{i + 1} = \mathcal{I}\mathcal{J}_{i + 1}$
and all other ideals stay the same.

\medskip\noindent
Thus we have the morphism $b : X' \to X$ agreeing locally with the
blowing ups in (5). Of course this immediately gives us the
effective Cartier divisor $D' = b^{-1}D$, its invertible ideal sheaf
$\mathcal{I}'$ and the object $Q = L\eta_{\mathcal{I}'}Lb^*E$.
See Remark \ref{remark-Leta} for the construction of $L\eta_{\mathcal{I}'}$.
Since the construction commutes with restricting to opens we find
that $Q|_{V'}$ is represented by the complex
$\mathcal{Q}^\bullet$ over the open $V' = b^{-1}(V)$ constructed
using $\mathcal{E}^\bullet$ over $V$.

\medskip\noindent
To finish the proof it suffices to show that the closed subschemes
$T_V \subset V'$ constructed in Lemma \ref{lemma-complex-and-divisor-blowup-T}
glue. Again by relative glueing, it suffices to show that the construction
of $T$ does not depend on the choice of the complex $\mathcal{E}^\bullet$
representing $E|_V$. Again we reduce to the case where
$$
(\mathcal{E}')^\bullet =
\mathcal{E}^\bullet \oplus ( \ldots \to 0 \to
\mathcal{O}_V \xrightarrow{1} \mathcal{O}_V \to 0 \to \ldots)
$$
with the two summands $\mathcal{O}_V$ placed in degrees $i$ and $i + 1$ say.
Note that in this case $(\mathcal{Q}')^\bullet$ and $\mathcal{Q}^\bullet$
differ as follows
$$
(\mathcal{Q}')^\bullet = \mathcal{Q}^\bullet \oplus
( \ldots \to 0 \to
(\mathcal{I}')^{i + 1}|_{V'} \xrightarrow{1}
(\mathcal{I}')^{i + 1}|_{V'} \to 0 \to \ldots)
$$
In the proof of Lemma \ref{lemma-complex-and-divisor-blowup-T}
we defined $T \subset D'$ as the largest closed subscheme of $D'$ such
that $\mathcal{Q}^i|_T$ is a direct sum of two parts compatible with
the restriction to $T$ of the canonical split injective maps
$$
c^i :
\mathcal{Q}^i
\longrightarrow
(\mathcal{I}')^ib^*\mathcal{E}^i \oplus
(\mathcal{I}')^{i + 1}b^*\mathcal{E}^{i + 1}
$$
for all $i$. The direct sum decomposition for $(\mathcal{Q}')^\bullet$
in terms of $\mathcal{Q}^\bullet$ and the explicit complex
$(\mathcal{I}')^{i + 1}|_{V'} \to (\mathcal{I}')^{i + 1}|_{V'}$
implies in a straightforward manner that $T$ plays the same
role for $(\mathcal{Q}')^\bullet$ and the proof is complete.
\end{proof}




@@ -13731,11 +13832,10 @@ \section{Blowing up complexes, III}
$(\mathbf{P}^1_X)_\infty \subset \mathbf{P}^1_X$ and the bounded complex
$p^*\mathcal{E}^\bullet$ of finite locally free modules. We also denote
$$
\mathcal{Q}^\bullet = \eta_\infty b^*p^*\mathcal{E}^\bullet
\mathcal{Q}^\bullet = \eta_\mathcal{I} b^*p^*\mathcal{E}^\bullet
$$
the complex considered in Lemma \ref{lemma-complex-and-divisor-blowup}
where $\eta_\infty = \eta_\mathcal{I}$ is the operator of
Section \ref{section-eta}
where $\eta_\mathcal{I}$ is the operator of Section \ref{section-eta}
associated to the ideal sheaf $\mathcal{I}$
of the effective Cartier divisor $W_\infty = b^{-1}(\mathbf{P}^1_X)_\infty$
on $W$.
Some details omitted.
\end{proof}

\begin{lemma}
\label{lemma-compare-representatives-perfect}
Let $R$ be a ring. Let $\mathfrak p \subset R$ be a prime. Let $M^\bullet$
and $N^\bullet$ be bounded complexes of finite projective $R$-modules
representing the same object of $D(R)$. Then there exists an $f \in R$,
$f \not \in \mathfrak p$ such that there is an isomorphism (!)
of complexes
$$
M^\bullet_f \oplus P^\bullet \cong N^\bullet_f \oplus Q^\bullet
$$
where $P^\bullet$ and $Q^\bullet$ are finite direct sums of
trivial complexes, i.e., complexes of the form
the form $\ldots \to 0 \to R_f \xrightarrow{1} R_f \to 0 \to \ldots$
(placed in arbitrary degrees).
\end{lemma}

\begin{proof}
If we have an isomorphism of the type described over the localization
$R_\mathfrak p$, then using that $R_\mathfrak p = \colim R_f$
(Algebra, Lemma \ref{algebra-lemma-localization-colimit}) we can
descend the isomorphism to an isomorphism over $R_f$ for some $f$.
Thus we may assume $R$ is local and $\mathfrak p$ is the maximal ideal.
In this case the result follows from the uniqueness of a ``minimal''
complex representing a perfect object, see Lemma \ref{lemma-lift-pseudo-coherent-from-residue-field}, and
the fact that any complex is a direct sum of a trivial complex
and a minimal one (Algebra, Lemma \ref{algebra-lemma-add-trivial-complex}).
\end{proof}

\begin{lemma}
\label{lemma-lift-complex-finite-projectives}
Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $E^\bullet$

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