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A calculation with loc chern classes

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aisejohan committed May 29, 2019
1 parent 347d2af commit d42062dc1af0c5efa7c7306ff5cc741b3998bb33
Showing with 273 additions and 9 deletions.
  1. +229 −9 chow.tex
  2. +44 −0 perfect.tex
238 chow.tex
@@ -6666,9 +6666,9 @@ \section{Additivity of chern classes}
$c_i(\mathcal{E}) = c_i(\mathcal{F}) + c_1(\mathcal{L})c_{i - 1}(\mathcal{F})$.
By Lemma \ref{lemma-get-rid-of-trivial-subbundle}
we have $c_j(\mathcal{E} \otimes \mathcal{L}^{\otimes -1})
= c_j(\mathcal{E} \otimes \mathcal{L}^{\otimes -1})$ for
$j = 0, \ldots, r$ (were we set $c_r(\mathcal{F}) = 0$ by
convention).
= c_j(\mathcal{F} \otimes \mathcal{L}^{\otimes -1})$ for
$j = 0, \ldots, r$ were we set
$c_r(\mathcal{F} \otimes \mathcal{L}^{-1}) = 0$ by convention.
Applying Lemma \ref{lemma-chern-classes-E-tensor-L} we deduce
$$
\sum_{j = 0}^i
@@ -9759,7 +9759,14 @@ \section{Blowing up at infinity}
\item
\label{item-cone-is-open}
$E \setminus X' = E \setminus (X' \cap E) =
\underline{\text{Spec}}_Z(\mathcal{C}_{Z/X, *}) = C_ZX$.
\underline{\text{Spec}}_Z(\mathcal{C}_{Z/X, *}) = C_ZX$,
\item
\label{item-find-Z-in-blowup}
there is a closed immersion $\mathbf{P}^1_Z \to W$ whose
composition with $b$ is the inclusion morphism
$\mathbf{P}^1_Z \to \mathbf{P}^1_X$ and whose base change by $\infty$
is the composition $Z \to C_ZX \to E \to W_\infty$ where the first
arrow is the vertex of the cone.
\end{enumerate}
We recall that $\mathcal{C}_{Z/X, *}$ is the conormal algebra of $Z$ in $X$,
see Divisors, Definition \ref{divisors-definition-conormal-sheaf} and
@@ -9791,6 +9798,16 @@ \section{Blowing up at infinity}
(for example because both are cut out by the pullback of the
ideal sheaf of $Z$ to $X'$). This proves (3).

\medskip\noindent
The intersection of $\infty(Z)$ with $\mathbf{P}^1_Z$ is the effective
Cartier divisor $(\mathbf{P}^1_Z)_\infty$ hence the strict transform
of $\mathbf{P}^1_Z$ by the blowing up $b$ maps isomorphically to
$\mathbf{P}^1_Z$ (see Divisors, Lemmas \ref{divisors-lemma-strict-transform}
and \ref{divisors-lemma-blow-up-effective-Cartier-divisor}).
This gives us the morphism $\mathbf{P}^1_Z \to W$ mentioned in (8).
It is a closed immersion as $b$ is separated, see
Schemes, Lemma \ref{schemes-lemma-section-immersion}.

\medskip\noindent
Suppose that $\Spec(A) \subset X$ is an affine open and that $Z \cap \Spec(A)$
corresponds to the finitely generated ideal $I \subset A$.
@@ -9801,7 +9818,7 @@ \section{Blowing up at infinity}
of $(A[s], J)$. Observe that
$$
J^n =
I^n \oplus sI^{n - 1} \oplus sI^{n - 2} \ldots \oplus s^nA
I^n \oplus sI^{n - 1} \oplus s^2I^{n - 2} \ldots \oplus s^nA
\oplus s^{n + 1}A \oplus \ldots
$$
as an $A$-submodule of $A[s]$ for all $n \geq 0$. Consider the open subscheme
@@ -9876,13 +9893,22 @@ \section{Blowing up at infinity}
This proves $D = X'$ as desired.

\medskip\noindent
It remains to prove (6). Observe that the zero scheme of $\frac{s}{f}$
Let us prove (6). Observe that the zero scheme of $\frac{s}{f}$
in the previous paragraph is the restriction of the zero scheme of $S$
on the affine open $D_+(f^{(1)})$. Hence we see that $S = 0$ defines
$X' \cap E$ on $E$. Thus (5) follows from (6).


$X' \cap E$ on $E$. Thus (6) follows from (5).

\medskip\noindent
Finally, we have to prove the last part of (8). This is clear
because the map $\mathbf{P}^1_Z \to W$ is affine locally
given by the surjection
$$
B \to B \otimes_{A[s]} A/I =
(A/I \oplus I/I^2 \oplus I^2/I^3 \oplus \ldots)[S] \to
A/I[S]
$$
and the identification $\text{Proj}(A/I[S]) = \Spec(A/I)$.
Some details omitted.



@@ -10253,6 +10279,200 @@ \section{Higher codimension gysin homomorphisms}



\section{Calculating some classes}
\label{section-calculate}

\noindent
To get further we need to compute the values of some of the
classes we've constructed above.

\begin{lemma}
\label{lemma-compute-koszul}
Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$
be a scheme locally of finite type over $S$. Let $\mathcal{E}$
be a locally free $\mathcal{O}_X$-module of rank $r$.
Then
$$
\prod\nolimits_{n = 0, \ldots, r} c(\wedge^n \mathcal{E})^{(-1)^n} =
1 - (r - 1)! c_r(\mathcal{E}) + \ldots
$$
\end{lemma}

\begin{proof}
By the splitting principle we can turn this into a calculation in the
polynomial ring on the chern roots $x_1, \ldots, x_r$ of $\mathcal{E}$. See
Section \ref{section-splitting-principle}. Observe that
$$
c(\wedge^n \mathcal{E}) =
\prod\nolimits_{1 \leq i_1 < \ldots < i_n \leq r}
(1 + x_{i_1} + \ldots + x_{i_n})
$$
Thus the logarithm of the left hand side of the equation in the lemma is
$$
-
\sum\nolimits_{p \geq 1}
\sum\nolimits_{n = 0}^r
\sum\nolimits_{1 \leq i_1 < \ldots < i_n \leq r}
\frac{(-1)^{p + n}}{p}(x_{i_1} + \ldots + x_{i_n})^p
$$
Please notice the minus sign in front. However, we have
$$
\sum\nolimits_{p \geq 0}
\sum\nolimits_{n = 0}^r
\sum\nolimits_{1 \leq i_1 < \ldots < i_n \leq r}
\frac{(-1)^{p + n}}{p!}(x_{i_1} + \ldots + x_{i_n})^p
=
\prod (1 - e^{-x_i})
$$
Hence we see that the first nonzero term in our chern class
is in degree $r$ and equal to the predicted value.
\end{proof}

\begin{lemma}
\label{lemma-compute-section}
Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$
be a scheme locally of finite type over $S$. Let $\mathcal{C}$
be a locally free $\mathcal{O}_X$-module of rank $r$. Consider the
morphisms
$$
X = \underline{\text{Proj}}_X(\mathcal{O}_X[T])
\xrightarrow{i}
E = \underline{\text{Proj}}_X(\text{Sym}^*(\mathcal{C})[T])
\xrightarrow{\pi}
X
$$
Then $p^* \circ \pi_* \circ c_r(i_*\mathcal{O}_X) = (-1)^{r - 1}(r - 1)! j^*$
where
$j : C \to E$ and $p : C \to X$ are the inclusion and structure
morphism of the vector bundle
$C = \underline{\Spec}(\text{Sym}^*(\mathcal{C}))$.
\end{lemma}

\begin{proof}
To prove the equality it suffices to assume that $X$ is integral
and prove that both sides give the same result when capping with
$[E]$, see Lemma \ref{lemma-bivariant-zero}.
The canonical map $\pi^*\mathcal{C} \to \mathcal{O}_E(1)$ vanishes
exactly along $i(X)$. Hence the Koszul complex on the map
$$
\pi^*\mathcal{C} \otimes \mathcal{O}_E(-1) \to \mathcal{O}_E
$$
is a resolution of $i_*\mathcal{O}_X$. In particular we see that
$i_*\mathcal{O}_X$ is a perfect object of $D(\mathcal{O}_E)$
whose chern classes are defined. By Lemma \ref{lemma-compute-koszul}
we conclude
$$
c_r(i_*\mathcal{O}_X) = - (r - 1)!
c_r(\pi^*\mathcal{C} \otimes \mathcal{O}_E(-1))
$$
On the other hand, by Lemma \ref{lemma-chern-classes-dual} we have
$$
c_r(\pi^*\mathcal{C} \otimes \mathcal{O}_E(-1)) =
(-1)^r c_r(\pi^*\mathcal{C}^\vee \otimes \mathcal{O}_E(1))
$$
and $\pi^*\mathcal{C}^\vee \otimes \mathcal{O}_E(1)$ has a section
vanishing exactly along $i(X)$. Thus
$c_r(\pi^*\mathcal{C}^\vee \otimes \mathcal{O}_E(1)) \cap [E]$
is $[i(X)]$ by Lemma \ref{lemma-top-chern-class}.
The result follows easily from this; details omitted.
\end{proof}

\begin{lemma}
\label{lemma-agreement-with-loc-chern}
Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $i : Z \to X$
be a regular closed immersion of codimension $r$
between schemes locally of finite type over $S$.
Let $\mathcal{N} = \mathcal{C}_{Z/X}^\vee$ be the normal sheaf. If $X$
is quasi-compact and has the resolution property, then
$$
c_r(Z \to X, i_*\mathcal{O}_Z) = (-1)^{r - 1} (r - 1)! c(Z \to X, \mathcal{N})
$$
in $A^r(Z \to X)$. The left hand side is the localized chern class
of Definition \ref{definition-localized-chern}.
\end{lemma}

\begin{proof}
For any $x \in Z$ we can choose an affine open neighbourhood
$\Spec(A) \subset X$ such that $Z \cap \Spec(A) = V(f_1, \ldots, f_r)$
where $f_1, \ldots, f_r \in A$ is a regular sequence.
See Divisors, Definition \ref{divisors-definition-regular-immersion} and
Lemma \ref{divisors-lemma-Noetherian-scheme-regular-ideal}.
Then we see that the Koszul complex on $f_1, \ldots, f_r$ is
a resolution of $A/(f_1, \ldots, f_r)$ for example by
More on Algebra, Lemma \ref{more-algebra-lemma-regular-koszul-regular}.
Hence $A/(f_1, \ldots, f_r)$ is perfect as an $A$-module.
It follows that $F = i_*\mathcal{O}_Z$ is a perfect object of
$D(\mathcal{O}_X)$ whose restriction to $X \setminus Z$ is zero.
Since $X$ is quasi-compact and quasi-separated
(Properties, Lemma \ref{properties-lemma-locally-Noetherian-quasi-separated})
and has the resolution property we see that $F = i_*\mathcal{O}_Z$ can be
represented by a bounded complex of finite locally free modules
(Derived Categories of Schemes, Lemma
\ref{perfect-lemma-construct-strictly-perfect}) and hence
the chern classes of $F = i_*\mathcal{O}_Z$ are defined
(Definition \ref{definition-defined-on-perfect}). All in all
we conclude that the statement makes sense.

\medskip\noindent
Denote $b : W \to \mathbf{P}^1_X$ the blowing up in $\infty(Z)$
as in Section \ref{section-blowup-Z-first}. By (\ref{item-find-Z-in-blowup})
we have a closed immersion
$$
i' : \mathbf{P}^1_Z \longrightarrow W
$$
We claim that $Q = i'_*\mathcal{O}_{\mathbf{P}^1_Z}$
is a perfect object of
$D(\mathcal{O}_W)$ and that $F$ and $Q$ satisfy the assumptions of
Lemma \ref{lemma-independent-loc-chern-bQ}.

\medskip\noindent
Assume the claim. The output of Lemma \ref{lemma-independent-loc-chern-bQ}
is that we have
$$
c_p(Z \to X, F) = c'_p(Q) = (E \to Z)_* \circ c'_p(Q|_E) \circ C
$$
On the other hand, by construction of $c(Z \to X, \mathcal{N})$ and
because $\mathcal{C}_{Z/X} = \mathcal{N}^\vee$ is locally free
we have
$$
j^* \circ C = p^* \circ c(Z \to X, \mathcal{N})
$$
where $j : C_ZX \to W_\infty$ and $p : C_ZX \to Z$ are the given maps.
Thus the relationship follows from Lemma \ref{lemma-compute-section}
applied to $E \to Z$.

\medskip\noindent
Proof of the claim. Let $A$ and $f_1, \ldots, f_r$ be as above.
Consider the affine open $\Spec(A[s]) \subset \mathbf{P}^1_X$
as in Section \ref{section-blowup-Z-first}. Recall that $s = 0$
defines $(\mathbf{P}^1_X)_\infty$ over this open. Hence over
$\Spec(A[s])$ we are blowing up in the ideal generated by
the regular sequence $s, f_1, \ldots, f_r$. By More on Algebra, Lemma
\ref{more-algebra-lemma-blowup-regular-sequence} the $r + 1$
affine charts are global complete intersections over $A[s]$.
The chart corresponding to the affine blowup algebra
$$
A[s][f_1/s, \ldots, f_r/s] = A[s, y_1, \ldots, y_r]/(sy_i - f_i)
$$
contains $i'(Z \cap \Spec(A))$ as the closed subscheme cut out by
$y_1, \ldots, y_r$. Since $y_1, \ldots, y_r, sy_1 - f_1, \ldots, sy_r - f_r$
is a regular sequence in the polynomial ring $A[s, y_1, \ldots, y_r]$
we find that $i'$ is a regular immersion. Some details omitted.
As above we conclude that $Q = i'_*\mathcal{O}_{\mathbf{P}^1_Z}$
is a perfect object of $D(\mathcal{O}_W)$. Since $W$ also has
the resolution property (Derived Categories of Schemes,
Lemma \ref{perfect-lemma-resolution-property-ample-relative})
we find that the chern classes of $Q$ are defined. All the
other assumptions on $F$ and $Q$ in Lemma \ref{lemma-independent-loc-chern-bQ}
(and Lemma \ref{lemma-localized-chern-pre}) are immediately verified.
\end{proof}







\section{Gysin maps for diagonals}
\label{section-gysin-for-diagonal}

@@ -7197,6 +7197,50 @@ \section{The resolution property}
\ref{properties-proposition-characterize-ample}.
\end{proof}

\begin{lemma}
\label{lemma-resolution-property-ample-relative}
Let $f : X \to Y$ be a morphism of schemes. Assume
\begin{enumerate}
\item $Y$ is quasi-compact and quasi-separated and has the resolution property,
\item there exists an $f$-ample invertible module on $X$.
\end{enumerate}
Then $X$ has the resolution property.
\end{lemma}

\begin{proof}
Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module.
Let $\mathcal{L}$ be an $f$-ample invertible module.
Choose an affine open covering $Y = V_1 \cup \ldots \cup V_m$.
Set $U_j = f^{-1}(V_j)$. By Properties, Proposition
\ref{properties-proposition-characterize-ample}
for each $j$ we know there exists finitely many maps
$s_{j, i} : \mathcal{L}^{\otimes n_{j, i}}|_{U_j} \to \mathcal{F}|_{U_j}$
which are jointly surjective. Consider the quasi-coherent
$\mathcal{O}_Y$-modules
$$
\mathcal{H}_n =
f_*(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n})
$$
We may think of $s_{j, i}$ as a section over $V_j$ of the sheaf
$\mathcal{H}_{-n_{j, i}}$. Suppose we can find finite locally
free $\mathcal{O}_Y$-modules $\mathcal{E}_{i, j}$ and maps
$\mathcal{E}_{i, j} \to \mathcal{H}_{-n_{j, i}}$ such that
$s_{j, i}$ is in the image. Then the corresponding maps
$$
f^*\mathcal{E}_{i, j}
\otimes_{\mathcal{O}_X}
\mathcal{L}^{\otimes n_{i, j}} \longrightarrow \mathcal{F}
$$
are going to be jointly surjective and the lemma is proved. By
Properties, Lemma \ref{properties-lemma-quasi-coherent-colimit-finite-type}
for each $i, j$ we can find a finite
type quasi-coherent submodule
$\mathcal{H}'_{i, j} \subset \mathcal{H}_{-n_{j, i}}$
which contains the section $s_{i, j}$ over $V_j$.
Thus using the resolution property of $Y$ to get surjections
$\mathcal{E}_{i, j} \to \mathcal{H}'_{j, i}$ and we conclude.
\end{proof}

\begin{lemma}
\label{lemma-resolution-property-goes-up-affine}
Let $f : X \to Y$ be an affine morphism of schemes with $Y$ quasi-compact

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