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K_0(Coh) otimes Q equals chow

First case we can prove directly using localized chern classes
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aisejohan committed May 30, 2019
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  1. +198 −16 chow.tex
  2. +12 −0 perfect.tex
214 chow.tex
@@ -10672,26 +10672,28 @@ \section{Calculating some classes}
\xrightarrow{\pi}
X
$$
Then $p^* \circ \pi_* \circ c_r(i_*\mathcal{O}_X) = (-1)^{r - 1}(r - 1)! j^*$
Then $c_t(i_*\mathcal{O}_X) = 0$ for $t = 1, \ldots, r - 1$ and in
$A^0(C \to E)$ we have
$$
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
whose chern classes are defined. The vanishing of $c_t(i_*\mathcal{O}_X)$
for $t = 1, \ldots, t - 1$ follows from Lemma \ref{lemma-compute-koszul}.
This lemma also gives
$$
c_r(i_*\mathcal{O}_X) = - (r - 1)!
c_r(\pi^*\mathcal{C} \otimes \mathcal{O}_E(-1))
@@ -10701,10 +10703,30 @@ \section{Calculating some classes}
c_r(\pi^*\mathcal{C} \otimes \mathcal{O}_E(-1)) =
(-1)^r c_r(\pi^*\mathcal{C}^\vee \otimes \mathcal{O}_E(1))
$$
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}.
and $\pi^*\mathcal{C}^\vee \otimes \mathcal{O}_E(1)$ has a section $s$
vanishing exactly along $i(X)$.

\medskip\noindent
After replacing $X$ by a scheme locally of finite type over $X$,
it suffices to prove that both sides of the equality have the
same effect on an element $\alpha \in A_*(E)$. Since $C \to X$
is a vector bundle, every cycle class on $C$ is of the form $p^*\beta$
for some $\beta \in A_*(X)$ (Lemma \ref{lemma-vectorbundle}).
Hence by Lemma \ref{lemma-restrict-to-open}
we can write $\alpha = \pi^*\beta + \gamma$ where $\gamma$
is supported on $E \setminus C$. Using the equalities above
it suffices to show that
$$
p^*(\pi_*(c_r(\pi^*\mathcal{C}^\vee \otimes \mathcal{O}_E(1)) \cap [W])) =
j^*[W]
$$
when $W \subset E$ is an integral closed subscheme which
is either (a) disjoint from $C$ or (b) is of the form $W = \pi^{-1}Y$
for some integral closed subscheme $Y \subset X$.
Using the section $s$ and Lemma \ref{lemma-top-chern-class} we find
in case (a) $c_r(\pi^*\mathcal{C}^\vee \otimes \mathcal{O}_E(1)) \cap [W] = 0$
and in case (b)
$c_r(\pi^*\mathcal{C}^\vee \otimes \mathcal{O}_E(1)) \cap [W] = [i(Y)]$.
The result follows easily from this; details omitted.
\end{proof}

@@ -10715,10 +10737,14 @@ \section{Calculating some classes}
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
$c_t(Z \to X, i_*\mathcal{O}_Z) = 0$ for $t = 1, \ldots, r - 1$ and
$$
c_r(Z \to X, i_*\mathcal{O}_Z) = (-1)^{r - 1} (r - 1)! c(Z \to X, \mathcal{N})
\quad\text{in}\quad
A^r(Z \to X)
$$
in $A^r(Z \to X)$. The left hand side is the localized chern class
where $c_t(Z \to X, i_*\mathcal{O}_Z)$
is the localized chern class
of Definition \ref{definition-localized-chern}.
\end{lemma}

@@ -10762,15 +10788,22 @@ \section{Calculating some classes}
$$
c_p(Z \to X, F) = c'_p(Q) = (E \to Z)_* \circ c'_p(Q|_E) \circ C
$$
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
for all $p \geq 1$. Observe that $Q|_E$ is equal to the pushforward of
the structure sheaf of $Z$ via the morphism $Z \to E$ which is the
base change of $i'$ by $\infty$.
Thus the vanishing of $c_t(Z \to X, F)$ for $1 \leq t \leq r - 1$
by Lemma \ref{lemma-compute-section} applied to $E \to Z$.
Because $\mathcal{C}_{Z/X} = \mathcal{N}^\vee$
is locally free the bivariant class $c(Z \to X, \mathcal{N})$
is characterized by the relation
$$
j^* \circ C = p^* \circ c(Z \to X, \mathcal{N})
$$
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 $E \to Z$.
(Recall $C \in A^0(W_\infty \to X)$ is the class of
Lemma \ref{lemma-gysin-at-infty}.)
Thus the displayed equation in the statement of the lemma
follows from the corresponding equation in Lemma \ref{lemma-compute-section}.

\medskip\noindent
Proof of the claim. Let $A$ and $f_1, \ldots, f_r$ be as above.
@@ -10798,6 +10831,155 @@ \section{Calculating some classes}
(and Lemma \ref{lemma-localized-chern-pre}) are immediately verified.
\end{proof}

\begin{lemma}
\label{lemma-actual-computation}
In the situation of Lemma \ref{lemma-agreement-with-loc-chern}
say $\dim_\delta(X) = n$. Then we have
\begin{enumerate}
\item $c_t(Z \to X, i_*\mathcal{O}_Z) \cap [X]_n = 0$ for
$t = 1, \ldots, r - 1$,
\item $c_r(Z \to X, i_*\mathcal{O}_Z) \cap [X]_n =
(-1)^{r - 1}(r - 1)![Z]_{n - r}$,
\item $ch_t(Z \to X, i_*\mathcal{O}_Z) \cap [X]_n = 0$ for
$t = 0, \ldots, r - 1$, and
\item $ch_r(Z \to X, i_*\mathcal{O}_Z) \cap [X]_n = [Z]_{n - r}$.
\end{enumerate}
\end{lemma}

\begin{proof}
Parts (1) and (2) follow immediately from
Lemma \ref{lemma-agreement-with-loc-chern}
combined with Lemma \ref{lemma-gysin-fundamental}.
Then we deduce parts (3) and (4) using the relationship
between $ch_p = (1/p!)P_p$ and $c_p$ given in
Lemma \ref{lemma-loc-chern-character}. (Namely,
$(-1)^{r - 1}(r - 1)!ch_r = c_r$ provided
$c_1 = c_2 = \ldots = c_{r - 1} = 0$.)
\end{proof}








\section{Chow groups and K-groups revisited}
\label{section-chow-and-K-II}

\noindent
This section is the continuation of Section \ref{section-chow-and-K}.
Let $(S, \delta)$ be as in Situation \ref{situation-setup}.
Let $X$ be locally of finite type over $S$. The K-group
$K_0(\textit{Coh}(X))$ has a canonical increasing filtration
$$
F_kK_0(\textit{Coh}(X)) =
\Im(K_0(\textit{Coh}_{\leq k}(X)) \to K_0(\textit{Coh}(X)))
$$
This is called the filtration by dimension of supports. Observe that
$$
\text{gr}_k K_0(\textit{Coh}(X)) \subset
K_0(\textit{Coh}(X))/F_{k - 1}K_0(\textit{Coh}(X)) =
K_0(\textit{Coh}(X)/\textit{Coh}_{\leq k - 1}(X))
$$
where the equality holds
by Homology, Lemma \ref{homology-lemma-serre-subcategory-K-groups}.
The discussion in Remark \ref{remark-good-cases-K-A} shows
that there are canonical maps
$$
A_k(X) \longrightarrow \text{gr}_k K_0(\textit{Coh}(X))
$$
defined by sending the class of an integral closed subscheme
$Z \subset X$ of $\delta$-dimension $k$ to the class of
$[\mathcal{O}_Z]$ on the right hand side.

\begin{proposition}
\label{proposition-K-tensor-Q}
Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Assume given a
closed immersion $i : X \to Y$ of schemes locally of finite type over $S$
with $Y$ regular, quasi-compact, affine diagonal, and
$\delta_{Y/S} : Y \to \mathbf{Z}$ bounded. Then the map
$$
A_k(X) \otimes \mathbf{Q}
\longrightarrow
\text{gr}_k K_0(\textit{Coh}(X)) \otimes \mathbf{Q}
$$
(see above) is an isomorphism of $\mathbf{Q}$-vector spaces.
\end{proposition}

\begin{proof}
We have the resolution property for $Y$ by Derived Categories of Schemes, Lemma
\ref{perfect-lemma-regular-resolution-property}. Thus every perfect object
$E$ of $D(\mathcal{O}_Y)$ can be represented by a bounded complex of finite
locally free modules, see Derived Categories of Schemes, Lemma
\ref{perfect-lemma-resolution-property-perfect-complex}. Hence the chern
classes of $E$ are defined (Definition \ref{definition-defined-on-perfect}).
Finally, because $\delta : Y \to \mathbf{Z}$ is bounded,
the perfect objects of $D(\mathcal{O}_Y)$ are exactly given by the objects of
$D^b_{\textit{Coh}}(\mathcal{O}_Y)$, see
Derived Categories of Schemes, Lemma \ref{perfect-lemma-perfect-on-regular}.
Please keep these facts in mind below.

\medskip\noindent
Given a coherent $\mathcal{O}_X$-module $\mathcal{F}$ we can
consider the localized chern class
$$
ch(X \to Y, i_*\mathcal{F}) \in A^*(X \to Y) \otimes \mathbf{Q}
$$
See Definition \ref{definition-localized-chern}
(and comments in the first paragraph).
This construction is additive in short exact sequences, see
Lemma \ref{lemma-additivity-loc-chern-P}. Hence we obtain
$$
\Psi : K_0(\textit{Coh}(X)) \longrightarrow A_*(X) \otimes \mathbf{Q},\quad
[\mathcal{F}] \longmapsto ch(X \to Y, i_*\mathcal{F}) \cap [Y]
$$
where $[Y] = \sum [Y_i] \in A_*(Y)$ is the sum of the classes of the
irreducible components of $Y$.
If $\mathcal{F}$ is (set theoretically) supported on a closed subscheme
$Z \subset X$, then we have
$$
ch(X \to Y, i_*\mathcal{F}) = (Z \to X)_* \circ ch(Z \to Y, i_*\mathcal{F})
$$
by Lemma \ref{lemma-loc-chern-shrink-Z}. We conclude that
$\Psi$ sends $F_kK_0(\textit{Coh}(X))$ into
$\bigoplus_{k' \leq k} A_{k'}(X) \otimes \mathbf{Q}$.
This already defines a map
$$
\Psi_k :
\text{gr}_k K_0(\textit{Coh}(X))
\longrightarrow
A_k(X) \otimes \mathbf{Q}
$$
for every $k \in \mathbf{Z}$. Thus to finish the proof, it suffices to show
that given an integral closed subscheme $Z \subset X$ of $\delta$-dimension $k$
we have $\Psi_k([\mathcal{O}_Z]) = [Z]$ in $A_k(X)$. By the
above, it suffices to show that
$$
ch(Z \to Y, i_*\mathcal{O}_Z) \cap [Y] = [Z]
\quad\text{in}\quad
A_k(Z) \otimes \mathbf{Q}
$$
Since $A_k(Z) = \mathbf{Z}$, in order to prove this we may replace $Y$
by an open neighbourhood of the generic point $\xi$ of $Z$. Since the maximal
ideal of the regular local ring $\mathcal{O}_{X, \xi}$ is generated by a
regular sequence (Algebra, Lemma \ref{algebra-lemma-regular-ring-CM})
we may assume the ideal of $Z$ is generated by a regular sequence, see
Divisors, Lemma \ref{divisors-lemma-Noetherian-scheme-regular-ideal}.
Thus we deduce the result from Lemma \ref{lemma-actual-computation}.
\end{proof}

\begin{remark}
\label{remark-K-tensor-Q}
In the situation of Proposition \ref{proposition-K-tensor-Q}
the map $\Psi$ of the proof defines an isomorphism
$$
K_0(\textit{Coh}(X)) \otimes \mathbf{Q} \cong A_*(X) \otimes \mathbf{Q}
$$
This isomorphism is not canonical: it depends on the choice of the
embedding $X \to Y$ and not just on $X$ itself.
\end{remark}




@@ -2015,6 +2015,18 @@ \section{Derived category of coherent modules}
\ref{lemma-direct-image-coherent-bdd-below}.
\end{proof}

\begin{lemma}
\label{lemma-perfect-on-regular}
Let $X$ be a Noetherian regular scheme of finite dimension. Then
every object of $D^b_{\textit{Coh}}(\mathcal{O}_X)$ is perfect
and conversely every perfect object of $D(\mathcal{O}_X)$
is in $D^b_{\textit{Coh}}(\mathcal{O}_X)$.
\end{lemma}

\begin{proof}
Combine More on Algebra, Lemma \ref{more-algebra-lemma-regular-perfect}
with Lemma \ref{lemma-perfect-affine}.
\end{proof}



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