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Diagonal is identity correspondence

Yep!
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aisejohan committed Aug 17, 2019
1 parent 20c8c7a commit eea7c43f33710c598b6a5a3b7546f5fbaa07c772
Showing with 77 additions and 9 deletions.
  1. +77 −9 weil.tex
@@ -642,14 +642,62 @@ \section{Correspondences}
\end{example}

\begin{lemma}
\label{lemma-identity-correspondence}
Let $X$ be a smooth projective scheme over $k$. Consider the diagonal
$\Delta \subset X \times X$ of $X$. The class of $\Delta$ is an
unit element of the graded algebra $\text{Corr}^*(X, X)$.
\label{lemma-category-correspondences}
Smooth projective schemes over $k$ with correspondences and composition
of correspondences as defined above form a graded category over
$\mathbf{Q}$
(Differential Graded Algebra, Definition \ref{dga-definition-graded-category}).
\end{lemma}

\begin{proof}
Omitted.
Everything is clear from the construction and
Lemma \ref{lemma-composition-correspondences}
except for the existence of identity morphisms.
Given a smooth projective scheme $X$ consider
the class $[\Delta]$ of the diagonal $\Delta \subset X \times X$
in $\text{Corr}^0(X, X)$. We note that $\Delta$ is
equal to the graph of the identity $\text{id}_X : X \to X$
which is a fact we will use below.

\medskip\noindent
To prove that $[\Delta]$ can serve as an identity we have to show that
$[\Delta] \circ c = c$ and $c' \circ [\Delta] = c'$ for any correspondences
$c \in \text{Corr}^r(Y, X)$ and $c' \in \text{Corr}^s(X, Y)$.
For the second case we have to show that
$$
c' = \text{pr}_{13, *}(\text{pr}_{12}^*[\Delta] \cdot \text{pr}_{23}^*c')
$$
where $\text{pr}_{12} : X \times X \times Y \to X \times X$ is the
projection and simlarly for $\text{pr}_{13}$ and $\text{pr}_{23}$.
We may write $c' = \sum a_i [Z_i]$ for some integral closed subschemes
$Z_i \subset X \times Y$ and rational numers $a_i$. Thus it clearly
suffices to show that
$$
[Z] = \text{pr}_{13, *}(\text{pr}_{12}^*[\Delta] \cdot \text{pr}_{23}^*[Z])
$$
in the chow group of $X \times Y$. After replacing $X$ and $Y$ by the
irreducible component containing the image of $X$ under the two projections
we may assume $X$ and $Y$ are integral as well. Then we have to show
$$
[Z] = \text{pr}_{13, *}([\Delta \times Y] \cdot [X \times Z])
$$
Denote $Z' \subset X \times X \times Y$ the image of $Z$ by the morphism
$(\Delta, 1) : X \times Y \to X \times X \times Y$. Then $Z'$
is a closed subscheme of $X \times X \times Y$ isomorphic to $Z$ and
$Z' = \Delta \times Y \cap X \times Z$ scheme theoretically.
By Chow Homology, Lemma \ref{chow-lemma-intersect-properly}\footnote{The
reader verifies that $\dim(Z') = \dim(\Delta \times Y) + \dim(X \times Z) -
\dim(X \times X \times Y)$ and that $Z'$ has a unique generic point
mapping to the generic point of $Z$ (where the local ring is CM)
and to some point of $X$ (where the local ring is CM). Thus all the
hypothese of the lemma are indeed verified.}
we conclude that
$$
[Z'] = [\Delta \times Y] \cdot [X \times Z]
$$
Since $Z'$ maps isomorphically to $Z$ by $\text{pr}_{13}$ also
we conclude. The verification that
$[\Delta] \circ c = c$ is similar and we omit it.
\end{proof}

\begin{lemma}
@@ -661,10 +709,30 @@ \section{Correspondences}
\end{lemma}

\begin{proof}
To see this we have to show if $g : Z \to Y$ is another morphism of
smooth projective schemes over $k$, then we have
$[\Gamma_g] \circ [\Gamma_f] = [\Gamma_{g \circ f}]$ in
$\text{Corr}^0(X, Z)$. Details omitted.
In the proof of Lemma \ref{lemma-category-correspondences}
we have seen that this construction sends identities to
identities. To finish the proof we have to show if $g : Z \to Y$
is another morphism of smooth projective schemes over $k$, then we have
$[\Gamma_g] \circ [\Gamma_f] = [\Gamma_{f \circ g}]$ in
$\text{Corr}^0(X, Z)$. Arguing as in the proof of
Lemma \ref{lemma-category-correspondences} we see that it
suffices to show
$$
[\Gamma_{f \circ g}] =
\text{pr}_{13, *}([\Gamma_f \times Z] \cdot [X \times \Gamma_g])
$$
in $\CH^*(X \times Z)$ when $X$, $Y$, $Z$ are integral.
Denote $Z' \subset X \times Y \times Z$ the image of the closed immersion
$(f \circ g, g, 1) : Z \to X \times Y \times Z$.
Then $Z' = \Gamma_f \times Z \cap X \times \Gamma_g$
scheme theoretically and we conclude using
Chow Homology, Lemma \ref{chow-lemma-intersect-properly}
that
$$
[Z'] = [\Gamma_f \times Z] \cdot [X \times \Gamma_g]
$$
Since it is clear that $\text{pr}_{13, *}([Z']) = [\Gamma_{f \circ g}]$
the proof is complete.
\end{proof}

\begin{remark}

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