# stacks/stacks-project

Diagonal is identity correspondence

Yep!
 @@ -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}