The Stacks project

Lemma 45.3.3. 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 22.25.1).

Proof. Everything is clear from the construction and Lemma 45.3.1 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.

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 similarly 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 numbers $a_ i$. 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 $X \times Y$ for any integral closed subscheme $Z$ of $X \times Y$. After replacing $X$ and $Y$ by the irreducible component containing the image of $Z$ 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 42.62.51 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. $\square$

[1] 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.

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