The Stacks project

Lemma 45.4.9. Let $X$ be a smooth projective scheme over $k$ which is equidimensional of dimension $d$. Then $h(X)(d)$ is a left dual to $h(X)$ in $M_ k$.

Proof. We will use Lemma 45.3.1 without further mention. We compute

\[ \mathop{\mathrm{Hom}}\nolimits (\mathbf{1}, h(X) \otimes h(X)(d)) = \text{Corr}^ d(\mathop{\mathrm{Spec}}(k), X \times X) = \mathop{\mathrm{CH}}\nolimits ^ d(X \times X) \]

Here we have $\eta = [\Delta ]$. On the other hand, we have

\[ \mathop{\mathrm{Hom}}\nolimits (h(X)(d) \otimes h(X), \mathbf{1}) = \text{Corr}^{-d}(X \times X, \mathop{\mathrm{Spec}}(k)) = \mathop{\mathrm{CH}}\nolimits _ d(X \times X) \]

and here we have the class $\epsilon = [\Delta ]$ of the diagonal as well. The composition of the correspondence $[\Delta ] \otimes 1$ with $1 \otimes [\Delta ]$ either way is the correspondence $[\Delta ] = 1$ in $\text{Corr}^0(X, X)$ which proves the required diagrams of Categories, Definition 4.43.5 commute. Namely, observe that

\[ [\Delta ] \otimes 1 \in \text{Corr}^ d(X, X \times X \times X) = \mathop{\mathrm{CH}}\nolimits ^{2d}(X \times X \times X \times X) \]

is given by the class of the cycle $\text{pr}^{1234, -1}_{23}(\Delta ) \cap \text{pr}^{1234, -1}_{14}(\Delta )$ with obvious notation. Similarly, the class

\[ 1 \otimes [\Delta ] \in \text{Corr}^{-d}(X \times X \times X, X) = \mathop{\mathrm{CH}}\nolimits ^{2d}(X \times X \times X \times X) \]

is given by the class of the cycle $\text{pr}^{1234, -1}_{23}(\Delta ) \cap \text{pr}^{1234, -1}_{14}(\Delta )$. The composition $(1 \otimes [\Delta ]) \circ ([\Delta ] \otimes 1)$ is by definition the pushforward $\text{pr}^{12345}_{15, *}$ of the intersection product

\[ [\text{pr}^{12345, -1}_{23}(\Delta ) \cap \text{pr}^{12345, -1}_{14}(\Delta )] \cdot [\text{pr}^{12345, -1}_{34}(\Delta ) \cap \text{pr}^{12345, -1}_{15}(\Delta )] = [\text{small diagonal in } X^5] \]

which is equal to $\Delta $ as desired. We omit the proof of the formula for the composition in the other order. $\square$

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