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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