Example 45.3.7. Let $X = \mathbf{P}^1_ k$. Then we have

$\text{Corr}^0(X, X) = \mathop{\mathrm{CH}}\nolimits ^1(X \times X) = \mathop{\mathrm{CH}}\nolimits _1(X \times X)$

Choose a $k$-rational point $x \in X$ and consider the cycles $c_0 = [x \times X]$ and $c_2 = [X \times x]$. A computation shows that $1 = [\Delta ] = c_0 + c_2$ in $\text{Corr}^0(X, X)$ and that we have the following rules for composition $c_0 \circ c_0 = c_0$, $c_0 \circ c_2 = 0$, $c_2 \circ c_0 = 0$, and $c_2 \circ c_2 = c_2$. In other words, $c_0$ and $c_2$ are orthogonal idempotents in the algebra $\text{Corr}^0(X, X)$ and in fact we get

$\text{Corr}^0(X, X) = \mathbf{Q} \times \mathbf{Q}$

as a $\mathbf{Q}$-algebra.

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