# Endomorphism of Chow group induced by a birational map

Let $\phi:X\dashrightarrow Y$ be a birational map between smooth projective $k$-varieties ($k=\bar k$) and $\Gamma$ be the closure of the graph of $\phi$. In Fulton's intersection theory example 16.1.11, it is said that $^t\Gamma\circ\Gamma$ is the sum of the identity correspondence and correspondences whose projections are contained in proper subvarieties of $Y$ but I cannot see why it is (formally) true.

• We see that the restriction of $^t\Gamma\circ \Gamma$ to $U\times U$ (where $U$ is an open subset on which $\phi$ is a morphism) at like $\phi_*\phi^*$ on the Chow ring of $X$ but a priori the homomorphism from correspondence to endomorphism of Chow ring is not injective (isn't it?) Oct 7 '15 at 14:42
• Related: mathoverflow.net/a/241907/82179 May 12 '20 at 18:47

Let $$U$$ be the largest open subset of $$X$$ where $$\phi$$ is defined. Let $$d=dim X$$. Then the sequence $$CH_d(X\times X-U\times U)\to CH_d(X\times X)\to CH_d(U\times U)\to 0$$ is exact. Now consider the cycle $$\alpha:=\Gamma^t\circ\Gamma-\Delta$$ on $$X\times X$$. Here $$\Delta$$ is the diagonal. Its restriction on $$U\times U$$ is 0. Hence $$\alpha$$ is supported on the complement $$X\times X-U\times U$$. Thus we get the required assertion.