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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.

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  • $\begingroup$ 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?) $\endgroup$
    – user3001
    Oct 7 '15 at 14:42
  • $\begingroup$ Related: mathoverflow.net/a/241907/82179 $\endgroup$ May 12 '20 at 18:47
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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.

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