The Stacks project

Lemma 45.10.1. Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let $X, Y$ be nonempty smooth projective schemes both equidimensional of dimension $d$ over $k$. Then $\int _{X \amalg Y} = \int _ X + \int _ Y$.

Proof. Denote $i : X \to X \amalg Y$ and $j : Y \to X \amalg Y$ be the coprojections. By Lemma 45.9.9 the map $(i^*, j^*) : H^*(X \amalg Y) \to H^*(X) \times H^*(Y)$ is an isomorphism. The statement of the lemma means that under the isomorphism $(i^*, j^*) : H^{2d}(X \amalg Y)(d) \to H^{2d}(X)(d) \oplus H^{2d}(Y)(d)$ the map $\int _ X + \int _ Y$ is transformed into $\int _{X \amalg Y}$. This is true because

\[ \int _{X \amalg Y} a = \int _{X \amalg Y} i_*(i^*a) + j_*(j^*a) = \int _ X i^*a + \int _ Y j^*a \]

where the equality $a = i_*(i^*a) + j_*(j^*a)$ was shown in the proof of Lemma 45.9.9. $\square$

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