The Stacks project

Lemma 45.9.6. Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let $i : X \to Y$ be a closed immersion of nonempty smooth projective equidimensional schemes over $k$. Then $\gamma ([X]) = i_*1$ in $H^{2c}(Y)(c)$ where $c = \dim (Y) - \dim (X)$.

Proof. This is true because $1 = \gamma ([X])$ in $H^0(X)$ by Lemma 45.9.5 and then we can apply axiom (C)(b). $\square$

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