The Stacks project

43.28 Pullback of cycles

Suppose that $X$ and $Y$ be nonsingular projective varieties, and let $f : X \to Y$ be a morphism. Suppose that $Z \subset Y$ is a closed subvariety. Let $f^{-1}(Z)$ be the scheme theoretic inverse image:

\[ \xymatrix{ f^{-1}(Z) \ar[r] \ar[d] & Z \ar[d] \\ X \ar[r] & Y } \]

is a fibre product diagram of schemes. In particular $f^{-1}(Z) \subset X$ is a closed subscheme of $X$. In this case we always have

\[ \dim f^{-1}(Z) \geq \dim Z + \dim X - \dim Y. \]

If equality holds in the formula above, then $f^*[Z] = [f^{-1}(Z)]_{\dim Z + \dim X - \dim Y}$ provided that the scheme $Z$ is Cohen-Macaulay at the images of the generic points of $f^{-1}(Z)$. This follows by identifying $f^{-1}(Z)$ with the scheme theoretic intersection of $\Gamma _ f$ and $X \times Z$ and using Lemma 43.16.5. Details are similar to the proof of part (4) of Lemma 43.27.1 above.

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