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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