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Given a morphism (not necessarily flat) $f \colon X \rightarrow Y$ between smooth varieties over $\mathbb{C}$ say, we have a pull-back homomophism of graded rings $f^* \colon A^*(Y) \rightarrow A^*(X)$ (See Fulton, 8.1 - 8.3). My question is:

Under what conditions can it be said of a closed subscheme (or subvariety) $Z \subset Y$ that $f^*([Z]) = [f^{-1}(Z)]$? I'm expecting some sort of transversality condition...

In Stacks, 0B0I, it's stated that this holds whenever $f^{-1}(Z)$ is of the expected codimension in $X$ AND in addition "$Z$ is Cohen-Macaulay at the images of the generic points of $f^{-1}(Z)$". Can we find a reference or proof for this fact, ideally in Fulton?

Can this latter condition (i.e., Cohen-Macaulayness, etc.) be shown to be superfluous if we assume in addition, say, that $f$ is a closed immersion and $f^{-1}(Z)$ is integral and smooth?

Thanks!

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    $\begingroup$ I'm not sure he ever explicitly puts the pieces together but I think that by looking at the graph of $f$ and combining some of the CM statements from Ch 7 and transversality statements from Ch 8 it should be doable with Fulton. I can try to find the precise numbers if you want. I know the new Eisenbud-Harris book uses this result but I don't think they prove it. $\endgroup$
    – Hoot
    Aug 22, 2016 at 23:26
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    $\begingroup$ Yes, in fact 3624 is definitely the way to go here. In fact, Thm 1.23 is exactly what I was looking for. Shocking how abstruse Fulton is! I don't know how anyone digests that book. $\endgroup$
    – BD107
    Aug 23, 2016 at 4:39

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