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The polynomial extension $R \rightarrow R[X]$ ($X$ an indeterminate) has many nice properties beyond faithful flatness. The one I'm most interested in at the moment is the following. Say that a flat $R$-algebra $S$ is intersection flat (terminology due to Hochster?) or a content module (terminology due to Ohm and Rush), or an algebra that has content (terminology of Eakin and Silver) if for any collection $\{I_\alpha\}_{\alpha \in \Lambda}$ of ideals $I_\alpha$ of $R$, one has $\bigcap_{\alpha \in \Lambda} (I_\alpha S) = (\bigcap_{\alpha \in \Lambda} I_\alpha)S$.

For this to make sense, of course, you don't need $S$ to be an $R$-algebra; it can be an $R$-module.

Does this property have any meaning in algebro-geometric terms? I ask because I want examples of it beyond the case where $S$ is projective as an $R$-module, or $S=R[[X]]$, or various kinds of localizations of same. Actually I want examples that satisfy some additional properties (e.g. I also require that primes extend to primes), but I'm probably getting ahead of myself.

By the way, I'm looking for pretty general constructions that will work over a large class of base rings. Any examples where $R$ has to be a field or a PID are useless to me, for instance.

Thanks!

EDIT: Maybe I should exhibit some additional cases where this condition holds. It holds, for instance, whenever $R$ is complete Noetherian and $R \rightarrow S$ is a flat local homomorphism (by Chevalley's theorem). Also, it always holds for the extension $R \rightarrow R[[x]]$ (though there, primes do not typically exend to primes), and when $R=k[X_1, \dotsc, X_n]$, with $k$ a perfect field of prime characteristic, then the Frobenius endomorphism is intersection flat (though there again, primes almost never extend to primes).

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As you may know, if M is flat and Mittag-Leffler as an R-module, then M satisfies the condition you are looking for. This follows via some diagram chasing from Proposition Tag 059M. This reminded me of the paper by Raynaud and Gruson. For example, they show, if R ---> S is smooth with geometrically irreducible fibres, then S is projective as an R-module. They generalize this to cases where \Spec(S) ---> \Spec(R) is relatively pure (and S is flat and of finite presentation over R). Maybe one could relate a version of purity of the morphism \Spec(S) ---> \Spec(R) to your condition (you'd have to modify the definitions to make it work for the level of generality you seem to want to work with).

Purity is discussed starting with Section Tag 05IW of the Stacks project. Of course I strongly urge you to read the original discussion of this in the paper by Raynaud and Gruson if you have not already done so (you can find the precise reference at the end of the corresponding chapter of the Stacks project).

Finally, I'd like to briefly mention that purity of a morphism X ---> Y means very roughly that if x is a generic point of an embedded component of the scheme theoretic fibre of X over y and if y -> y' is a specialization, then x specializes to a point of X lying over y'. This does seem like a type of condition that morally speaking might (jointly with flatness) imply something like your "content" condition.

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  • $\begingroup$ There are some interesting ideas here! It's not exactly geometric, except for the purity stuff which is only a guess on your part. Thus, I'm not accepting it as the answer. But it is at least potentially useful, so +1, and thank you! $\endgroup$ – Neil Epstein Dec 23 '13 at 18:17

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