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I know that the following question is somewhat vague; I'm trying to understand what flatness "buys".

Suppose $A, B$ are rings (commutative with identity) with $A$ a PID. Suppose we are given a flat homomorphism $\rho: A \to B$. What can be inferred about the ideal structure of $B$?

Recall $\rho$ being flat means that if $M \to N$ is an injection of $A$-modules, then $M \otimes B \to N \otimes B$ is also injective.

Example $\mathbb{Z}$ is a PID and so is $\mathbb{Z}_p$. The canonical homomorphism $\mathbb{Z} \to \mathbb{Z}_p$ is flat (a general result on formal completion of Noetherian rings). Concidence? A similar remark applies to $\mathbb{Z} \to \mathbb{Z}^n$ (diagonal embedding). [Of course both of these statements are very easy to prove, and I have no idea how to use flatness here...]

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I think very bad things may happen. For example take a PID $A$ and a domain $B$ such that $A \subseteq B$. Since $A$ is a PID, $B$ is flat over $A$ because it is torsion-free. – Andrea Aug 18 '11 at 16:18
thanks, good point – Tom Bachmann Aug 18 '11 at 16:49
up vote 4 down vote accepted

As Andrea observes above, the condition you list is quite weak: every domain is flat over a PID (either a prime field or $\mathbb{Z}$). Consequently not too much can be inferred about the ideal structure, as integral domains tend to vary pretty wildly...

But since your question is about the intuitive idea behind flatness, let me try to offer a few random comments. It seems that your interest is in, given a flat morphism $A \to B$, deducing properties of $B$ from those of $A$. In general, these tend to be somewhat restricted unless you make further hypotheses.

For instance, there is the notion of an etale morphism: these are those that are flat, of finite presentation, and satisfy $\Omega_{B/A} = 0$ (i.e. any $A$-derivation of $B$ is trivial). This means that $B$ is close to $A$ in some sense, certainly in a much stronger sense than if $B$ was simply flat over $A$ . Given an etale morphism $A \to B$ of, say, local rings, one can show that many properties of $A$ ascend to properties of $B$. For instance, the dimension of $A$ is equal to the dimension of $B$, and if $A$ is a normal domain, so is $B$. This means that if $A \to B$ is an etale morphism and $A$ is a PID (not necessarily local, now) and $B$ a domain, then $B$ is itself normal, noetherian, and of dimension one -- hence a Dedekind domain. (Is $B$ necessarily a PID? I think not, but don't have a good example at the moment.)

With flatness I think in general one cannot "go up" from $A$ to $B$ as one can for an etale morphism. However, one can sometimes go "down" from properties of $B$ to properties of $A$; this is a small part of the story of faithfully flat descent (at least if $B$ is faithfully flat over $A$). For instance, here is an interesting consequence: given a flat morphism of noetherian local rings $A \to B$, then if $B$ is regular, so is $A$ (the proof uses Serre's characterization of local rings in terms of global dimension).

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