Sufficient conditions for $\operatorname {Spec} f$ to be closed

Let $f:A\to B$ be a ring homomorphism and suppose that $\operatorname {Spec}B$ is a Noetherian space. Show that (1) the prime ideals of $B$ satisfy the ascending chain condition, and (2) $f$ has the going-up property if and only if $f^*:=\operatorname {Spec}f$ is a closed map.

(1) $\operatorname {Spec}B$ Noetherian means that any ascending chain of radical (hence of prime) ideals stabilize; I would say that the converse is false in general, but I couldn't think of a counterexample.

(2) In general a closed map has the going-up property; let's prove the converse, assuming that $\operatorname {Spec}B$ is Noetherian. Suppose that $f^*$ isn't closed: then there is an ideal $J\subseteq B$ (whose contraction we call $I$) such that $f^*V(J)\subsetneq V (I)$. In other words, there is a prime ideal $p\supseteq I$ that is not the contraction of any prime ideal in $B$. $f$ having the going-up property means that $J$ cannot be prime. My idea was to find an ascending chain of radical ideals in $B$ which doesn't stabilize. However I've been thinking for a while now and I don't see how to recover a chain in $B$. Would you give me a hint?

Solution 1:

For (2), note first that going-up property is the same as $f^\ast(V(\mathfrak{p}))=V(\mathfrak{p}^c)$ for each $\mathfrak{p}\in\operatorname{Spec}B$.

In general, since $\overline{f^\ast(V(\mathfrak{p}))}=V(\mathfrak{p}^c)$ (you can check this from the definitions), going-up is essentially the same as the assertion that $f^\ast$ preserves closedness of irreducible closed subsets.

When $\operatorname{Spec}B$ is assumed to be Noetherian, each closed subset of $\operatorname{Spec}B$ is a finite union of irreducible closed, so we see the equivalence between $f^\ast$ being closed and $f$ having the going-up property in this case.

For converse of (1), what you are looking for are examples of non-noetherian rings with noetherian specs. These examples are not hard to find on this site.

The thing is that noetherianess on spec is the same as ascending condition for radical ideals (due to the bijective inclusion-reversing correspondence between radical ideals and closed subsets), while noetherianess for rings is a stronger condition (namely on general ideals).