Let $f:(A,m)\rightarrow (B,n)$ be a local homomorphism of Noetherian local rings of finite flat dimension. Does the goingdown theorem hold for $f$? If yes, then by Theorem 15.1 in Matsumura’s Commutative Ring Theory,we have $dim(A)\geq dim(B)$. The motivation for this question comes from my attempt to show that pullback by morphisms of finite Tordimension between Noetherian schemes of finite Krull dimension preserves the codimension of closed subschemes, so that the BGQ spectral sequence for a Noetherian scheme of finite Krull dimension is contravariant with respect to morphisms of finite Tordimension.

$\begingroup$ The same counterexample I gave for your previous question ($\mathbb{Z}_p \to \mathbb{F}_p$) seems to show the answer is no $\endgroup$– Vik78Dec 11, 2023 at 2:19

$\begingroup$ @ Vik78 Thank you very much for your kind help. I noticed that in the counterexample you gave, the Krull dimension drops as we follow the local homomorphism. Is this always true? Thanks. $\endgroup$– BorisDec 11, 2023 at 13:24

$\begingroup$ The miracle flatness theorem implies that in quite general situations such a map is already flat: en.m.wikipedia.org/wiki/Local_criterion_for_flatness . It's also worth noting that a finitely presented local map of local rings satisfies going down iff the induced map on specs is open. This leads me to think that there is probably a counterexample, though I'm not sure what yet $\endgroup$– BmaDec 11, 2023 at 15:20

$\begingroup$ @Bma Thank you very much for your information and thoughts. $\endgroup$– BorisDec 11, 2023 at 20:49
1 Answer
Here's a counterexample with $A$ and $B$ of the same Krull dimension. Take $A = B = \mathbb{C}[x,y]_{(x,y)}$, and let $f: A \to B$ be the composition of the quotient of $A$ by $(y)$ with the embedding $\mathbb{C}[x]_{(x)} \to \mathbb{C}[x,y]_{(x,y)}$. $A$ is a regular local ring, so by the AuslanderBuchsbaum theorem $B$ has finite projective dimension as an $A$module. However, the map cannot satisfy the goingdown property because the induced map $f^*:$ Spec $B \to $ Spec $A$ is not surjective. Geometrically, $f^*$ projects the plane onto the $x$axis, then embeds the $x$axis into another plane (with everything localized at the origin). Clearly its image is the closure of the $x$axis in the codomain.

$\begingroup$ Thank you so much for your detailed answer. I would like to ask is it true in general that if $f:A\rightarrow B$ is a local homomorphism of finite flat dimension between Noetherian local rings, then dimension of $A$ is always at least dimension of $B$? Thanks. $\endgroup$– BorisDec 12, 2023 at 13:41

$\begingroup$ No, by the AuslanderBuchsbaum theorem the map $\mathbb{C}[x]_{(x)} \to \mathbb{C}[x, y]_{(x,y)}$ is a counterexample $\endgroup$– Vik78Dec 12, 2023 at 15:56

$\begingroup$ I see. Thank you very much for the counterexample. $\endgroup$– BorisDec 12, 2023 at 17:54