# Does going-down theorem hold for local homomorphism of finite flat dimension?

Let $$f:(A,m)\rightarrow (B,n)$$ be a local homomorphism of Noetherian local rings of finite flat dimension. Does the going-down 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 Tor-dimension 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 Tor-dimension.

• The same counterexample I gave for your previous question ($\mathbb{Z}_p \to \mathbb{F}_p$) seems to show the answer is no Dec 11, 2023 at 2:19
• @ Vik78 Thank you very much for your kind help. I noticed that in the counter-example you gave, the Krull dimension drops as we follow the local homomorphism. Is this always true? Thanks. Dec 11, 2023 at 13:24
• 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
– Bma
Dec 11, 2023 at 15:20
• @Bma Thank you very much for your information and thoughts. Dec 11, 2023 at 20:49

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 Auslander-Buchsbaum theorem $$B$$ has finite projective dimension as an $$A$$-module. However, the map cannot satisfy the going-down 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.
• 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. Dec 12, 2023 at 13:41
• No, by the Auslander-Buchsbaum theorem the map $\mathbb{C}[x]_{(x)} \to \mathbb{C}[x, y]_{(x,y)}$ is a counterexample Dec 12, 2023 at 15:56