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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.

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  • $\begingroup$ The same counterexample I gave for your previous question ($\mathbb{Z}_p \to \mathbb{F}_p$) seems to show the answer is no $\endgroup$
    – Vik78
    Dec 11, 2023 at 2:19
  • $\begingroup$ @ 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. $\endgroup$
    – Boris
    Dec 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 $\endgroup$
    – Bma
    Dec 11, 2023 at 15:20
  • $\begingroup$ @Bma Thank you very much for your information and thoughts. $\endgroup$
    – Boris
    Dec 11, 2023 at 20:49

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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.

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  • $\begingroup$ 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. $\endgroup$
    – Boris
    Dec 12, 2023 at 13:41
  • $\begingroup$ No, by the Auslander-Buchsbaum theorem the map $\mathbb{C}[x]_{(x)} \to \mathbb{C}[x, y]_{(x,y)}$ is a counterexample $\endgroup$
    – Vik78
    Dec 12, 2023 at 15:56
  • $\begingroup$ I see. Thank you very much for the counter-example. $\endgroup$
    – Boris
    Dec 12, 2023 at 17:54

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