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

Question

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.

Answer 1

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.