Let $f:A\rightarrow B$ be a local homomorphism of Noetherian local rings, such that the $A$-module $B$ has finite flat dimension. Is it true that the Krull dimensions of $A$ and $B$ agree? If yes, then Srinivas’ argument for the preservation of codimension of closed subschemes by the pullback of a flat morphism between Noetherian schemes of finite Krull dimension might be modified to show that the BGQ spectral sequence for a Noetherian scheme of finite Krull dimension is contravariant with respect to morphisms of finite Tor-dimension.

Ultimately, I am trying to use this conjectured functoriality with respect to any morphism of finite Tor-dimension to show that the natural surjection $G_0(X)\rightarrow \mathbb{Z}\oplus Cl(X)$ for a toric 3-fold $X$ induced by the filtration on $G_0(X)$ by codimension of support for coherent sheaves on $X$ has kernel $A^2(X)$.