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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 $A^2(X)$.

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  • $\begingroup$ This is true at least in the special case where the extension of residue fields is separable. In that case the module of Kahler differentials must be zero, so the morphism is etale, and etale maps preserve dimension. $\endgroup$
    – Bma
    Dec 10, 2023 at 15:04
  • $\begingroup$ Wait sorry, I misread the question as saying the map was flat. That is necessary for it to be etale. $\endgroup$
    – Bma
    Dec 10, 2023 at 15:18
  • $\begingroup$ @Bma Thank you very much for your kind help. $\endgroup$
    – Boris
    Dec 10, 2023 at 18:35

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If I am reading your question correctly this is false. For instance take the quotient from $\mathbb{Z}_p$ to its residue field.

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    $\begingroup$ Or even $\mathbb{C}\to \mathbb{C}[[x]]$. $\endgroup$ Dec 10, 2023 at 15:36
  • $\begingroup$ @LaurentMoret-Bailly Thank you very much for your kind help. $\endgroup$
    – Boris
    Dec 10, 2023 at 18:36

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