Let $(A,m\_A)$ and $(B,m\_B)$ be noetherian local rings and $f:A\rightarrow B$ a local homomorphism. Let $F = B/m\_AB$ be the fiber ring and assume that
$$\mathrm{dim}(B) = \mathrm{dim}(A) + \mathrm{dim}(F).$$


The following Theorem (23.1 in Matsumura's CRT) is really quite a miracle:


Theorem: If $A$ is regular and $B$ is Cohen-Macaulay then $f$ is flat.


I am wondering to what extent this theorem can be generalized. What I have in mind is a statement of the type:


"Theorem": If $A$ is $X$ and $B$ is $Y$ then $f$ is of finite Tor-dimension
(i.e. $\mathrm{Tor}^i\_A(B,A)=0$ for all $i$ sufficiently large).


Here, $X$ and $Y$ are ring-theoretic conditions which should be *strictly weaker* than 
"regular" and "CM" respectively. Is the "Theorem" above true just requiring $A$ and $B$ to be normal?
How about both CM? Or maybe CM plus finitely many $(R\_i)$? 


Any thoughts/ counterexamples? 



Generalizing miracle flatness (Matsumura 23.1) via finite Tor-dimension
The "Theorem" isn't true with both rings just normal, or just CM, or even normal *and* CM. Let $A = k[[x,y,z]]/(xz-y^2) \cong k[[a^2,ab,b^2]]$ and let $B = k[[a,b]]$, with $f$ the natural inclusion. The dimensions add up as they must, since $f$ is module-finite. In this case finite flat dimension is the same as finite projective dimension, but $B$ does not have finite projective dimension over $A$. 


I don't expect that any addition of assumptions $(R\_i)$ would help.


