Hi Andrea, I don't think one can prove much without flatness. Let's assume the simplest case, that , , and is a finite local homomorphism with regular. Then I claim what you want is equivalent to flatness.
Your condition amounts to
for -modules . There is a well-known result that the first such that is the length of the longest -regular sequence in . Let , , then by the Ext condition we can conclude that contains a -regular sequence of length equals to . So is Cohen-Macaulay.
But then "miracle flatness" implies that is flat! This also provides counter-examples: if is not Cohen-Macaulay, choose to be .
ADDED: for the sake of completeness, here is a class of examples to show the second nice situation ( is locally free) can't be generalized to much.
Let , such that vanishes at the origin. Let be locally free on minus the origin, but not free at the origin. Let be any torsion-free coherent sheaf on . Then I claim the condition you want (let's call it ) will not hold.
From the short exact sequence (the first map is multiplication by , exact because is torsion-free) one can take to get a long exact sequence. Looking at such l.e.s, means precisely that the maps by multiplication by :
must be injective for all . But as all these vanish away from the origin and vanishes at the origin,
they have to be . Now localize at the origin, take a minimal free resolution of the stalk of to compute Ext and use Nakayama, one can show that must also be free there, contradiction.
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