Hi Andrea, I don't think one can prove much without flatness. Let's assume the simplest case, that X=textSpec(S)X=textSpec(S), Y=textSpec(R)Y=textSpec(R), and RtoSRtoS is a finite local homomorphism with RR regular. Then I claim what you want is equivalent to flatness.
Your condition amounts to
textExtiR(M,N)otimesRScongtextExtiS(MotimesS,NotimesS)textExtiR(M,N)otimesRScongtextExtiS(MotimesS,NotimesS)
for RR-modules M,NM,N. There is a well-known result that the first ii such that textExtiR(R/I,R)neq0textExtiR(R/I,R)neq0 is the length of the longest RR-regular sequence in II. Let M=R/mRM=R/mR, N=RN=R, then by the Ext condition we can conclude that mRSmRS contains a SS-regular sequence of length equals to textdimStextdimS. So SS is Cohen-Macaulay.
But then "miracle flatness" implies that ff is flat! This also provides counter-examples: if SS is not Cohen-Macaulay, choose ii to be textdepthStextdepthS.
ADDED: for the sake of completeness, here is a class of examples to show the second nice situation (mathcalFmathcalF is locally free) can't be generalized to much.
Let Y=mathbbAnY=mathbbAn, X=V(f)subsetYX=V(f)subsetY such that ff vanishes at the origin. Let mathcalFmathcalF be locally free on YY minus the origin, but not free at the origin. Let mathcalGmathcalG be any torsion-free coherent sheaf on YY. Then I claim the condition you want (let's call it (∗)(∗)) will not hold.
From the short exact sequence 0tomathcalGtomathcalGtomathcalG/(f)to00tomathcalGtomathcalGtomathcalG/(f)to0 (the first map is multiplication by ff, exact because mathcalGmathcalG is torsion-free) one can take mathcalHom(mathcalF,−)mathcalHom(mathcalF,−) to get a long exact sequence. Looking at such l.e.s, (∗)(∗) means precisely that the maps by multiplication by ff:
mathcalExti(mathcalF,mathcalG)tomathcalExti(mathcalF,mathcalG)mathcalExti(mathcalF,mathcalG)tomathcalExti(mathcalF,mathcalG)
must be injective for all i>0i>0. But as all these mathcalExtmathcalExt vanish away from the origin and ff vanishes at the origin,
they have to be 00. Now localize at the origin, take a minimal free resolution of the stalk of mathcalFmathcalF to compute Ext and use Nakayama, one can show that mathcalFmathcalF must also be free there, contradiction.
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