Saturday, 1 December 2007

ag.algebraic geometry - Restriction of Ext sheaves

Hi Andrea, I don't think one can prove much without flatness. Let's assume the simplest case, that X=textSpec(S), Y=textSpec(R), and RtoS is a finite local homomorphism with R regular. Then I claim what you want is equivalent to flatness.



Your condition amounts to
textExtRi(M,N)otimesRScongtextExtSi(MotimesS,NotimesS)
for R-modules M,N. There is a well-known result that the first i such that textExtRi(R/I,R)neq0 is the length of the longest R-regular sequence in I. Let M=R/mR, N=R, then by the Ext condition we can conclude that mRS contains a S-regular sequence of length equals to textdimS. So S is Cohen-Macaulay.



But then "miracle flatness" implies that f is flat! This also provides counter-examples: if S is not Cohen-Macaulay, choose i to be textdepthS.



ADDED: for the sake of completeness, here is a class of examples to show the second nice situation (mathcalF is locally free) can't be generalized to much.



Let Y=mathbbAn, X=V(f)subsetY such that f vanishes at the origin. Let mathcalF be locally free on Y minus the origin, but not free at the origin. Let mathcalG be any torsion-free coherent sheaf on Y. Then I claim the condition you want (let's call it ()) will not hold.



From the short exact sequence 0tomathcalGtomathcalGtomathcalG/(f)to0 (the first map is multiplication by f, exact because mathcalG is torsion-free) one can take mathcalHom(mathcalF,) to get a long exact sequence. Looking at such l.e.s, () means precisely that the maps by multiplication by f:
mathcalExti(mathcalF,mathcalG)tomathcalExti(mathcalF,mathcalG)
must be injective for all i>0. But as all these mathcalExt vanish away from the origin and f vanishes at the origin,
they have to be 0. Now localize at the origin, take a minimal free resolution of the stalk of mathcalF to compute Ext and use Nakayama, one can show that mathcalF must also be free there, contradiction.

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