Wednesday, 7 February 2007

cohomology - Cech to derived spectral sequence and sheafification

Yes, this is true in general.



It suffices to show the stalks vanish. Pick xinX and take an injective resolution 0tocalFtoI0tocdots. For any open U containing x, we get a chain complex



0toI0(U)toI1(U)tocdots



whose cohomology groups are Hp(U,calF|U).



Taking direct limits of these sections gives the chain complex



0toIx0toIx1tocdots



of stalks, which has zero cohomology in positive degrees because the original complex was a resolution. However, direct limits are exact and so we find



0=rmcolimxinUHp(U,calF|U)=underlineHp(calF)x



as desired.



Generally, cohomology tells you the obstructions to patching local solutions into global solutions, and this says that locally those obstructions vanish.

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