Cohomological Descent
نویسنده
چکیده
Introduction In classical C̆ech theory, we “compute” (or better: filter) the cohomology of a sheaf when given an open covering. Namely, if X is a topological space, U = {Ui} is an indexed open covering, and F is an abelian sheaf on X, then we get a C̆ech to derived functor spectral sequence E 2 = H (U,H(F ))⇒ H(X,F ), where H(F ) is the presheaf whose value on an open U is H(U,F |U ) (and we use the contravariant pullback functoriality). In particular, H(F ) = F and H(F ) sheafifies to be zero if q > 0. Of course, if F has vanishing cohomology on the finite overlaps of the Ui’s then this degenerates to give an edge isomorphism H(U,F ) ' H(X,F ).
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