Sheaf Cohomology
It was noted above that the functor preserves isomorphisms and monomorphisms, but not epimorphisms. If F is a sheaf of abelian groups, or more generally a sheaf with values in an abelian category, then is actually a left exact functor. This means that it is possible to construct derived functors of . These derived functors are called the cohomology groups (or modules) of F and are written . Grothendieck proved in his Tohoku paper that every category of sheaves of abelian groups contains enough injective objects, so these derived functors always exist.
However, computing sheaf cohomology using injective resolutions is nearly impossible. In practice, it is much more common to find a different and more tractable resolution of F. A general construction is provided by Godement resolutions, and particular resolutions may be constructed using soft sheaves, fine sheaves, and flabby sheaves (also known as flasque sheaves from the French flasque meaning flabby). As a consequence, it can become possible to compare sheaf cohomology with other cohomology theories. For example, the de Rham complex is a resolution of the constant sheaf on any smooth manifold, so the sheaf cohomology of is equal to its de Rham cohomology. In fact, comparing sheaf cohomology to de Rham cohomology and singular cohomology provides a proof of de Rham's theorem that the two cohomology theories are isomorphic.
A different approach is by Čech cohomology. Čech cohomology was the first cohomology theory developed for sheaves and it is well-suited to concrete calculations. It relates sections on open subsets of the space to cohomology classes on the space. In most cases, Čech cohomology computes the same cohomology groups as the derived functor cohomology. However, for some pathological spaces, Čech cohomology will give the correct but incorrect higher cohomology groups. To get around this, Jean-Louis Verdier developed hypercoverings. Hypercoverings not only give the correct higher cohomology groups but also allow the open subsets mentioned above to be replaced by certain morphisms from another space. This flexibility is necessary in some applications, such as the construction of Pierre Deligne's mixed Hodge structures.
A much cleaner approach to the computation of some cohomology groups is the Borel–Bott–Weil theorem, which identifies the cohomology groups of some line bundles on flag manifolds with irreducible representations of Lie groups. This theorem can be used, for example, to easily compute the cohomology groups of all line bundles on projective space.
In many cases there is a duality theory for sheaves that generalizes Poincaré duality. See Grothendieck duality and Verdier duality.
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