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Especially in the context of sheaves on manifolds, sheaf cohomology can often be computed using resolutions by soft sheaves, fine sheaves, and flabby sheaves (also known as flasque sheaves from the French flasque meaning flabby). For example, a partition of unity argument shows that the sheaf of smooth functions on a manifold is soft.
Integration over simplices give us a morphism of sheaves of complexes :. Since both objects admit partition of unities, it is a standard fact that the second pages of the hypercohomology spectral sequences for both of them only have one nonzero column each, thus the hypercohomologies of the two complexes of sheaves indeed calculates the de Rham ...
For an n-manifold X, the dualizing complex D X is isomorphic to the shift o X [n] of the orientation sheaf. As a result, Verdier duality includes Poincaré duality as a special case. Alexander duality is another useful generalization of Poincaré duality. For any closed subset X of an oriented n-manifold M and any field k, there is an ...
Moreover, if X is path-connected, locally path-connected and semi-locally simply connected (so X has a universal cover), then every functor is of the above form; i.e., the functor category (,) is equivalent to the category of locally constant sheaves on X.
Unlike vector bundles, coherent sheaves (in the analytic or algebraic case) form an abelian category, and so they are closed under operations such as taking kernels, images, and cokernels. On a scheme, the quasi-coherent sheaves are a generalization of coherent sheaves, including the locally free sheaves of infinite rank.
This construction is a global version of the construction of the divisor class group, or ideal class group, and is much used in algebraic geometry and the theory of complex manifolds. Alternatively, the Picard group can be defined as the sheaf cohomology group (,).
Fine sheaves are usually only used over paracompact Hausdorff spaces X. Typical examples are the sheaf of germs of continuous real-valued functions over such a space, or smooth functions over a smooth (paracompact Hausdorff) manifold, or modules over these sheaves of rings. Also, fine sheaves over paracompact Hausdorff spaces are soft and acyclic.
In mathematics, the exponential sheaf sequence is a fundamental short exact sequence of sheaves used in complex geometry. Let M be a complex manifold, and write O M for the sheaf of holomorphic functions on M. Let O M * be the subsheaf consisting of the non-vanishing holomorphic functions. These are both sheaves of abelian groups.