Қосымшалы когомологиядағы қабықшалар және ациклдік қабықшалар

Mathematical object in sheaf cohomology
In mathematics, injective sheaves of abelian groups are used to construct the resolutions needed to define sheaf cohomology (and other derived functors, such as sheaf Ext). There is a further group of related concepts applied to sheaves: flabby (flasque in French), fine, soft (mou in French), acyclic. In the history of the subject they were introduced before the 1957 "Tohoku paper" of Alexander Grothendieck, which showed that the abelian category notion of injective object sufficed to found the theory. The other classes of sheaves are historically older notions. The abstract framework for defining cohomology and derived functors does not need them. However, in most concrete situations, resolutions by acyclic sheaves are often easier to construct. Acyclic sheaves therefore serve for computational purposes, for example the Leray spectral sequence.

An acyclic sheaf over X is one such that all higher sheaf cohomology groups vanish. The cohomology groups of any sheaf can be calculated from any acyclic resolution of it (this goes by the name of De Rham Weil theorem).

A fine sheaf over X is one with "partitions of unity"; more precisely for any open cover of the space X we can find a family of homomorphisms from the sheaf to itself with sum 1 such that each homomorphism is 0 outside some element of the open cover. 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. One can find a resolution of a sheaf on a smooth manifold by fine sheaves using the Alexander–Spanier resolution. As an application, consider a real manifold X. There is the following resolution of the constant sheaf by the fine sheaves of (smooth) differential forms:

This is a resolution, i. e. an exact complex of sheaves by the Poincaré lemma. The cohomology of X with values in can thus be computed as the cohomology of the complex of globally defined differential forms:

A soft sheaf over X is one such that any section over any closed subset of X can be extended to a global section. Soft sheaves are acyclic over paracompact Hausdorff spaces.