Injective sheaf

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 sufficed to found the theory. Most of these other classes of sheaves are obsolescent, at least for that application.

=Injective sheaves=

An injective sheaf F is just a sheaf that is an injective element of the category of abelian sheaves; in other words, homomorphisms from A to F can always be lifted to any sheaf B containing A .

The category of abelian sheaves has enough injective elements: this means that any sheaf is a subsheaf of an injective sheaf. This result of Grothendieck follows from the existence of a generator of the category (it can be written down explicitly, and is related to the subobject classifier). This is enough to show that derived functors of any functor exist and are unique up to canonical isomorphism.

Injective sheaves are usually superior to the other classes of sheaves mentioned above: they can do almost anything the other classes can do, and their theory is simpler and more general.

Injective sheaves are flabby ( flasque ), soft, and acyclic.

=Fine sheaves=

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 homomophisms from the sheaf to itself with sum 1 such that each homomorphims 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 continous real functions over such a space, or smooth functions over a smooth (paracompact Hausdorff) manifold, or modules over these sheaves of rings.

Fine sheaves over paracompact Hausdorff spaces are soft and acyclic.

=Soft sheaves=

A soft sheaf F 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.

=Flasque or flabby sheaves=

A flasque sheaf (also called a flabby sheaf) is a on which the sheaf is defined and

:U subset V subset X

are open subsets, then the restriction map

:r_{V subset U} : Gamma(V, mathcal{F}) o Gamma(U, mathcal{F})

is surjective, as a map of group (mathematics) (ring (mathematics), module (mathematics), etc.).

Flasque sheaves are useful because (by definition) sections of them extend. This means that they are some of the simplest sheaves to handle in terms of homological algebra. Any sheaf has a canonical embedding into the flasque sheaf of all possibly discontinuous sections of the étale space, and by repeating this we can find a canonical flasque resultion for any sheaf. Flasque resolutions, that is, resolution (homological algebra)s by means of flasque sheaves, are one approach to defining sheaf cohomology.

Flasque is a French language word, that has sometimes been translated into English as flabby .

Flabby sheaves are soft and acyclic.

=Acyclic sheaves=

An acyclic sheaf F 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.

=References=

*Godement, Théorie des faisceaux ISBN 2705612521