Sheaf (mathematics)

A sheaf is a term from different areas of mathematics such as algebraic geometry and function theory. A sheaf of abelian groups on a topological space shall consist of an abelian group on any open subset of the base space and compatible Einschränkungshomomorphismen between these abelian groups. Accordingly, there is a sheaf of rings from one ring for each open subset and ring homomorphisms. The simplest example of a sheaf is the sheaf of continuous real-valued functions on open subsets of a topological space together with the restriction of functions to smaller open subsets. Prägarben can be defined in any category. Sheaves can be on any situs (which is a category on which a Grothendieck topology is explained ) define.

Definitions

To understand the definition of the sheaf, it is advisable to keep the example of the sheaf of continuous functions in mind: is the set of continuous functions, the restriction maps ( images of inclusion pictures under the functor ) are simply the restrictions of the functions on smaller areas.

Presheaf on a topological space

A presheaf on a topological space consists of a set (or abelian group, module, ring) for each open subset, together with restriction maps for two open subsets; while the restriction maps in the " obvious " way must match:

• For open subsets.

The elements of are called ( local ) section of more than, the elements of global sections. Instead we also write

For the restriction of a section on an open subset to write well.

Sheaf on a topological space

A sheaf is a presheaf, in which the data is " local", i.e. the following two conditions are met:

• Local correspondence implies global compliance: Are and cuts of over and a surplus of, and is

• Matching local data can be " glued ": Are cuts added so that the limitations of and on the same, so there is a section, so that

From the first condition it follows that is uniquely determined in the second condition by the.

Category Theoretical definition of a sheaf on a topological space

It should be a topological space. Have the category as objects open subsets of a morphism for each inclusion of open sets. A presheaf on with values ​​in a category is a contravariant functor. A presheaf is called sheaf if the following chart for each open subset and every cover of is exact:

That is, the difference between the core of the two right arrows.

(The concept of sheaf is defined only if the product possesses. )

Presheaf on a category sheaf on a situs

A presheaf on a category C is a contravariant functor: CA in a category A, for example, the category of sets or the category of abelian groups. If C has a Grothendieck topology, it is called a presheaf a sheaf if for every covering family {? I: Vi U} iI the sequence: is exact, that is, if the difference kernel of the two right arrows.

As in the case of a topological space can be vergarben Prägarben. Similarly, one can develop various cohomology theories, such as Cech cohomology.

The totality of all sheaves on a situs forms a topos.

Morphisms

As a sheaf is a collection of objects, a morphism between sheaves is a collection of morphisms of these objects. This must be compatible with the restriction maps.

Let and sheaves on with values ​​in the same category. A morphism is a collection of morphisms, one for each open subset of, such that for every inclusion of open subsets satisfies the condition. Herein, the restriction mapping and by.

Summing up the sheaves described above as functors, so is a morphism between the sheaves the same as a natural transformation of functors.

For each category, the form -valued sheaves with this Morphismenbegriff a category.

Stems and seeds

It is a category of algebraic structures that are defined by finite projective Limites, eg ( abelian ) groups, rings, modules. In particular, there pseudofiltrierende colimits in, and their underlying quantities are consistent with the colimits of the underlying rates of the individual objects match.

For each point of the stalk of a presheaf is defined as the point

Elements of the stems, called germs.

Germs are thus equivalence classes of local sections over open environments, where cuts are equivalent if they are equal in restriction to a smaller environment.

Vergarbung

Is a presheaf on a topological space, so there is a sheaf of Vergarbung or associated sheaf, such that for every sheaf

Applies. So linksadjungiert for forgetful

There is no standard notation for the Vergarbungsfunktor.

Direct images and inverse image sheaves

Is a sheaf on a topological space and a continuous map, then

A sheaf, which is designated and direct image or image sheaf of means under.

Is a sheaf, so is the associated sheaf to

A sheaf on, the archetype sheaf is denoted by.

Is a more continuous mapping, then the functors

And the functors

Of course equivalent.

The functors and adjoint: Is a sheaf on and a sheaf, so is

Straws are special sheaf archetypes: Identifies the inclusion of a point, then

While the sheaf was identified in the one-point space with its global sections. Consequently, the inverse image sheaf is compatible with straws:

This relationship is also the reason that, despite the complicated definition of the easier -to-understand functor is: in a sense, is the study of the cohomology functor.

The étale space of a sheaf

For a sheaf of sets on a topological space is defined as follows:

• The underlying set is the disjoint union of all stalks of; the picture from fancy to.
• The topology is the most powerful topology for the images

Then there is a bijection between the cuts of over an open set and the cuts of over, that is, the continuous maps for which is equal to the inclusion.

Examples

• The continuous functions with compact support not form a presheaf, because the restriction of a function with compact support on an open subset in general not again has compact support.

• The presheaf which assigns to each open subset of the Abelian group, is not a sheaf: Includes and so the cut can be over and the cut over not " stick " to a cut above.

• The sheaf of holomorphic functions on a sheaf of rings ( a ring sheaf ): the stalk at the origin can be identified with the ring of convergent power series, that is, the power series whose radius of convergence is not zero. The other blades caused by change of coordinates (ie replace by ).

• It is the topological space of two points, one of which has been completed and does not, i.e., the Sierpinski space. Then a sheaf is determined by the two quantities and accompanied by an image, and vice versa, you can specify these data freely and receives a sheaf. The stalks are of

• It should be open and to be the set of all functions that locally have a slope of 1, which are all equipped with, as long as both sides are defined and is sufficiently small. This is a sheaf in which each stalk isomorphic to and for each connected open proper subset. However, there is no global sections. Thus, this is "just" a set-valued and no abelian groups -valued sheaf.

Generalization

The concept of cooking can generally hold in the context of Grothendieck topologies.