Grothendieck topology

A Grothendieck topology is a mathematical concept that makes it possible to develop in an abstract categorical framework of a sheaf theory and cohomology theory one. A category on a Grothendieck topology is explained, is called a situs. On a situs a sheaf can be explained. The concept of Grothendieck topology was developed around 1960 by Alexander Grothendieck to in positive characteristic to have a replacement for the topological cohomology theories such as the singular cohomology in algebraic geometry. The motivation for this came from the conjectures of André Weil, which ( as the Betti numbers) of one variety and the number of points predicted a close relationship between the topological shape on her over a finite field ( Weil conjectures ). Introduced in this context étale topology together with the étale cohomology and l -adic cohomology finally allowed the proof of the Weil conjectures by Deligne.

Definition of a basis of a Grothendieck topology

A base of a Grothendieck topology in a category C is given by distinguished for each object U of C families of morphisms, as covering family of U. These families must meet the following axioms:

• An isomorphism is a covering family of U.

• If a covering family of U and a morphism, then there exists the pullback for each i from I and the induced family is a covering family for V.

• If overlapping family of subway and if, for each i from I, a family of overlapping, then a covering family of U.

The simplest example of a Grothendieck topology is given by the category of open sets of a topological space ( with inclusions as morphisms ), where a family is a covering family if the union is quite underground.

Sheaves on a Grothendieck topology

A presheaf on a category C is a contravariant functor 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 the sequence

Is exact, that is, when the difference between the core 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 the Čech cohomology.

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

• Algebraic Geometry
• Category theory