Leray's theorem

The set of Leray, named after Jean Leray, is a mathematical theorem in the field of algebraic topology and function theory. There is a possibility to determine Garbenkohomologien easily.

Wording of the sentence

Below is a sheaf of abelian groups over a paracompact Hausdorff space. The cohomology result is known as the inductive limit of groups, where the open coverings of passes that are directed with respect to the refinement. This raises the question of whether there are open overlaps with, so that the inductive limit must not be performed. This is the case, in fact, since the following applies:

Set of Leray: It is a sheaf of abelian groups over a paracompact Hausdorff space. Next is an open cover of, such that for all and coverage amounts with the equation

Applies. Then

If the overlap is therefore such that the sheaf is trivial kohomologisch on the average of the coverage amounts, so the cohomology over the total space does already match the cohomology of coverage. The proof uses the existence of finer resolutions of a sheaf.

Application

On a typical example is to show how the set of Leray can be used for the calculation of cohomology groups. Whether the complex plane excluding zero. Then we have

With the stand on the left side of the equation for the sheaf of - valued functions. These are

Then an open cover is of. The coverage amounts are star-shaped slotted as levels, ie simply connected, ie homotopisch and therefore kohomologisch trivial. Thus the conditions of the theorem of Leray are met and obtained. The latter can now be recognized as isomorphic because of the finite nature of light as is in running. Thus, the cohomology is determined by the set of Leray.