Cohomological dimension

From Wikipedia, the free encyclopedia

The introduction to this article provides insufficient context for those unfamiliar with the subject matter.
Please help improve the introduction to meet Wikipedia's layout standards. You can discuss the issue on the talk page.

A discrete group G has cohomological dimension less than or equal to n, if the trivial ZG-module Z has a projective resolution of length n, i.e. there are projective ZG-modules P0,..,Pn and ZG-module homomoprhisms dk: Pk\toPk-1 (k=1,..,n) and d0: P0\to Z, such that the image of dk coincides with the kernel of dk-1 for k=1,..,n.

Equivalently, the cohomological dimension is less than or equal to n, if for an arbitrary ZG-module M, the cohomology of G with coeffients in M vanishes in degrees k>n: H^k(G,M)=0 whenever k>n.

The smallest n such that the cohomological dimension of G is less than or equal to n is the cohomological dimension of G.

Retrieved from "http://en.wikipedia.org../../../c/o/h/Cohomological_dimension.html"