Group Cohomology - Formal Constructions

Formal Constructions

In this article, G is a finite group. The collection of all G-modules is a category (the morphisms are group homomorphisms f with the property f(gx) = g(f(x)) for all g in G and x in M). This category of G-modules is an abelian category with enough injectives (since it is isomorphic to the category of all modules over the group ring ℤ).

Sending each module M to the group of invariants MG yields a functor from this category to the category of abelian groups. This functor is left exact but not necessarily right exact. We may therefore form its right derived functors; their values are abelian groups and they are denoted by Hn(G,M), "the n-th cohomology group of G with coefficients in M". H0(G,M) is identified with MG.

Read more about this topic:  Group Cohomology

Famous quotes containing the word formal:

    The conviction that the best way to prepare children for a harsh, rapidly changing world is to introduce formal instruction at an early age is wrong. There is simply no evidence to support it, and considerable evidence against it. Starting children early academically has not worked in the past and is not working now.
    David Elkind (20th century)