Lecture Notes for Course 88-909 Cohomology of Groups

Before starting, we’ll say a few words of motivation. We wish to understand the infinite Galois groupG = Gal(Q/Q), which is very important in number theory. In particular, we wish to understand G-modules, namely modules over some ring that are endowed with an action of G. Such objects tend to be complicated. Recall that in algebraic topology, we define the homology and cohomology of simplicial complexes. These are collections of invariants that can be used, for instance, to prove that two simplicial complexes are not homeomorphic. One loses information when passing from a complex to its cohomology – two different complexes can have the same cohomology groups in every dimension – but retains enough to do some useful things. On the other hand, the cohomology groups have the advantage of sometimes being computable; for simple complexes they can be computed directly, and for more complicated ones they can often be deduced with tools such as the long exact cohomology sequence, the Mayer-Vietoris sequence, etc. The aim of this course is to develop an analogous theory in the setting of G-modules.