Group cohomology and quadratic forms

We give some basic results of the cohomology theory of algebraic groups in terms of quadratic forms. In particular, we give cohomological interpretations of the classification of quadratic forms over finite fields and p-adic fields, and also give an interpretation of the Hasse Principle. We then investigate the integral case and indicate why in that setting the Hasse Principle fails.; After developing some of the basics of the theory, we prove our main result on the vanishing of certain cohomology sets of SL_n($RM$) where $RM$ is the $M$-adic completion of a Noetherian ring R. We further require the ideal $M$ to be such that $R/M$ is a finite field of characteristic not 2. The vanishing of these cohomology sets may be interpreted as the equivalence of non-degenerate quadratic forms of a fixed discriminant over $RM$. We then indicate how to generalize our result to other groups and derive the equivalence of symplectic forms over $RM$.; Lastly, we give a summary of some recent results of T. Ono which relates certain cohomology sets of congruence groups with class groups of certain quadratic extensions.