Some topological properties of quotients modulo semisimple algebraic groups

We will prove a general result in Invariant Theory, viz. for a quotient C(n)//G, where G is a connected complex semisimple algebraic group, the local first homology group at any point in the quotient C(n)//G is trivial and the local second homology group is finite. Using this we will prove that the completion of the local ring of any point in C(n)//G is a unique factorization domain (UFD).