Homology of \(Gl(R)\) with coefficients in a functor of finite degree
In the present note we study the homology groups of \(Gl_ n R\) with twisted coefficients for \(R\) a commutative ring with unit. Let \(i_ n\) be the group homomorphism from \(Gl_ n R\) to \(Gl_{n + 1} R\) which sends \(X\) to \(\left | \begin{smallmatrix} X\;0\\0\;I \end{smallmatrix} \right |\). Moreover, assume that for every \(n\) we have a \(Gl_ n R\)-module \(M_ n\) and a \(Gl_ n R\)-homomorphism \(F_ n : M_ n \to M_{n + 1}\). The set \(\{M_ n\}\) will be called a coefficient system for \(Gl_ n R\). Let \(Gl R\) denote \(\lim_{n \to \infty} Gl_ n R\) and \(M = \lim_{n \to \infty} M_ n\). We are especially interested in finding \(H_ * (Gl R;M)\).
scientific article; zbMATH DE number 63895