Twisted (co)homological stability for monoids of endomorphisms (Q1318037)

Let \(R\) be a ring with 1 and \(M_ n(R)\) be the multiplicative monoid of \(n \times n\)-matrices over \(R\). Let \(\mathbb{Z} [M_ n(R)]\) be its monoid ring. Let \(D:(R\)-\(\bmod) ^{op} \times (R\)-\(\bmod) \to Ab\) be a functor. This paper is devoted to studying the Hochschild homology \(HH_ *(\mathbb{Z} [M_ n(R)], D(R^ n,R^ n))\), where \(M_ n(R)\) acts on \(D(R^ n,R^ n)\) from both sides by an obvious action. It is shown that these groups stabilize when \(n \to \infty\) for all rings \(R\) and almost any functor \(D\). This work was motivated by the recent development of topological Hochschild homology theory. It has turned out that the homotopy groups of \(THH(R)\) are the same as \(HH_ *(\mathbb{Z} [M_ \infty],-)\) groups with correctly chosen coefficients. Here \(M_ \infty=colim M_ n(R)\).