COHEN-MACAULAY INVARIANT SUBALGEBRAS OF HOPF DENSE GALOIS EXTENSIONS

Abstract

Let H be a semisimple Hopf algebra, and let R be a noetherian left H-module algebra. If R=RH is a right H -dense Galois extension, then the invariant subalgebra RH will inherit the AS-Cohen-Macaulay property from R under some mild conditions, and R, when viewed as a right RH-module, is a Cohen-Macaulay module. In particular, we show that if R is a noetherian complete semilocal algebra which is AS-regular of global dimension 2 and H = kG for some nite subgroup G Aut(R), then all the indecomposable Cohen- Macaulay module of RH is a direct summand of RRH, and hence RH is Cohen- Macaulay- nite, which generalizes a classical result for commutative rings. The main tool used in the paper is the extension groups of objects in the corresponding quotient categories.