Hopf Extensions of CM-finite Artin Algebras

Abstract

Let H be a finite-dimensional Hopf algebra over a field k, and A a left \(H\mbox{-}\)module \(k\mbox{-}\)algebra. We show that A#H is a CM-finite algebra if and only if A is a CM-finite algebra preserving global dimension of their relative Auslander algebras when A/AH is an \(H^{*}\mbox{-}\)Galois extension and A#H/A is separable. As application, we describe all the finitely-generated Gorenstein-projective modules over a triangular matrix artin algebra \(\Lambda=\left(\begin{smallmatrix} A^{H}& A\\ 0&A\#H \end{smallmatrix}\right)\), and obtain a criteria for Λ being Gorenstein. We also show that Hopf extensions can induce recollements between categories \(A\#H\mbox{-}{\rm Mod}\) and \(A^{H}\mbox{-}{\rm Mod}\).