Branched covers and matrix factorizations

Abstract

AbstractLet be a regular local ring and a non‐zero element of . A theorem due to Knörrer states that there are finitely many isomorphism classes of maximal Cohen–Macaulay (CM) ‐modules if and only if the same is true for the double branched cover of , that is, the hypersurface ring which is defined by in . We consider an analogue of this statement in the case of the hypersurface ring defined instead by for . In particular, we show that this hypersurface, which we refer to as the ‐fold branched cover of , has finite CM representation type if and only if, up to isomorphism, there are only finitely many indecomposable matrix factorizations of with factors. As a result, we give a complete list of polynomials with this property in characteristic zero. Furthermore, we show that reduced ‐fold matrix factorizations of correspond to Ulrich modules over the ‐fold branched cover of .