Branched covers and matrix factorizations

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