Branched covers and matrix factorizations

TL;DR

This paper generalizes Kn"orrer's theorem by relating the finite Cohen-Macaulay representation type of $d$-fold branched covers to the finiteness of indecomposable matrix factorizations with $d$ factors, providing a complete classification in characteristic zero.

Contribution

It extends Kn"orrer's theorem to $d$-fold branched covers, linking Cohen-Macaulay modules to matrix factorizations with multiple factors and classifying polynomials with finite type in characteristic zero.

Findings

Finite Cohen-Macaulay representation type corresponds to finitely many indecomposable matrix factorizations with $d$ factors.

Complete classification of such polynomials in characteristic zero.

Reduced $d$-fold matrix factorizations relate to Ulrich modules.

Abstract

Let $(S,\mathfrak n)$ be a regular local ring and $f$ a non-zero element of $\mathfrak n^2$. A theorem due to Kn\"orrer states that there are finitely many isomorphism classes of maximal Cohen-Macaulay $R=S/(f)$-modules if and only if the same is true for the double branched cover of $R$, that is, the hypersurface ring defined by $f+z^2$ in $S[[ z ]]$. We consider an analogue of this statement in the case of the hypersurface ring defined instead by $f+z^d$ for $d\ge 2$. In particular, we show that this hypersurface, which we refer to as the $d$-fold branched cover of $R$, has finite Cohen-Macaulay representation type if and only if, up to isomorphism, there are only finitely many indecomposable matrix factorizations of $f$ with $d$ factors. As a result, we give a complete list of polynomials $f$ with this property in characteristic zero. Furthermore, we show that reduced $d$-fold matrix factorizations of $f$ correspond to Ulrich modules over the $d$-fold branched cover of $R$.