Matrix factorizations with more than two factors

Given an element $f$ in a regular local ring, we study matrix factorizations of $f$ with $d \ge 2$ factors, that is, we study tuples of square matrices $(\varphi_1,\varphi_2,\dots,\varphi_d)$ such that their product is $f$ times an identity matrix of the appropriate size. Several well known properties of matrix factorizations with $2$ factors extend to the case of arbitrarily many factors. For instance, we show that the stable category of matrix factorizations with $d\ge 2$ factors is naturally triangulated and we give explicit formula for the relevant suspension functor. We also extend results of Kn\"orrer and Solberg which identify the category of matrix factorizations with the full subcategory of maximal Cohen-Macaulay modules over a certain non-commutative algebra $\Gamma$. As a consequence of our findings, we observe that the ring $\Gamma$ behaves, homologically, like a "non-commutative hypersurface ring" in the sense that every finitely generated module over $\Gamma$ has an eventually $2$-periodic projective resolution.Comment: 36 pages, comments welcom

Branched covers and matrix factorizations

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 $R$.Comment: 17 pages, comments welcome. v2: correction to a mistake in Example 3.6 as well as other minor change