Use DG-methods to build a matrix factorization

TL;DR

This paper develops a method using differential graded algebra to construct explicit matrix factorizations for certain hypersurface rings, advancing the algebraic tools for resolving complex modules.

Contribution

It introduces a novel approach to build resolutions of hypersurface rings via DG-methods, providing explicit matrix factorizations and homotopy maps.

Findings

Constructed a resolution N of Pbar/K Pbar using DG-algebra techniques.

Provided explicit matrix factorizations for the infinite tail of the resolution.

Demonstrated the method's applicability to specific Gorenstein ideals.

Abstract

Let P be a commutative Noetherian ring, K be an ideal of P which is generated by a regular sequence of length four, f be a regular element of P, and Pbar be the hypersurface ring P/(f). Assume that K:f is a grade four Gorenstein ideal of P. We give a resolution N of Pbar/K Pbar by free Pbar-modules. The resolution N is built from a Differential Graded Algebra resolution of P/(K:f) by free P-modules, together with one homotopy map. In particular, we give an explicit form for the matrix factorization which is the infinite tail of the resolution N.