# 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.

## Key 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.

---
Source: https://tomesphere.com/paper/1905.11435