Deformations of matrix factorisations with Macaulay2

Abstract

We describe a Macaulay2 package which computes versal deformations of matrix factori- sations, or equivalently maximal Cohen-Macaulay modules on hypersurfaces. 1. INTRODUCTION. We present here a Macaulay2 (M2) package, called ModuleDeformations, which computes versal deformations of matrix factorisations of polynomials. We begin with some notation and definitions. Definition 1.1. Let S be a regular ring and f2 S a regular element. A matrix factorisation of f is a pair (A; B) of square matrices of the same size over S such that AB = f I = BA, where I is the identity matrix of the same dimension. Matrix factorisations of f are equivalent to maximal Cohen-Macaulay (MCM) modules on the hypersurface defined by f : given a matrix factorisation (A; B) of f , the cokernel of A is an MCM module over S=( f), and all MCM modules over S=( f) arise in this way; see (E). For the course of this paper, we work with germs of varieties whose distinguished point is denoted by 0. All morphisms X! Y map 02 X to 02 Y. Our package represents a germ as the coordinate ring of an affine variety of which the germ is the localisation (or completion) at the ideal generated by the variables of the representing ring. It is an error if that ideal is not prime (e.g., if it is in fact the whole ring). With this in mind, we now define deformations of MCM modules.