Given a regular element * f * in * R *, * R * a regular ring, a *matrix factorisation* of * f * is a pair * (A,B) * of square matrices of the same size over * S * such that * AB = fI = 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.

Let * X * be a germ of a hypersurface at a point 0 and * M _{0} * be a maximal Cohen-Macaulay module over

A deformation * (T,M’) * is *induced from** (S,M) * if there exists a map * φ:T → S * commuting with the maps * T → Σ* and * S → Σ* and such that * M’ * is isomorphic to * (φ× _{Σ} id_{Y})*M *.

The deformation * (S,M) * is *versal* if any deformation of * M _{0} * is induced from

In this package, a germ is represented as the coordinate ring R of an affine variety and the distinguished point 0 of a germ is assumed to be the ideal generated by the variables of R. It is an error if the ideal generated by the variables of R is not prime, e.g. if it is all of R.

This package contains two entry points. One interface may be used only in the *absolute* case, where * Σ* is a reduced point, and the module * M _{0} * is deformed over

The implementation has been designed to take advantage of graded structure when available. Thus the user is highly recommended to assign degrees to the inputs so as to make the rings, homomorphism, and deformation module homogeneous when possible. Doing so may greatly improve performance.

- Bradford Hovinen <hovinen@math.uni-hannover.de>

Version **1.0** of this package was accepted for publication in volume 2 of the journal The Journal of Software for Algebra and Geometry: Macaulay2 on 2010-04-17, in the article Deformations of Matrix Factorisations with Macaulay2. That version can be obtained from the journal or from the *Macaulay2* source code repository, `svn://svn.macaulay2.com/Macaulay2/trunk/M2/Macaulay2/packages/ModuleDeformations.m2`, release number 11367.

- Functions and commands
- deformMCMModule -- versal deformation of MCM-module on hypersurface

- Other things
- xi -- Indexed variable of deformation parameters