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# ModuleDeformations -- versal deformations of modules on hypersurfaces

## Description

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 $ X $. Let $ Y \rightarrow{} \Sigma $ a flat map of germs of affine varieties at 0 whose fibre over 0 of $\Sigma$ is isomorphic to $X$. A deformation of $ M_0 $ (in the deformation theory defined by the map $ Y \rightarrow{} \Sigma $) is a pair $ (S,M) $ with $ S $ a germ equipped with a map $ S \rightarrow{} \Sigma $ and $ M $ a module over $ S \times_\Sigma Y $ flat over $ S $, whose restriction to $ 0 \times_\Sigma X $ is isomorphic to $ M_0 $.

A deformation $ (T,M') $ is induced from$ (S,M) $ if there exists a map $ \phi:T \rightarrow{} S $ commuting with the maps $ T \rightarrow{} \Sigma $ and $ S \rightarrow{} \Sigma $ and such that $ M' $ is isomorphic to $ (\phi \times_\Sigma id_Y)*M $.

The deformation $ (S,M) $ is versal if any deformation of $ M_0 $ is induced from $ (S,M) $.

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 $ \Sigma $ is a reduced point, and the module $ M_0 $ is deformed over $ Y $ only. The other interface may be used when $ \Sigma $ is the base space of a nontrivial deformation of $ X $ and $ Y $ is the total space of the deformation. In such a case, $ M_0 $ is deformed also along the given deformation of $ X $.

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.

## Author

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

## Certification

Version 1.0 of this package was accepted for publication in volume 2 of The Journal of Software for Algebra and Geometry: Macaulay2 on 2010-04-17, in the article Deformations of Matrix Factorisations with Macaulay2 (DOI: 10.2140/jsag.2010.2.6). That version can be obtained from the journal.

## Version

This documentation describes version 1.0 of ModuleDeformations, released 28 October 2009.

## Citation

If you have used this package in your research, please cite it as follows:

    @misc{ModuleDeformationsSource,
      title = {{ModuleDeformations: versal deformations of maximal Cohen-Macaulay modules. Version~1.0}},
      author = {Bradford Hovinen},
      howpublished = {A \emph{Macaulay2} package available at
        \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}}
    }

    @article{ModuleDeformationsArticle,
      title = {{Deformations of Matrix Factorisations with \emph{Macaulay2}}},
      author = {Bradford Hovinen},
      journal = {The Journal of Software for Algebra and Geometry: Macaulay2},
      volume = {2},
      year = {2010},
    }

## Exports

- Functions and commands
  - deformMCMModule -- versal deformation of MCM-module on hypersurface
- Methods
  - deformMCMModule(Module) -- deformation in the absolute case
  - deformMCMModule(Module,RingMap) -- deformation in the relative case
- Other things
  - xi -- Indexed variable of deformation parameters

## For the programmer

The object ModuleDeformations is a package, defined in ModuleDeformations.m2.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/ModuleDeformations.m2:361:0.