Isomorphism -- probabilistic test for isomorphism of modules

Description

Two modules are isomorphic if there a homomorphism $f:M \to N$ which is both injective and surjective, or equivalently if there is a surjection in each direction. These routines produce random combinations of the generators of Hom and test for a homomorphism which is both injective and surjective.

Note that it suffices to check surjectivity after tensoring with the residue field, so in a future version we may instead check for surjections in both directions.

Contributors

Mike Stillman and Devlin Mallory contributed to this package.

Authors

Version

This documentation describes version 2.0 of Isomorphism.

Citation

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

@misc{IsomorphismSource,
  title = {{Isomorphism: probabilistic test of isomorphism between modules. Version~2.0}},
  author = {David Eisenbud and Mahrud Sayrafi},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

Exports

Functions and commands
- checkDegrees -- compares the degrees of generators of two modules
- isIsomorphic -- probabilistic test for isomorphism of modules
- isomorphism -- retrieve an isomorphism of modules
Methods
- checkDegrees(Matrix,Matrix) -- see checkDegrees -- compares the degrees of generators of two modules
- checkDegrees(Module,Module) -- see checkDegrees -- compares the degrees of generators of two modules
- isIsomorphic(Matrix,Matrix) -- see isIsomorphic -- probabilistic test for isomorphism of modules
- isIsomorphic(Module,Module) -- see isIsomorphic -- probabilistic test for isomorphism of modules
- isomorphism(Module,Module) -- see isomorphism -- retrieve an isomorphism of modules

For the programmer

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

The source of this document is in Isomorphism.m2:528:0.