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In Macaulay2, modules are implemented as subquotient modules. Submodules and quotients of free modules are perhaps the most common and important modules, and subquotients form the smallest class of modules that naturally includes these cases.

See Tutorial: Modules in Macaulay2 for an overview tutorial on modules.

For additional common operations and a comprehensive list of all routines in Macaulay2 which return or use modules, see Module.

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• Module -- the class of all modules
• module -- make or get a module
• isModule -- whether something is a module

construction of modules

• free modules
• matrices to and from modules -- including kernel, cokernel and image
• making modules from matrices
• submodules and quotients
• subquotient modules -- the way Macaulay2 represents modules

properties of modules

• manipulating modules
• computing syzygies
• extracting elements
• minimal presentations and generators
• equality and containment of modules

homomorphisms (maps) between modules

• module homomorphisms
• maps between modules
• information about a map of modules
• constructing maps between modules
• right modules or left modules?
• module quotients, saturation, and annihilator

graded modules

• gradedModule -- make a graded module
• Hilbert functions and free resolutions -- including degree and betti numbers
• basis -- basis or generating set of all or part of a ring, ideal or module

multilinear algebra

• exterior power of a module

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