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ExteriorModules -- a package for working with modules over exterior algebra

Description

ExteriorModules is a package for creating and manipulating modules over exterior algebra

Let $K$ be a field, $E$ the exterior algebra of a finite dimensional, $K$-vector space, and $F$ a finitely generated graded free $E$-module with homogeneous basis $g_1, \ldots, g_r$ such that $\mathrm{deg}(g_1) \le \mathrm{deg}(g_2) \le \cdots \le \mathrm{deg}(g_r)$. We present a Macaulay2 package to manage some classes of monomial submodules of $F$. The package is an extension of the one on monomial ideals, and contains some algorithms for computing stable, strongly stable and lexicograhic $E$-submodules of $F$. Such a package also includes some methods to check whether a sequence of nonnegative integers is the Hilbert function of a graded $E$-module of the form $F/M$, with $M$ graded submodule of $F$. Moreover, if $H_{F/M}$ is the Hilbert function of a graded $E$-module $F/M$, some routines are able to compute the unique lexicograhic submodule $L$ of $F$ such that $H_{F/M} = H_{F/L}.$

Authors

Certification a gold star

Version 1.0 of this package was accepted for publication in volume 11 of The Journal of Software for Algebra and Geometry on 3 June 2021, in the article ExteriorModules: a package for computing monomial modules over an exterior algebra (DOI: 10.2140/jsag.2021.11.71). That version can be obtained from the journal or from the Macaulay2 source code repository.

Version

This documentation describes version 1.0 of ExteriorModules.

Source code

The source code from which this documentation is derived is in the file ExteriorModules.m2.

Exports

  • Functions and commands
    • almostLexModule -- compute an almost lex module with the same Hilbert sequence of the module in input
    • almostStableModule -- compute the smallest almost stable module containing a given monomial module
    • almostStronglyStableModule -- compute the smallest almost strongly stable module containing a given monomial module
    • bassNumbers -- compute the Bass numbers of a given graded module
    • createModule -- create a module over an exterior algebra from a list of ideals in input
    • getIdeals -- get component ideals from a monomial module
    • initialModule -- compute the initial module of a given module
    • isAlmostLexModule -- whether a monomial module over an exterior algebra is almost lex
    • isAlmostStableModule -- whether a monomial module over an exterior algebra is almost stable
    • isAlmostStronglyStableModule -- whether a monomial module over an exterior algebra is almost strongly stable
    • isLexModule -- whether a monomial module over an exterior algebra is lex
    • isMonomialModule -- whether a module is monomial
    • isStableModule -- whether a monomial module over an exterior algebra is stable
    • isStronglyStableModule -- whether a monomial module over an exterior algebra is strongly stable
    • lexModule -- compute the lex submodule with a given Hilbert sequence in a free module
    • lexModuleBySequences -- alternative algorithm to compute the lex submodule with a given Hilbert sequence in a free module
    • stableModule -- compute the smallest stable module containing a given monomial module
    • stronglyStableModule -- compute the smallest strongly stable module containing a given monomial module
  • Methods
    • almostLexModule(Module) -- see almostLexModule -- compute an almost lex module with the same Hilbert sequence of the module in input
    • almostStableModule(Module) -- see almostStableModule -- compute the smallest almost stable module containing a given monomial module
    • almostStronglyStableModule(Module) -- see almostStronglyStableModule -- compute the smallest almost strongly stable module containing a given monomial module
    • bassNumbers(Module) -- see bassNumbers -- compute the Bass numbers of a given graded module
    • createModule(List,Module) -- see createModule -- create a module over an exterior algebra from a list of ideals in input
    • getIdeals(Module) -- see getIdeals -- get component ideals from a monomial module
    • hilbertSequence(Module) -- compute the Hilbert sequence of a given module over an exterior algebra
    • initialModule(Module) -- see initialModule -- compute the initial module of a given module
    • isAlmostLexModule(Module) -- see isAlmostLexModule -- whether a monomial module over an exterior algebra is almost lex
    • isAlmostStableModule(Module) -- see isAlmostStableModule -- whether a monomial module over an exterior algebra is almost stable
    • isAlmostStronglyStableModule(Module) -- see isAlmostStronglyStableModule -- whether a monomial module over an exterior algebra is almost strongly stable
    • isHilbertSequence(List,Module) -- whether the given sequence is a Hilbert sequence
    • isLexModule(Module) -- see isLexModule -- whether a monomial module over an exterior algebra is lex
    • isMonomialModule(Module) -- see isMonomialModule -- whether a module is monomial
    • isStableModule(Module) -- see isStableModule -- whether a monomial module over an exterior algebra is stable
    • isStronglyStableModule(Module) -- see isStronglyStableModule -- whether a monomial module over an exterior algebra is strongly stable
    • lexModule(List,Module) -- see lexModule -- compute the lex submodule with a given Hilbert sequence in a free module
    • lexModule(Module) -- see lexModule -- compute the lex submodule with a given Hilbert sequence in a free module
    • lexModuleBySequences(List,Module) -- see lexModuleBySequences -- alternative algorithm to compute the lex submodule with a given Hilbert sequence in a free module
    • minimalBettiNumbers(Module) -- compute the minimal Betti numbers of a given graded module
    • stableModule(Module) -- see stableModule -- compute the smallest stable module containing a given monomial module
    • stronglyStableModule(Module) -- see stronglyStableModule -- compute the smallest strongly stable module containing a given monomial module

For the programmer

The object ExteriorModules is a package.