Quasidegrees -- a package to compute quasidegrees

Description

Quasidegrees is a package that enables the user to construct multigraded rings and look at the graded structure of multigraded finitely generated modules over a polynomial ring. The quasidegree set of a $\ZZ^d$-graded module $M$ is the Zariski closure in $\CC^d$ of the degrees of the nonzero homogeneous components of $M$. This package can compute the quasidegree set of a finitely generated module over a $\ZZ^d$-graded polynomial ring. This package also computes the quasidegree sets of local cohomology modules supported at the maximal irrelevant ideal of modules over a $\ZZ^d$-graded polynomial ring.

The motivation for this package comes from $A$-hypergeometric functions and the relation between the rank jumps of $A$-hypergeometric systems and the quasidegree sets of non-top local cohomology modules supported at the maximal irrelevant ideal of the associated toric ideal as described in the paper:

Laura Felicia Matusevich, Ezra Miller, and Uli Walther. Homological methods for hypergeometric families. J. Am. Math. Soc., 18(4):919-941, 2005.

This package requires FourTiTwo, Depth, and Polyhedra.

Caveat

This package is written when the ambient ring of the modules in question are positively graded and are presented by a monomial matrix, that is, a matrix whose entries are monomials. This is due to the algorithms depending on finding standard pairs of monomial ideals generated by rows of a presentation matrix.

Author

Roberto Barrera <[email protected]>

Certification

Version 1.0 of this package was accepted for publication in volume 9 of The Journal of Software for Algebra and Geometry on 26 February 2019, in the article Computing quasidegrees of A-graded modules. That version can be obtained from the journal or from the Macaulay2 source code repository.

Version

This documentation describes version 1.0 of Quasidegrees.

Source code

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

Exports

Functions and commands
- exceptionalSet -- returns the exceptional set of a matrix
- makeGradedRing -- makes a polynomial ring graded by a matrix
- quasidegrees -- compute the quasidegree set of a module
- quasidegreesAsVariables -- represents the quasidegree set in variables
- quasidegreesLocalCohomology -- returns the quasidegree sets of local cohomology modules
- removeRedundancy -- removes redundancies from a list of planes
- toGradedRing -- grade a polynomial ring by a matrix
- toricIdeal -- returns a toric ideal
Methods
- exceptionalSet(Matrix) -- see exceptionalSet -- returns the exceptional set of a matrix
- makeGradedRing(Matrix,Symbol) -- see makeGradedRing -- makes a polynomial ring graded by a matrix
- quasidegrees(Ideal) -- see quasidegrees -- compute the quasidegree set of a module
- quasidegrees(Module) -- see quasidegrees -- compute the quasidegree set of a module
- quasidegreesAsVariables(Ideal) -- see quasidegreesAsVariables -- represents the quasidegree set in variables
- quasidegreesAsVariables(Module) -- see quasidegreesAsVariables -- represents the quasidegree set in variables
- quasidegreesLocalCohomology(Ideal) -- see quasidegreesLocalCohomology -- returns the quasidegree sets of local cohomology modules
- quasidegreesLocalCohomology(Module) -- see quasidegreesLocalCohomology -- returns the quasidegree sets of local cohomology modules
- quasidegreesLocalCohomology(ZZ,Ideal) -- see quasidegreesLocalCohomology -- returns the quasidegree sets of local cohomology modules
- quasidegreesLocalCohomology(ZZ,Module) -- see quasidegreesLocalCohomology -- returns the quasidegree sets of local cohomology modules
- removeRedundancy(List) -- see removeRedundancy -- removes redundancies from a list of planes
- toGradedRing(Matrix,PolynomialRing) -- see toGradedRing -- grade a polynomial ring by a matrix
- toricIdeal(Matrix,Ring) -- see toricIdeal -- returns a toric ideal

For the programmer

The object Quasidegrees is a package.