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 ℤ^{d}-graded module M is the Zariski closure in C^{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 ℤ^{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 ℤ^{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 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.
Version 1.0 of this package was accepted for publication in volume 9 of the journal 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, http://github.com/Macaulay2/M2/blob/master/M2/Macaulay2/packages/Quasidegrees.m2, commit number d76252d2c8d38f0ec55212eb458869503b1f0312.
This documentation describes version 1.0 of Quasidegrees.
The source code from which this documentation is derived is in the file Quasidegrees.m2.