MonomialAlgebras -- Decompose a monomial algebra as a module over a subalgebra.

Overview:

Consider a monoid B in \mathbb{N}^m and a submonoid A \subseteq B (both finitely generated) such that K[B] is a finitely generated K[A]-module (with the module structure given by inclusion, and K being a field).

Note that this is equivalent to the condition that the corresponding cones C(A) and C(B) spanned by the monoids are equal. From this it follows that G(B)/G(A) is finite, where G(A) and G(B) are the groups generated by the monoids.

This package provides functions to decompose the corresponding monomial algebra K[B] as a direct sum of monomial ideals in K[A]. In

Le Tuan Hoa, Juergen Stueckrad: Castelnuovo-Mumford regularity of simplicial toric rings, Journal of Algebra, Volume 259, Issue 1, 1 January 2003, pages 127-146,

it is shown that this decomposition exists in the case that B is homogeneous and simplicial and A is generated by minimal generators of B on the extremal rays of C(B). In particular then K[A] is a Noether normalization of K[B].

For the existence of the decomposition in the general (non-simplicial) case, and for algorithms computing the decomposition and the regularity see:

J. Boehm, D. Eisenbud, M. Nitsche: Decomposition of semigroup algebras, Exper. Math. 21(4) (2012) 385-394, http://arxiv.org/abs/1110.3653.

Using the decomposition algorithm, the package also provides fast functions to test ring-theoretic properties of monomial algebras. For details on the corresponding algorithms see:

J. Boehm, D. Eisenbud, M. Nitsche: Decomposition of Monomial Algebras: Applications and Algorithms, 2012, http://arxiv.org/abs/1206.1735.

Key user functions:

Decomposition:

decomposeMonomialAlgebra -- Decomposition of one monomial algebra over a subalgebra.

decomposeHomogeneousMA -- Decomposition of one homogeneous monomial algebra over a subalgebra.

Ring-theoretic properties:

isCohenMacaulayMA -- Test whether a simplicial monomial algebra is Cohen-Macaulay.

isGorensteinMA -- Test whether a simplicial monomial algebra is Gorenstein.

isBuchsbaumMA -- Test whether a simplicial monomial algebra is Buchsbaum.

isNormalMA -- Test whether a simplicial monomial algebra is normal.

isSeminormalMA -- Test whether a simplicial monomial algebra is seminormal.

isSimplicialMA -- Test whether a monomial algebra is simplicial.

Regularity:

regularityMA -- Compute the regularity via the decomposition.

degreeMA -- Compute the degree via the decomposition.

codimMA -- Compute the codimension of a monomial algebra.

Options:

Computation of the binomial ideal associated to B via the function binomialIdeal can be done via two different methods:

The standard method uses the Macaulay2 function kernel and requires no configuration.

Alternatively, the external program 4ti2, http://www.4ti2.de/ can be called, which is delivered with the Macaulay2 distribution. For large examples this is typically much faster. To use this option load the MonomialAlgebras package with the following configuration option:

loadPackage("MonomialAlgebras",Configuration=>\{"Use4ti2"=>true\})

Note that you can change the standard option by editing the file init-MonomialAlgebras.m2 in the .Macaulay directory in your home directory.

In order to use 4ti2, on some systems you may have to configure the Macaulay2 package FourTiTwo first.

Setup:

Install this Package by doing

installPackage("MonomialAlgebras")

Tests:

The following files contain the tuples (regularity, degree, codim) for every semigroup B in \mathbb{N}^d with fixed d and fixed coordinate sum a. Using this data we have verified the Eisenbud-Goto conjecture in these cases.

d = 3:

a = 3: proveEG_33

a = 4: proveEG_43

a = 5: proveEG_53

d = 4:

a = 2: proveEG_24

a = 3: proveEG_34

d = 5:

a = 2: proveEG_25

More functions related to testing the Eisenbud-Goto conjecture can be found in the MonomialAlgebraExtras package.

Diagrams:

We illustrate the tests with Diagrams showing projections of the set of all possible (regularity, degree, codim).

Authors

• David Eisenbud <[email protected]>
• Janko Boehm <[email protected]>
• Max Nitsche <[email protected]>

Certification

Version 2.3 of this package was accepted for publication in volume 5 of The Journal of Software for Algebra and Geometry on 2013-04-07, in the article Decomposition of Monomial Algebras: Applications and Algorithms. That version can be obtained from the journal or from the Macaulay2 source code repository.

Version

This documentation describes version 2.3 of MonomialAlgebras.

Source code

The source code from which this documentation is derived is in the file MonomialAlgebras.m2. The auxiliary files accompanying it are in the directory MonomialAlgebras/.

Exports