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MonomialAlgebras :: MonomialAlgebras

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

Description

Overview:

Consider a monoid B in ℕm and a submonid A ⊆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 an algorithm for computing the decomposition see:

J.Boehm, D. Eisenbud, M. Nitsche: Decomposition of semigroup algebras, 2011, preprint.

Using the decomposition algorithm, the package also provides fast functions to test ring-theoretic properties of monomial algebras, and to compute the regularity.

Key user functions:

Decomposition:

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

decomposeHomogeneousMonomialAlgebra -- 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.

Options:

Computation of the binomial ideal associated to B via the function monomialAlgebraIdeal 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 you have to configure the Macaulay2 package FourTiTwo first. On most systems it should work right out of the box.

Setup:

Install this Package by doing

installPackage("MonomialAlgebras")

Tests:

The following files contain the tuples (regularity, degree, codim) for every semigroup B in ℕd with fixed d and fixed coodinate sum a. This gives a computer proof of 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

Version

This documentation describes version 2.1 of MonomialAlgebras.

Source code

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

Exports