infinitelyNearModules -- The sequence of blowups of a semigroup ring as fractional ideals

Usage:

MM = infinitelyNearModules R
Inputs:
- R, a ring, a semigroup ring, as produced by semigroupRing
Outputs:
- MM, a list, of R-modules, one for each semigroup in infinitelyNearSemigroups applied to the underlying semigroup of R

Description

Let R be the semigroup ring of a numerical semigroup S, and let S = S_0 \subseteq S_1 \subseteq \dots \subseteq S_r = N be the sequence of blowups of S returned by infinitelyNearSemigroups. Each S_i is a finitely generated R-submodule of the fraction field frac R: namely, the R-module generated (in frac R) by the monomials t^s for s in a generating set of S_i.

The function infinitelyNearModules returns the sequence of these R-modules. The first entry is (an R-module isomorphic to) R itself.

i1 : R = semigroupRing {3,5,7}

o1 = R

o1 : QuotientRing

i2 : MM = infinitelyNearModules R

o2 = {image | x_0 |, image | x_0 x_2 x_1 |, image | x_2 x_0^2 x_1 |}

o2 : List

i3 : MM/numgens

o3 = {1, 3, 3}

o3 : List

Caveat

The construction goes through fractionalIdeal, which clears denominators by multiplying through by a high enough power of the multiplicity element so as to produce an honest ideal of R. As a result, the modules returned by infinitelyNearModules are isomorphic to the actual infinitely near modules sitting inside frac R, but their internal grading is shifted by the chosen denominator. In other words, what these modules represent is correct \emph{up to a degree shift}; if you need the natural grading inherited from frac R, you must compensate for that shift by hand.

Ways to use infinitelyNearModules:

infinitelyNearModules(Ring)

For the programmer

The object infinitelyNearModules is a method function.

The source of this document is in NumericalSemigroups.m2:2842:0.

infinitelyNearModules -- The sequence of blowups of a semigroup ring as fractional ideals

Description

Caveat

See also

Ways to use infinitelyNearModules:

For the programmer