ResidualIntersections -- Tests for the conditions used in the theory of residual intersections

Description

Definition: If I \subset S is an ideal in a polynomial ring (or Gorenstein ring) and a_1..a_s are elements of I, then K = (a_1..a_s):I is called an s-residual intersection of I if the codimension of K is at least s.

In the simplest case, s == codim I, the ideal K is said to be linked to I if also I = (a_1..a_s):K; this is automatic when S/I is Cohen-Macaulay, and in this case S/K is also Cohen-Macaulay; see Peskine-Szpiro, Liaison des variétés algébriques. I. Invent. Math. 26 (1974), 271–302).

The theory for s>c, which has been used in algebraic geometry since the 19th century, was initiated in a commutative algebra setting by Artin and Nagata in the paper Residual intersections in Cohen-Macaulay rings. J. Math. Kyoto Univ. 12 (1972), 307–323.

Craig Huneke (Strongly Cohen-Macaulay schemes and residual intersections, Trans. Amer. Math. Soc. 277 (1983), no. 2, 739–763) proved that an s-residual intersection K is Cohen-Macaulay if I satisfies the G_d condition and is strongly Cohen-Macaulay, and successive authors have weakened the latter condition to sliding depth, and, most recently, Bernd Ulrich (Artin-Nagata properties and reductions of ideals. Commutative algebra: syzygies, multiplicities, and birational algebra, Contemp. Math., 159, 1994) showed that the weaker condition depth( S/(I^t) ) >= dim(S/I) - (t-1) for t = 1..s-codim I +1 suffices. All these properties are true if I is licci.

This package implements tests for most of these properties.

Authors

Version

This documentation describes version 1.1 of ResidualIntersections.

Citation

If you have used this package in your research, please cite it as follows:

@misc{ResidualIntersectionsSource,
  title = {{ResidualIntersections: conditions associated to residual intersection theory. Version~1.1}},
  author = {Katie Ansaldi and David Eisenbud and Robert Krone and Jay Yang},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

Exports

Functions and commands
- depthsOfPowers -- Computes depth of powers of an ideal
- genericArtinNagata -- Generic Artin nagata
- genericResidual -- Computes generic residual intersections of an ideal
- hasSlidingDepth -- Checks if an ideal has the sliding depth property
- isLicci -- Tests whether an ideal is licci
- isStronglyCM -- Checks if the given ideal is Strongly Cohen Macaulay
- koszulDepth -- Computes the depths of the Koszul homology
- linkageBound -- computes a bound on the number of general links of an ideal to test the licci property
- maxGs -- maximum G_s of a monomial ideal
- numgensByCodim -- maximum number of generators of localizations of a monomial ideal
- residualCodims -- a list of possible residual intersection codimensions
Methods
- depthsOfPowers(ZZ,Ideal) -- see depthsOfPowers -- Computes depth of powers of an ideal
- depthsOfPowers(ZZ,ZZ,Ideal) -- see depthsOfPowers -- Computes depth of powers of an ideal
- genericArtinNagata(ZZ,Ideal) -- see genericArtinNagata -- Generic Artin nagata
- genericResidual(ZZ,Ideal) -- see genericResidual -- Computes generic residual intersections of an ideal
- hasSlidingDepth(ZZ,Ideal) -- see hasSlidingDepth -- Checks if an ideal has the sliding depth property
- isLicci(Ideal) -- see isLicci -- Tests whether an ideal is licci
- isLicci(ZZ,Ideal) -- see isLicci -- Tests whether an ideal is licci
- isLicci(ZZ,ZZ,Ideal) -- see isLicci -- Tests whether an ideal is licci
- isStronglyCM(Ideal) -- see isStronglyCM -- Checks if the given ideal is Strongly Cohen Macaulay
- koszulDepth(Ideal) -- see koszulDepth -- Computes the depths of the Koszul homology
- koszulDepth(ZZ,Ideal) -- see koszulDepth -- Computes the depths of the Koszul homology
- linkageBound(Ideal) -- see linkageBound -- computes a bound on the number of general links of an ideal to test the licci property
- maxGs(MonomialIdeal) -- see maxGs -- maximum G_s of a monomial ideal
- numgensByCodim(MonomialIdeal) -- see numgensByCodim -- maximum number of generators of localizations of a monomial ideal
- numgensByCodim(MonomialIdeal,ZZ) -- see numgensByCodim -- maximum number of generators of localizations of a monomial ideal
- residualCodims(MonomialIdeal) -- see residualCodims -- a list of possible residual intersection codimensions
Symbols
- UseNormalModule -- option for linkageBound and isLicci

For the programmer

The object ResidualIntersections is a package, defined in ResidualIntersections.m2.

The source of this document is in ResidualIntersections.m2:439:0.