Definition: If I ⊂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.

- isLicci -- Tests whether an ideal is licci
- residualCodims -- a list of possible residual intersection codimensions
- genericResidual -- Computes generic residual intersections of an ideal
- residualCodims -- a list of possible residual intersection codimensions
- genericArtinNagata -- Generic Artin nagata
- numgensByCodim -- maximum number of generators of localizations of a monomial ideal
- maxGs -- maximum G_s of a monomial ideal
- koszulDepth -- Computes the depths of the Koszul homology
- hasSlidingDepth -- Checks if an ideal has the sliding depth property
- isStronglyCM -- Checks if the given ideal is Strongly Cohen Macaulay
- depthsOfPowers -- Computes depth of powers of an ideal

- Katie Ansaldi <kansaldi@gmail.com>
- Jay Yang <jkelleyy@gmail.com>

- 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

- Symbols
- UseNormalModule -- option for linkageBound and isLicci