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.
This documentation describes version 1.1 of ResidualIntersections, released 07 May 2016.
If you have used this package in your research, please cite it as follows:
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The object ResidualIntersections is a package, defined in ResidualIntersections.m2.
The source of this document is in /build/reproducible-path/macaulay2-1.25.06+ds/M2/Macaulay2/packages/ResidualIntersections.m2:439:0.