ResLengthThree -- Computation of multiplicative structures on free resolutions of length three

Description

Let I be a homogeneous ideal contained in the irrelevant maximal ideal of a graded ring Q (obtained as a quotient of a polynomial ring). If the length of the minimal free resolution F of $R=Q/I$ is 3, then the resolution admits the structure of a differential graded algebra. The induced algebra structure on $A = Tor^Q(R,k)$ is unique and provides for a classification of such quotient rings. The package determines a multiplicative structure on the free resolution F as well as the unique induced structure on A and the class of the quotient R according to the classification scheme of Avramov, Kustin, and Miller.

Authors

Version

This documentation describes version 1.0 of ResLengthThree.

Citation

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

@misc{ResLengthThreeSource,
  title = {{ResLengthThree: Multiplication in free resolutions of length three. Version~1.0}},
  author = {Lars Winther Christensen and Luigi Ferraro and Francesca Gandini and Frank Moore and Oana Veliche},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

Exports

Functions and commands
- makeRes -- creates a resolution starting from three matrices
- multTableOneOne -- the multiplication table for products of elements in degree one
- multTableOneTwo -- the multiplication table for products of elements in degree one with elements in degree two
- resLengthThreeAlg -- the minimal free resolution presented as a graded-commutative ring
- resLengthThreeTorAlg -- the Tor algebra presented as a graded-commutative ring
- resLengthThreeTorAlgClass -- the class (w.r.t. multiplication in homology) of an ideal
Methods
- multTableOneOne(Ring) -- see multTableOneOne -- the multiplication table for products of elements in degree one
- multTableOneTwo(Ring) -- see multTableOneTwo -- the multiplication table for products of elements in degree one with elements in degree two
- resLengthThreeAlg(Complex) -- see resLengthThreeAlg -- the minimal free resolution presented as a graded-commutative ring
- resLengthThreeAlg(Complex,List) -- see resLengthThreeAlg -- the minimal free resolution presented as a graded-commutative ring
- resLengthThreeTorAlg(Complex) -- see resLengthThreeTorAlg -- the Tor algebra presented as a graded-commutative ring
- resLengthThreeTorAlg(Complex,List) -- see resLengthThreeTorAlg -- the Tor algebra presented as a graded-commutative ring
- resLengthThreeTorAlgClass(Complex) -- see resLengthThreeTorAlgClass -- the class (w.r.t. multiplication in homology) of an ideal
- resLengthThreeTorAlgClass(Ideal) -- see resLengthThreeTorAlgClass -- the class (w.r.t. multiplication in homology) of an ideal
Symbols
- Labels -- an optional argument for multTableOneOne and MultTableOneTwo determining whether to label rows and columns
- Compact -- see multTableOneOne(...,Compact=>...) -- an optional argument for multTableOneOne that prints dots below the diagonal

For the programmer

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

The source of this document is in ResLengthThree.m2:386:0.