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# 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

- Lars Winther Christensen <lars.w.christensen@ttu.edu>
- Luigi Ferraro <lferraro@ttu.edu>
- Francesca Gandini <fra.gandi.phd@gmail.com>
- Frank Moore <moorewf@wfu.edu>
- Oana Veliche <o.veliche@northeastern.edu>

## Version

This documentation describes version 1.0 of ResLengthThree, released 3 December 2020.

## 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/stable/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 /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/ResLengthThree.m2:386:0.