# 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.

## Version

This documentation describes version 1.0 of ResLengthThree.

## Source code

The source code from which this documentation is derived is in the file ResLengthThree.m2.

## 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(ChainComplex)" -- see resLengthThreeAlg -- the minimal free resolution presented as a graded-commutative ring
• "resLengthThreeAlg(ChainComplex,List)" -- see resLengthThreeAlg -- the minimal free resolution presented as a graded-commutative ring
• "resLengthThreeTorAlg(ChainComplex)" -- see resLengthThreeTorAlg -- the Tor algebra presented as a graded-commutative ring
• "resLengthThreeTorAlg(ChainComplex,List)" -- see resLengthThreeTorAlg -- the Tor algebra presented as a graded-commutative ring
• "resLengthThreeTorAlgClass(ChainComplex)" -- 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 .