TorAlgebra -- Classification of local rings based on multiplication in homology

Description

Let $I$ be an ideal of a regular local ring $Q$ with residue field $k$. The length of the minimal free resolution of $R=Q/I$ is called the codepth of $R$; if it is at most $3$, then the resolution carries a structure of a differential graded algebra. While the DG algebra structure may not be unique, the induced algebra structure on Tor$^Q$ ($R,k$) is unique and provides for a classification of such local rings.

According to the multiplicative structure on Tor$^Q$ ($R,k$), a non-zero local ring $R$ of codepth at most 3 belongs to exactly one of the (parametrized) classes designated B, C(c), G(r), H(p,q), S, or T. An overview of the theory can be found in L.L. Avramov, A cohomological study of local rings of embedding codepth 3.

There is a similar classification of Gorenstein local rings of codepth 4, due to A.R. Kustin and M. Miller. There are four classes, which in the original paper Classification of the Tor-Algebras of Codimension Four Gorenstein Local rings, are called A, B, C, and D, while in the survey Homological asymptotics of modules over local rings by L.L. Avramov, they are called CI, GGO, GTE, and GH(p), respectively. Here we denote these classes C(c), GS, GT, and GH(p), respectively.

The package implements an algorithm for classification of local rings in the sense discussed above. For rings of codepth at most 3 it is described in L.W. Christensen and O. Veliche, Local rings of embedding codepth 3: A classification algorithm. The classification of Gorenstein rings of codepth 4 is analogous.

The package also recognizes Golod rings, Gorenstein rings, and complete intersection rings of any codepth. To recognize Golod rings the package implements a test found in J. Burke, Higher homotopies and Golod rings.

Certification

Version 1.0 of this package was accepted for publication in volume 6 of The Journal of Software for Algebra and Geometry on 2014-07-11, in the article Local rings of embedding codepth 3: A classification algorithm. That version can be obtained from the journal or from the Macaulay2 source code repository.

Version

This documentation describes version 2.1 of TorAlgebra.

Source code

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

Exports

• Functions and commands
• isCI -- whether the ring is complete intersection
• isGolod -- whether the ring is Golod
• isGorenstein -- whether the ring is Gorenstein
• setAttemptsAtGenericReduction -- control the number of attempts to compute Bass numbers via a generic reduction
• torAlgClass -- the class (w.r.t. multiplication in homology) of a local ring
• torAlgData -- invariants of a local ring and its class (w.r.t. multiplication in homology)
• torAlgDataList -- list invariants of a local ring
• torAlgDataPrint -- print invariants of a local ring
• Methods
• "isCI(QuotientRing)" -- see isCI -- whether the ring is complete intersection
• isCI(Ideal) -- whether the ring is complete intersection
• "isGolod(QuotientRing)" -- see isGolod -- whether the ring is Golod
• isGolod(Ideal) -- whether the ring is Golod
• "isGorenstein(QuotientRing)" -- see isGorenstein -- whether the ring is Gorenstein
• isGorenstein(Ideal) -- whether the ring is Gorenstein
• "torAlgClass(QuotientRing)" -- see torAlgClass -- the class (w.r.t. multiplication in homology) of a local ring
• torAlgClass(Ideal) -- the class (w.r.t. multiplication in homology) of a local ring
• "torAlgData(QuotientRing)" -- see torAlgData -- invariants of a local ring and its class (w.r.t. multiplication in homology)
• torAlgData(Ideal) -- invariants of a local ring and its class (w.r.t. multiplication in homology)
• "torAlgDataList(QuotientRing,List)" -- see torAlgDataList -- list invariants of a local ring
• torAlgDataList(Ideal,List) -- list invariants of a local ring
• "torAlgDataPrint(QuotientRing,List)" -- see torAlgDataPrint -- print invariants of a local ring
• torAlgDataPrint(Ideal,List) -- print invariants of a local ring
• Symbols

For the programmer

The object TorAlgebra is .