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TorAlgebra :: TorAlgebra

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


Let I be an ideal of a regular local ring Q with residue field k. The minimal free resolution of R=Q/I carries a structure of a differential graded algebra. If the length of the resolution, which is called the codepth of R, is at most 3, then the induced algebra structure on TorQ* (R,k) is unique and provides for a classification of such local rings.

According to the multiplicative structure on TorQ* (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 a gold star

Version 1.0 of this package was accepted for publication in volume 6 of the journal 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,, commit number 4b2e83cd591e7dca954bc0dd9badbb23f61595c0.


This documentation describes version 2.0 of TorAlgebra.

Source code

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