A classification of generic type 2 artinian rings

Abstract: The commutative local rings are usually placed in the following hierarchy, based on the character of their singularity: regular, hypersurface, complete intersection, and Gorenstein. These classes would be enough to describe all the rings of codepth 0 and 1. However, a new class is needed to describe all the rings of codepth 2. This is the class of Golod rings; an example of such a ring is the quotient of any local ring by the square of the maximal ideal. Such a classification is still possible for all codepth 3 rings if one considers the multiplicative structure of the Tor-algebra of the ring. The Golod rings are exactly the rings with trivial multiplication.

In a joint work with Lars W. Christensen we completely classify the Artinian compressed rings of type 2 of codepth 3 that are obtained from two compressed Gorenstein rings (rings of type 1). We prove that the class of all generic Artinian rings of type 2 is exactly determined by only two easily computable numbers, namely the socle degrees of the two Gorenstein rings.

commutative algebraalgebraic geometrycombinatoricsrings and algebrasrepresentation theory

Audience: researchers in the topic

Geometry, Algebra, Singularities, and Combinatorics

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