Commutative Algebra Seminar

Title

Generic local rings interpolate between Gorenstein and Golod

Abstract

Let Q be a power series ring, for example $\mathbb{Q}[![x,y,z]!]$, and let I be an ideal in Q. If the drop in depth from Q to the quotient R=Q/I is at most 3, then R can be classified based on a multiplicative structure on the finite free resolution of R as a Q-module. The identification of the possible multiplicative structures was done 25 years ago, but the question of which structures can actually be realized has only recently seen progress.

In joint work with Veliche and Weyman we construct families of rings $\mathbb{Q}[![x,y,z]!]/I$ that realize multiplicative structures which had been conjectured not to occur. In fact, rings with this structure seem to be everywhere, and what emerges is a picture that describes generic local rings on a one-dimensional scale whose end points are Gorenstein rings and Golod rings.