Commutative Algebra and Algebraic Geometry: The Fellowship of the Ring: Quadratic Gorenstein rings and the Koszul property

Seminar | April 2 | 3:45-4:45 p.m. | 939 Evans Hall

 Michael Stillman, Cornell University

 Department of Mathematics

An artinian local ring $(R,m)$ is called Gorenstein if it has a unique minimal ideal. If $R$ is graded, then it is called Koszul if $R/m$ has a linear $R$-free resolution. Any Koszul algebra is defined by quadratic relations, but the converse is false, and no one knows a finitely computable criterion. Both types of rings occur in many situations in algebraic geometry and commutative algebra, and in many cases, a Gorenstein quadratic algebra coming from geometry is often Koszul (e.g. homogeneous coordinate rings of most canonical curves).

In 2001, Conca, Rossi, and Valla asked the question: must a (graded) quadratic Gorenstein algebra of regularity 3 be Koszul?

I will talk about techniques for deciding whether a quadratic Gorenstein algebra is Koszul and methods for generating many examples which are not Koszul. We will explain how these methods provide a negative answer to the above question, as well as a complete picture in the case of regularity at least 4. (This is joint work with Hal Schenck and Matt Mastroeni).