# Michael Stillman: Quadratic Gorenstein rings and the Koszul property

**Time: **
Wed 2019-06-26 13.15 - 14.15

**Lecturer: **
Michael Stillman (Cornell)

**Location: ** Room 3418, KTH

Abstract: 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 Matt Mastroeni and Hal Schenck.