Quadratic Gorenstein rings and the Koszul property

  • Michael Stillman (Cornell University, Ithaca, USA)
  • Guest
E1 05 (Leibniz-Saal)


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 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.

(Joint work with Matt Mastroeni and Hal Schenck)

01.07.19 04.07.19

Summer School on Randomness and Learning in Non-Linear Algebra

MPI für Mathematik in den Naturwissenschaften Leipzig (Leipzig) E1 05 (Leibniz-Saal)
Universität Leipzig (Leipzig) Felix-Klein-Hörsaal

Saskia Gutzschebauch

Max-Planck-Institut für Mathematik in den Naturwissenschaften Contact via Mail

Paul Breiding

Technische Universität Berlin

Jesus De Loera

University of California at Davis

Despina Stasi

Illinois Institute of Technology

Sonja Petrovic

Illinois Institute of Technology