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# Fellowship of the Ring: Stillman, “Quadratic Gorenstein rings and the Koszul property”

## April 21, 2020 @ 8:00 am - 5:00 pm EDT

This will be a weekly Commutative Algebra webinar, hosted by MSRI, that will take place at 1:30pm Pacific time every Thursday. The organizing committee is listed below. Each talk will have two parts: the first part should be accessible to graduate students after a first course in commutative algebra, and should survey the basic facts and the goals of some area of commutative algebra; the second part will typically cover recent research.

The second Seminar in the series:

**Date/Time:** Thursday, April 23, 1:30-3pm Pacific time

**Speaker:** Mike Stillman, Cornell University

**Title:** Quadratic Gorenstein rings and the Koszul property

**Abstract:** (joint work with Matt Mastroeni and Hal Schenck)

A graded ring R = S/I is Gorenstein (S = polynomial ring, I = homogeneous ideal) if the length of its free resolution over S is its codimension in S, and the top betti number is one. R is called Koszul if the free resolution of k = R/(maximal homogeneous ideal) over R is linear. 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 have duality properties, and 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?

In the first 45 minutes, we will define these notions, and give examples of quadratic Gorenstein algebras and Koszul algebras. We will give methods for their construction, e.g. via inverse systems. After a short break, we will use these techniques to answer negatively the above question, as well as see how to construct many otherexamples of quadratic Gorenstein algebras which are not Koszul.

**Zoom link:** https://msri.zoom.us/webinar/register/WN_RZUNkiShRQS4PVtZa8P9jw

**How to join:** The goal of the seminar is to reach as wide as possible a group of students and faculty interested in commutative algebra, but some protection is necessary to avoid annoying interference on Zoom. Therefore, to be on the mailing list of the seminar and receive login information for each webinar, one must be a member, vetted by one of the organizers. Students and faculty are invited to send an email requesting membership *from a university account* to any one of the organizers.

**Organizing Committee:**

- David Eisenbud <de@msri.org>, chair
- Mel Hochster <hochster@umich.edu>
- Craig Huneke <huneke@virginia.edu>
- Srikanth Iyengar <iyengar@math.utah.edu>
- Claudia Miller <clamille@syr.edu>
- Irena Peeva <ivp1@cornell.edu>
- Steven Sam <ssam@ucsd.edu>
- Karl E Schwede <schwede@math.utah.edu>
- Bernd Ulrich <bulrich@purdue.edu>