BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Mike Stillman (Cornell University) DTSTART:20200423T203000Z DTEND:20200423T220000Z DTSTAMP:20260423T021404Z UID:FOTR/1 DESCRIPTION:Title: Qua dratic Gorenstein rings and the Koszul property\nby Mike Stillman (Cor nell University) as part of Fellowship of the Ring\n\n\nAbstract\nA graded ring R = S/I is Gorenstein (S = polynomial ring\, I =\nhomogeneous ideal) if the length of its free resolution over S is its\ncodimension in S\, an d the top betti number is one. R is called Koszul\nif the free resolution of k = R/(maximal homogeneous ideal) over R is\nlinear. Any Koszul algebra is defined by quadratic relations\, but the\nconverse is false\, and no o ne knows a finitely computable criterion.\nBoth types of rings have dualit y properties\, and occur in many\nsituations in algebraic geometry and com mutative algebra\, and in many\ncases\, a Gorenstein quadratic algebra com ing from geometry is often\nKoszul (e.g. homogeneous coordinate rings of m ost canonical curves).\n\nIn 2001\, Conca\, Rossi\, and Valla asked the qu estion: must a (graded)\nquadratic Gorenstein algebra of regularity 3 be K oszul?\n\nIn the first 45 minutes\, we will define these notions\, and giv e\nexamples of quadratic Gorenstein algebras and Koszul algebras. We\nwill give methods for their construction\, e.g. via inverse systems.\nAfter a short break\, we will use these techniques to answer negatively\nthe above question\, as well as see how to construct many other\nexamples of quadra tic Gorenstein algebras which are not Koszul.\n LOCATION:https://researchseminars.org/talk/FOTR/1/ END:VEVENT END:VCALENDAR