UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

April 23, 2020

939 Evans Hall

1:30PM - 3:30PM: Quadratic Gorenstein rings and the Koszul property (joint work with Matt Mastroeni and Hal Schenck)

Mike Stillman

Location: Online via Zoom

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 other examples of quadratic Gorenstein algebras which are not Koszul.

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