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Alessio D'Alì: On the Koszul property for quadratic Artinian Gorenstein graded rings

Time: Mon 2022-05-02 15.00 - 16.00

Location: Zoom

Video link: Meeting ID: 634 5223 2191

Participating: Alessio D'Alì (Osnabrück)

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Quadratic Artinian Gorenstein graded rings have been the subject of several recent papers in commutative algebra: in this talk I want to focus on two topics that involve such rings. The first goal is to introduce a combinatorial construction introduced by Gondim and Zappalà and developed in later independent work by myself and Lorenzo Venturello (KTH Stockholm). Such a construction takes a pure flag simplicial complex \(\Delta\) and associates with it an Artinian Gorenstein ring \(A_{\Delta}\); similarly to Stanley-Reisner theory, there exists a rich two-way vocabulary between the combinatorics of \(\Delta\) and the algebraic properties of \(A_{\Delta}\). In particular, \(A_{\Delta}\) turns out to be Koszul (respectively, to admit a Gröbner basis of quadrics) if and only if \(\Delta\) is Cohen-Macaulay (respectively, shellable). The second goal is to introduce a problem which is - to the best of my knowledge - still open. Conca, Rossi and Valla asked in 2001 whether every quadratic Artinian Gorenstein algebra of socle degree 3 has the Koszul property. Using Nagata idealization, in 2019 Mastroeni, Schenck and Stillman answered this question in the negative and constructed non-Koszul objects with codimension 9 and higher. In 2020, McCullough and Seceleanu further exhibited a non-Koszul object in codimension 8 and proved that this is the lowest achievable codimension when working with Nagata idealization. It is currently unknown whether Koszulness needs to hold in codimensions 6 and 7; I'd like to present some of the challenges that come with investigating such a question, and possibly to have some brainstorming with the audience.