UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

November 22, 2016

939 Evans Hall


3:45PM: STABILITY OF BETTI NUMBERS UNDER SOME REDUCTION PROCESSES WITH APPLICATION TO CHORDALITY OF CLUTTERS

Mina Bigdeli

One of the most challenging problems in combinatorial commutative algebra is to give a characterization of the square free monomial ideals with linear resolution independent of the characteristic of the base field. Thanks to Fröberg, the problem is solved for quadratic ideals. In trying to generalize the result of Fröberg, in this talk, we first introduce a reduction process on a d-uniform clutter C under which the non-linear Betti numbers of the ideal associated to the complement of C do not change. Motivated by this result, we define a class of d-uniform clutters, called chordal, with the property that the ideal I(C) has a d-linear resolution over any field, if C is the complement of an element in this class. We also introduce a class of d-uniform clutters whose associated ideals have linear quotients. Finally, we state some open problems regarding these two classes.

5:00PM: Tor as a module over an exterior algebra

David Eisenbud

Let S is a regular local ring with residue field k, let f = f1..fc in S be a regular sequence, set R = S/(f), and suppose that M is a finitely generated R-module. In this setting Tor^S(M,k) is naturally a module over an exterior algebra on c generators, and Ext_R(M,k) is naturally a module over the “dual” polynomial ring on c generators. I will explain these structures and something about their relationship. This is joint work with Irena Peeva and Frank-Olaf Schreyer.

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