UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

August 28, 2007

939 Evans Hall

3:45PM: When does I/I^2 have a free summand? A theorem

David Eisenbud

If I is an ideal in a local Noetherian ring A and I can be generated by a regular sequence, then I/I^2 is free as a module over A/I and I has finite projective dimension as an A-module. The converse of this statement is a famous result of Ferrand and Vasconcelos, who prove a little more: if I/I^2 has an (A/I)-free summand of rank k, and I has finite projective dimension, then I contains a regular sequence of length k. I'll explain the proof of this result and some of its applications. In particular, I'll explain its connection with recent joint work with Roya Beheshti on the singularities of generic projections of smooth projective varieties.

5:00PM: Relations between the Principal Minors of a 4 x 4 Matrix

Shaowei Lin

A principal minor of an n x n-matrix is a minor whose rows and columns are indexed by the same subset I of {1, ..., n}. One problem of interest is to find a finite set of generators for the prime ideal Pn of all polynomial relations among the principal minors of an n x n-matrix. While this elimination-type problem is in theory solvable by Groebner bases techniques, it is computationally expensive even for the case n = 4. Recently, a set of generators for the prime ideal in the symmetric 4 x 4 case was found by Holtz and Sturmfels. In this talk, we discuss how to find generators for the general 4 x 4 case, using classical results involving so-called devertebrated minors. The question of whether this set generates P4 remains open. This is work in progress.

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