Algebra and Geometry Seminar

Jason McCullough – Quadratic and Koszul Licci Ideals
Abstract: Licci stands for LInkage Class of a Complete Intersection. This notion was introduced by Peskine-Szpiro as a convenient way to study Cohen-Macaulay varieties. While defined in a larger context, I will stick to ideals in a polynomial ring. First C = (f_1,…,f_c) in a polynomial ring S is a complete intersection if each f_i is a nonzero divisor on S/(f_1,…,f_{i-1}). They define the varieties X in affine or projective space defined by exactly codim(X) many equations. Two ideals I,J in S are linked by C if J = C:I and I = C:J. We will see that linked ideals share many nice properties. One can extend this notion by saying that I and J are in the same linkage class if there is a sequence of ideals I = L_0,…,L_t=J complete intersections C_1,…,C_t such that L_(i-1) and L_i are linked by C_i. Then I is licci if it is in the linkage class of a complete intersection. I will review these definitions and go over some elementary examples.