Xianglong Ni (Notre Dame)
Title: Grade three perfect ideals and length four self-dual resolutions
Abstract: Let d_3 be the last differential in a minimal free resolution of a grade three perfect ideal. If f is a submatrix of d_3 consisting of all but one column, then it turns out that the ideal of maximal minors I(f) has grade at most four. Furthermore, if the grade is four, then f is the last differential of a length four self-dual resolution, and every such resolution arises in this manner. In particular, every grade four Gorenstein ideal "appears in" the resolution of some grade three perfect ideal of type two.This construction is elementary to state, and can be proven without much fancy machinery, as we'll see in the first half of the talk after surveying some background results on structure theorems for free resolutions and perfect ideals. But it is actually one instance of a much deeper "triality" phenomenon that we do not fully understand yet. The second half of the talk focuses on explaining the connection to representation theory, how our seemingly ad-hoc construction is natural from this perspective, and how it enables us to deduce a structure theorem for certain grade four Gorenstein ideals from earlier work done in the setting of grade three perfect ideals. Time permitting, we will also discuss some (work in progress) connections with the theory of linkage. This is joint work with Tymoteusz Chmiel, Lorenzo Guerrieri, and Jerzy Weyman.
