Algebraic Geometry Seminar
Let $I$ be a homogeneous ideal in a polynomial ring $S = K[x_1,...,x_n]$ over a field $K$. Many of the properties of the homogeneous ideal associated to I can be determined from the minimal graded free resolution of $I$. Lately there has been a lot of progress in understanding the structure of free resolutions of modules and ideals over $S$ including resolutions of the Boij-Soederberg, Stillman and Eisenbud-Goto Conjectures. In my talk I will briefly survey some of the recent progress on free resolutions and then discuss in more detail some progress on understanding the maximal graded shifts (i.e. degrees of syzygy modules) of ideals and modules over $S$. I will present some new restrictions on the maximal graded shifts of ideals, which can be thought of as restrictions on the shapes of the nonzero entries in the graded Betti table of an ideal.