Seminar

3:45 Jesse Burke: Higher Homotopies and Golod Rings

Let Q be a commutative ring, R = Q/I a quotient ring, and M an R-module. Let A be a Q-projective resolution of R and G a Q-projective resolution of M. I will explain the Q-linear A∞-algebra structure on A and an A∞ A-module structure on G using higher homotopies. From these one can: build an R-projective resolution of M from A and G, modify A and G to obtain Q-free resolutions for all R-syzygies of M, and, when Q is local and A and G are min- imal, describe the differentials in the Eilenberg-Moore spectral sequence for M. These methods work especially well for Golod modules, and show that if the inequality traditionally used to define Golod modules is an equality in the first dim Q+1 degrees, then the module is Golod. Finally, we give an explicit construction of the minimal R-free resolution of every finitely generated module, when R is Golod.



5:00 Justin Chen: The Green-Lazarsfeld Gonality Conjecture (after Ein-Lazarsfeld)

Abstract from the Ein-Lazarsfeld paper:

We show that a small variant of the methods used by Voisin in her study of canonical curves leads to a surprisingly quick proof of the gonality conjecture of Green and the second author, asserting that one can read off the gonality of a curve C from its resolution in the embedding defined by any one line bundle of sufficiently large degree. More generally, we establish a necessary and sufficient condition for the asymptotic vanishing of the weight one syzygies of the module associated to an arbitrary line bundle on C.