UBC Algebra and Algebraic Geometry: Sebastian Gant

We study the question of when a generic stably free module splits off a free summand of a given rank. This question has the following geometric interpretation due to M. Raynaud. Let V(r,n) denote the Stiefel variety GL(n)/GL(n-r) over a field k. There is a projection map V(r,n) -> V(1,n) given by “forgetting frames.” Raynaud showed that V(r,n) -> V(1,n) has a section if and only if the following holds: if P is any module over any k-algebra R with the property that P+R is isomorphic to R^n, then P has a free factor of rank r-1. Using machinery from A1-homotopy theory, we characterize those n for which the map V(r,n) -> V(1,n) has a section in the cases r <= 4 and under some restrictions on the base field. This is joint work with Ben Williams.