Algebraic Geometry Seminar: "Distinguishing objects with separable algebras"

Abstract: Given an algebraic object X defined over a field k, such as a variety, its k-forms are the objects which may not be isomorphic over the original field but become isomorphic to X after a field extension.  Among the most famous examples are the Severi-Brauer varieties and the central simple algebras, which are the k-forms of projective spaces and matrix algebras, respectively.  Indeed, to each Severi-Brauer variety one can associate a central simple algebra (and a corresponding class in the Brauer group) and this invariant can be used to distinguish isomorphism classes of the original varieties.

Separable algebras are a generalization of central simple algebras. It is natural to attach certain separable algebras to k-forms of algebraic objects in order to distinguish them.  As with Severi-Brauer varieties, these invariants often suffice to completely determine the original varieties.  Sometimes, features like rationality or the existence of k-points can be determined from the structure of the corresponding separable algebras.  However, there are cases where we can prove that any natural invariants of this type are insufficient.

I will overview k-forms, separable algebras, and the successes and limitations of invariants as above.  A major source of these invariants are exceptional collections for the derived categories of varieties.  The main class of varieties considered will be arithmetic toric varieties, which are computationally manageable, but still exhibit interesting behavior in this setting.

Host: Humberto Diaz