Number of generators: an algebraic-geometry approach

The Primitive Element Theorem says that a separable field extension is generated by one element, and a well-known folklore result says that a central simple algebra (CSA) is generated by two elements over its center. The globalization of separable field extensions and CSAs are finite etale algebras and Azumaya algebras, respectively, and so one could ask if something could be said about their number of generators if the base scheme has dimension at most d. The same question can be asked for vector bundles and other types of algebra bundles. I will discuss some recent works with Reichstein, Williams and others where we study this question by turning it into a geometric question, thus finding both upper bounds and examples requiring arbitrarily many generators. For example, if X is a d-dimensional affine algebraic scheme over an infinite field, then any finite etale algebra over X can be generated by d+1 elements, and this cannot be improved in general.