On the minimal number of generators of an \'etale algebra

O. Forster proved that over a ring R of Krull dimension d a finite module M of rank at most n can be generated by n + d elements. Generalizing this in great measure U. First and Z. Reichstein showed that any finite R-algebra A can be generated by n + d elements if each localization of A at a maximal ideal of R can be generated by n elements. It is natural to ask if the upper bounds can be improved. For modules over rings R. Swan produced examples to match the upper bound. Recently the second author obtained weaker lower bounds in the context of Azumaya algebras. In this paper we investigate this question for etale algebras. We show that the upper bound is indeed sharp. Our main result is a construction of universal varieties for degree-2 etale algebras equipped with a set of r generators and explicit examples realizing the upper bound of First & Reichstein.