On the number of generators of an algebra over a commutative ring

A theorem of O. Forster says that if R R is a noetherian ring of Krull dimension d d , then every projective R R -module of rank n n can be generated by d + n d+n elements. S. Chase and R. Swan subsequently showed that this bound is sharp: there exist examples that cannot be generated by fewer than d + n d+n elements. We view rank- n n projective R R -modules as R R -forms of the non-unital R R -algebra R n R^n where the product of any two elements is 0 0 . The first two authors generalized Forster’s theorem to forms of other algebras (not necessarily commutative, associative or unital); A. Shukla and the third author then showed that this generalized Forster bound is optimal for finite étale algebras. In this paper, we prove new upper and lower bounds on the number of generators of an R R -form of a k k -algebra, where k k is an infinite field and R R is a finitely generated k k -ring of Krull dimension d d . In particular, we show that, contrary to expectations, for most types of algeb