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 Krull dimension alttext="d"> d encoding="application/x-tex">d , then every projective -module rank alttext="n"> n encoding="application/x-tex">n can be generated by alttext="d plus n"> + encoding="application/x-tex">d+n elements. S. Chase and R. Swan subsequently showed this bound sharp: there exist examples cannot fewer than We view rank--modules as -forms the non-unital -algebra R Superscript encoding="application/x-tex">R^n where product any two elements alttext="0"> 0 encoding="application/x-tex">0 . The first authors generalized Forster’s to forms other algebras (not necessarily commutative, associative or unital); A. Shukla third author optimal for finite étale algebras. In paper, we prove new upper lower bounds on number generators an -form alttext="k"> k encoding="application/x-tex">k -algebra, infinite field finitely -ring particular, show that, contrary expectations, most types algebras, far from optimal. Our results are particularly detailed in case Azumaya proofs based reinterpreting problem question about approximating classifying stack B G"> B G encoding="application/x-tex">BG encoding="application/x-tex">G automorphism group algebra question, algebraic spaces certain type.