  A theorem of O. Forster says that if $R$ is a noetherian ring of Krull
dimension $d$, then any projective $R$-module of rank $n$ can be generated by
$d+n$ elements. S. Chase and R. Swan subsequently showed that this bound is
sharp: there exist examples that cannot be generated by fewer than $d+n$
elements. We view projective $R$-modules as $R$-forms of the non-unital
$R$-algebra where the product of any two elements is $0$. 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 \'etale algebras.
  In this paper, we prove new upper and lower bound on the number of generators
of an $R$-form of a $k$-algebra, where $k$ is an infinite field and $R$ has
finite transcendence degree $d$ over $k$. In particular, we show that, contrary
to expectations, for most types of algebras, the generalized Forster bound is
far from optimal. Our results are particularly detailed in the case of Azumaya
algebras. Our proofs are based on reinterpreting the problem as a question
about approximating the classifying stack $BG$, where $G$ is the automorphism
group of the algebra in question, by algebraic spaces of a certain form.
