论文标题
关于代数的发电机数量
On the number of generators of an algebra over a commutative ring
论文作者
论文摘要
O. Forster的定理说，如果$ r $是Krull Dimension $ d $的Noetherian环，那么任何投影$ r $ r $ - $ n $的模块都可以由$ d+n $元素产生。 S. Chase和R. Swan随后表明，这种界限很清晰：存在少于$ d+n $元素无法生成的示例。我们将投影$ r $ - 模型视为$ r $ - 非潜水$ r $ -r $ - 代数，其中任何两个元素的产品均为$ 0 $。前两个作者将福斯特定理概括为其他代数的形式（不一定是可交换，联想或联合的）；舒克拉（A. Shukla）和第三作者随后表明，这种广义的福斯特（Forster Bound）对典型代数是最佳的。 在本文中，我们证明了$ k $ -Algebra的发电机数量的新上限和下限，其中$ k $是无限的字段，$ r $具有有限的超越度$ D $ d $ abo $ k $。特别是，我们表明，与期望相反，对于大多数类型的代数，广义的福斯特绑定远非最佳。对于Azumaya代数，我们的结果尤其详细。我们的证明是基于将问题重新解释为近似分类堆栈$ bg $的问题，其中$ g $是所讨论的代数的自动形态组，该组是某种形式的代数空间。
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 étale 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.
