We construct a scheme $B(r; {\mathbb {A}}^n)$ such that a map $X \to B(r; {\mathbb {A}}^n)$ corresponds to a degree-n étale algebra on X equipped with r generating global sections. We then show that when $n=2$ , i.e., in the quadratic étale case, the singular cohomology of $B(r; {\mathbb {A}}^n)({\mathbb {R}})$ can be used to reconstruct a famous example of S. Chase and to extend its application to showing that there is a smooth affine $r-1$ -dimensional ${\mathbb {R}}$ -variety on which there are étale algebras ${\mathcal {A}}_n$ of arbitrary degrees n that cannot be generated by fewer than r elements. This shows that in the étale algebra case, a bound established by U. First and Z. Reichstein in [6] is sharp.

我们构造了一个方案 $B(r; {\mathbb {A}}^n)$ 使得映射 $X \to B(r; {\mathbb {A}}^n)$ 对应于一个度 - n étale X 上的代数配备r生成全局部分。然后我们证明当 $n=2$ 时 ，即在二次 étale 情况下， $B(r; {\mathbb {A}}^n)({\mathbb {R}})$ 的奇异上同调可以是用于重建 S. Chase 的一个著名例子，并将其应用扩展到表明存在平滑仿射 $r-1$ - 维 ${\mathbb {R}}$ -variety 上有 étale 代数 ${\数学{A}}_n$ 不能由少于r 个元素生成的任意度数n。这表明在 étale 代数情况下，U. First 和 Z. Reichstein 在 [6] 中建立的界限是尖锐的。