There is the following Theorem of Chevalley:
Let $k$ be a field and $A$ a finite-type reduced $k$-algebra. Then for any $\mathfrak p\in \mathrm{Spec}(A)$ the ring $\hat{A_{\mathfrak p}}$ is reduced.
I would like to learn why this theorem is important, and how it is related to other results. Is it important because it shows that $\mathrm{Spec}(\hat{A_{\mathfrak p}})$ is a variety "in the classical sense"?