Basic Ring Theory - 18 (Regular Local Rings)

[Tsakanikas Nickos]

Prove the following assertions.

1. Let $ \left(A,\mathfrak{m}\right) $ be a regular local ring and let $ I $ be a proper ideal of $ A $. Then the quotient $ A / I $ is regular if and only if $ I $ is generated by $ r = \dim(A) - \dim(A/I) $ elements of a coordinate system for $ A $.
2. Let $ \left(A,\mathfrak{m}\right) $ be a noetherian local ring. If $ \mathfrak{p} $ is a prime ideal of $ A $ such that $ A_{\mathfrak{p}} $ and $ A / \mathfrak{p} $ are regular, and $ \mathfrak{p} $ is generated by $ \dim(A_{\mathfrak{p}}) $ elements, then $ A $ is regular.