Remark 2

Let k be a field and A a k-algebra such that the ring $A_{(k')}$ is regular for every purely inseparable extension $k'$ of k, of finite degree and of height $\leq 1$; then A is absolutely regular. In fact, let $k'$ be such an extension; for every maximal ideal m of A, the ring $k' \otimes_k A_m$ is identified with a ring of fractions of $A_{(k')}$, and is therefore regular. By Prop. 8 above and Prop. 6 of § 6, No. 4, the algebra A is absolutely regular.