Proposition 7

— Let L be an extension of a field K and M an L-algebra (for example, an extension of L). If L is regular over K and M is a regular L-algebra, then M is regular as K-algebra.

Let E be an extension of K; since L is regular over K, $E \otimes_K L$ is an integral domain. By Prop. 2 (V, p. 141) the ring $(E \otimes_K L) \otimes_L M$ is thus an integral domain and the same is true of the ring $E \otimes_K M$ isomorphic to it (II. p. 278, Prop. 2). Hence the result.