Proposition 1

Let $A$ and $B$ be local noetherian rings and $\rho : A \to B$ a local homomorphism making $B$ a flat $A$-module. Suppose that one has m_B = B.\rho(m_A). Then dim(A) = dim(B) and B is regular if and only if A is regular.

The first assertion follows from Cor. 1 of Prop. 7 of § 3, No. 4. Since B is flat over A, one can identify m_B^k = B.\rho(m_A^k) with B \otimes_A m_A^k for every integer k \geqslant 0, hence m_B/m_B^2 with B \otimes_A (m_A/m_A^2), or again with \kappa_B \otimes_{\kappa_A} (m_A/m_A^2). One therefore has

(2)

whence the proposition at once.