Remark

Let $A$ be a regular noetherian local ring and $q \subset m_A$ an ideal of $A$ such that $A/q$ is regular. Let $(x_1, ..., x_r)$ be a sequence of elements of $q$ whose classes modulo $m_A^2$ generate the vector space $(q + m_A^2)/m_A^2$ over the field $\kappa_A$. The proof of Cor. 2 shows that the ideal $q$ of $A$ is generated by $(x_1, ..., x_r)$.