A Note on p-Bases of a Regular Affine Domain Extension

Let $R^{p}\subseteq R^{\prime}\subseteq R$ be a tower of commutative rings where R is a regular affine domain over an algebraically closed field of prime characteristic p and R′ is a regular domain. Suppose R has a p-basis {φ₁,..., $\varphi _{r}$ } over $R^{p}$ and $[Q(R^{\prime})\colon Q(R^{p})]=p^{l}(1\leq l\leq r-1)$ . For a subset $\Gamma _{r-l}$ of R whose elements satisfy a certain condition on linear independence, let $M_{\Gamma _{r-l}}$ be a set of maximal ideals m of R such that $\Gamma _{r-l}$ is a p-basis of $R_{\germ{m}}$ over $R_{\germ{m}^{\prime}}^{\prime}$ ( $\germ{m}^{\prime}=\germ{m}\cap R^{\prime}$ ). We shall characterize this set in a geometrical aspect.