A note on p-bases of a regular affine domain extension

Let R^p\subseteq R'\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 \{\varphi _1,\dots ,\varphi _r\} over R^p and [Q(R') : 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 \mathfrak m of R such that \Gamma _{r-l} is a p-basis of R_{\mathfrak m} over R'_{\mathfrak m'} (\mathfrak m'=\mathfrak m\cap R'). We shall characterize this set in a geometrical aspect.