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

Abstract:

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.