Using a local monomialization result of Knaf and Kuhlmann, which was generalized by Cutkosky, we prove that the valuation ring of an Abhyankar valuation of a function field over an $F$-finite ground field of prime characteristic is Frobenius split. We show that a Frobenius splitting of a sufficiently well-behaved center lifts to a Frobenius splitting of the valuation ring. We also investigate properties of valuations centered on arbitrary Noetherian domains of prime characteristic. In contrast to our work with Smith (Algebra Number Theory 10:5 (2016), 1057–1090 and its correction in 11:4 (2017), 1003–1007), this paper emphasizes the role of centers in controlling properties of valuation rings in prime characteristic.

使用由 Cutkosky 推广的 Knaf 和 Kuhlmann 的局部单项化结果，我们证明了函数场的 Abhyankar 估值的估值环在$F$- 主要特征的有限地面场是 Frobenius 分裂。我们证明了一个行为良好的中心的 Frobenius 分裂提升为估值环的 Frobenius 分裂。我们还研究了以主要特征的任意 Noetherian 域为中心的估值属性。与我们与 Smith（代数数论10 :5 (2016), 1057-1090 及其在11 :4 (2017), 1003-1007）中的工作相比，本文强调中心在控制估值环属性中的作用在主要特征。