The Lemma on b-functions in Positive Characteristic

Abstract

Let $X$ be an $F$-finite smooth scheme of essentially finite type over a perfect field. This article proves the existence of $b$-functions for locally finitely generated unit $F$-modules when equipped with their induced $\mathbb{D}_X$-module structure. It is shown that the $b$-function has rational roots and is determined locally in the \'etale topology.