Test exponents for modules with finite phantom projective dimension

Abstract

Let $(R,\mathfrak{m})$ be an equidimensional excellent local ring of prime characteristic $p>0$. We give an alternate proof of the existence of a uniform test exponent for any given $c\in R^{\circ}$ and all ideals generated by (full or partial) systems of parameters. This follows from a more general result about the existence of a test exponent for any given Artinian $R$-module. If we further assume $R$ is Cohen–Macaulay, then there exists a test exponent for any given $c\in R^{\circ}$ and all perfect modules with finite projective dimension.