Test exponents for modules with finite phantom projective dimension (Q2452313)

The authors exhibit a new proof of a theorem of \textit{R. Y. Sharp} [Mich. Math. J. 54, No. 2, 307--317 (2006; Zbl 1116.13004)] which states: For an equidimensional excellent local ring of characteristic \(p\) and an element \(c\) of \(R\) not in the union of minimal primes, there exists a test exponent for \(c\) and all parameter ideals. The new proof utilizes colon capturing and the Artinian property of the top local cohomology module of \(R\) with respect to the maximal ideal. They further prove that if \(R\) is additionally either Cohen Macaulay or of dimension at most two then there is a test exponent for \(c\) and all \(R\)-modules \(M\) of finite length and finite phantom projective dimension.