We prove two theorems concerning the test properties of the Frobenius endomorphism over commutative Noetherian local rings of prime characteristic p.Our first theorem generalizes a result of Funk and Marley on the vanishing of Ext and Tor modules, while our second theorem generalizes one of our previous results on maximal Cohen-Macaulay tensor products.In these earlier results, we replace e R with a more general module e M, where R is a Cohen-Macaulay ring, M is a Cohen-Macaulay R-module with full support, and e M is the module viewed as an R-module via the e-th iteration of the Frobenius endomorphism.We also provide examples and present applications of our results, yielding new characterizations of the regularity of local rings.
Celikbas et al. (Thu,) studied this question.