Characterizations of Gorenstein Rings Using Frobenius

Let R be a local commutative Noetherian ring of characteristic p \u3e 0 and f : R → R the Frobenius ring homomorphism sending r ∈ R to its pth power rp. Denote by Rf the R – R-bimodule with additive group R and left and right R-actions given by r·u = ru and u·r = urp for all r ∈ R, u ∈ Rf. The Frobenius functor FR takes a left R-module M to the left R-module FR( M) := Rf⊗RM. Motivated by work of Peskine and Szpiro, Marley, and Webb, we give various characterizations of Gorenstein local rings of prime characteristic through investigations into how the Frobenius functor interacts with injective dimension, injective resolutions, and certain Tor modules. We also include separate work, in the setting of commutative Noetherian rings (with no prime characteristic assumption), which establishes when certain maps in the direct limit definition of local cohomology are injective