Math 711: Lecture of October 24, 2007 The action of Frobenius on the injective hull of the residue class field of a Gorenstein local ring

Let (R, m, K) be a Gorenstein local ring of prime characteristic p > 0, and let x1, . . . , xn be a system of parameters. Let It = (x1, . . . , x t n)R for all t ≥ 1, and let u ∈ R represent a socle generator in R/I, where I = I1 = (x1, . . . , xn)R. Let y = x1 · · · xn. We have seen that E = lim −→ t R/It is an injective hull of K = R/m over R, where the map R/It → R/It+1 is induced by multiplication by y acting on the numerators. Each of these maps is injective. Note that the map from R/It → R/It+k in the direct limit system is induced by multiplication by y acting on the numerators.