Math 615 (Winter 2005)  Homework Set #3  Due: 03/18/05 (Fri.)

Problem 1. Let R be a Noetherian ring of prime characteristic p. Show that R has a weak test element if and only if R/\sqrt{0} has a weak test element. (Here a weak test element is, by definition, a q-weak test element for some q.) Problem 2. Let R be a Noetherian ring of prime characteristic p and M an R-module. Recall that eM is the derived R-module structure on M via the Frobenius homomorphism F^{e} : R → R.

(1) If ^{e_{0}}M is a faithful R-module for some e_{0} > 0, then R is reduced and ^{e}M (including M = ^{0}M) are faithful for all e ∈ ℕ.

(2) Show that Ass_{R}(M) = Ass_{R}(^{e}M) for every e ∈ ℕ. Problem 3. Let (R, 𝔪, k) be a Noetherian equidimensional catenary local ring of prime characteristic p with dim(R) = d. Suppose q^{d} < ℓ_{R}(R/𝔪^{[q]}) < q^{d} + q for some q = p^{e} ≥ p. Prove Sing(R) = {𝔪}, where Sing(R) = {P ∈ Spec(R) | R_{P} is not regular} is the singular locus of R. Problem 4. Let R be a ring (not necessarily of characteristic p). Given R-modules M, N and f ∈ Hom_{R}(M, N), we say f is pure if the induced map f ⊗_{R} 1_{L} : M ⊗_{R} L → N ⊗_{R} L is injective for every R-module L. (Denote by m-Spec(R) the set consisting of all maximal ideals of R.)

(1) If f ∈ Hom_{R}(M, N) is pure, then f is injective. (Therefore, f ∈ Hom_{R}(M, N) is pure if and only if f is injective and the inclusion map f(M) ⊆ N is pure.)

(2) f ∈ Hom_{R}(M, N) is pure if and only if f_{P} : M_{P} → N_{P} is pure for every P ∈ Spec(R) if and only if f_{𝔪} : M_{𝔪} → N_{𝔪} is pure for every 𝔪 ∈ m-Spec(R).

(3) Show (A) f ∈ Hom_{R}(M, N) is pure if and only if f ⊗_{R} 1_{L} : M ⊗_{R} L → N ⊗_{R} L is injective for every finitely generated R-module L; and (B) If R is Noetherian and M is finitely generated, then f ∈ Hom_{R}(M, N) is pure if and only if f ⊗_{R} 1_{L} : M ⊗_{R} L → N ⊗_{R} L is injective for every finitely generated R-module L such that Ass_{R}(L) = {𝔪} for some 𝔪 ∈ m-Spec(R).

(4) Suppose R is Noetherian and M, N are finitely generated R-modules. Then f ∈ Hom_{R}(M, N) is pure if and only if f̂_{𝔪} : M̂_{𝔪} → N̂_{𝔪} is pure for every 𝔪 ∈ m-Spec(R) if and only if f : M → N splits (meaning there exists g ∈ Hom_{R}(N, M) such that g ∘ f = 1_{M}.)

(5) Suppose R is Noetherian and F is a free R-module. Then f ∈ Hom_{R}(F, N) is pure if and only

if the induced map f⊗_{R}1_{E} : F ⊗_{R}E → N⊗_{R}E is injective, where E = ⊕_{𝔪∈m-Spec(R)}E_{R}(R/𝔪). Problem 5. Given a local Noetherian ring (R, 𝔪, k) of prime characteristic p (not necessarily F - finite), one could define R to be strongly F -regular if, for any c ∈ R^{∘}, there exists an integer e ≥ 1 such that the R-linear map R → ^{e}R sending 1 to c is pure. In general, one could define R is strongly F -regular if R_{𝔪} is strongly F -regular for every 𝔪 ∈ m-Spec(R). (By Problem 4, we see that the above definition agrees with the one given in class when R is F -finite.)

(1) If there exists a pure R-linear map R → ^{e}R sending 1 to c with e ≥ 1, then R is reduced and, for every e′ ≥ e, the R-linear map R → ^{e′}R sending 1 to c is pure. (Thus the above definition of strong F -regularity forces R to be reduced.)

(2) Show that (R, 𝔪, k) is strongly F -regular if and only if 0_{E_{R}(k)}^{∗} = 0.