Math 615 (Winter 2005)  Homework Set #4  Due: 04/18/05 (Mon.)

Problem 1. Let R be a Noetherian ring of characteristic p, I an ideal, and M an R-module. (1) For any given e ∈ ℕ, show that H_{I}^{i}(^{e}M) ≅ ^{e}(H_{I}^{i}(M)) for every i.

(2) Suppose that ^{e_{0}}M is finitely generated over R for some e_{0} ≥ 1. Show that R/ Ann_{R}(M) is

an F -finite ring. Consequently, ^{e}M is finitely generated over R for every e. Problem 2. Let (R, 𝔪, k) be a Noetherian local ring of prime characteristic p and P ∈ Spec(R) be any prime ideal of R. Suppose R is F -finite and say [k : k^{p}] = p^{a}.

(1) Prove that dim(R̂/Q) = dim(R/P) for every Q ∈ AssR̂(R̂/P R̂) = minR̂(R̂/P R̂).

(2) Show that [(R/P)_{P} : ((R/P)_{P})^{p}] = p^{a+dim(R/P)}. (We have proved this when R is complete.) Problem 3. Let R be a Noetherian ring of prime characteristic p, M a finitely generated R-module such that Ann_{R}(M) ⊆ \sqrt{0}. For any R-modules N ⊆ L and x ∈ L, prove x ∈ N_{L}^{∗} ⟺ there exists c ∈ R^{∘} such that Image(x ⊗_{R} ^{e}(cM) → L ⊗_{R} ^{e}M) ⊆ Image(N ⊗_{R} ^{e}M → L ⊗_{R} ^{e}M) for all e ≫ 0. Problem 4. Let R be a Noetherian F -finite ring of prime characteristic p, M a finitely generated R-module with FFRT by finitely generated R-modules M_{1}, M_{2}, …, M_{r}, and L a finitely generated R-module. Show that ∪_{e∈ℕ} Ass(L⊗_{R} ^{e}M) is a finite set and, moreover, there exists an integer k ∈ ℕ such that the following are satisfied.

(1) For every e ∈ ℕ, there exists a primary decomposition

0 = Q_{e1} ∩ Q_{e2} ∩ ⋯ ∩ Q_{es_{e}} of 0 in L ⊗_{R} ^{e}M,

where Ass(L⊗^{e}M) = {P_{ej} | 1 ≤ j ≤ s_{e}} and Q_{ej} are P_{ej}-primary components of 0 ⊆ L⊗_{R} ^{e}M satisfying P_{ej}^{k}(L ⊗_{R} ^{e}M) ⊆ Q_{ej} for all 1 ≤ j ≤ s_{e}.

(2) We have J^{k}(0 :_{L⊗_{R} ^{e}M} J^{∞}) = 0, i.e., J^{k} H_{J}^{0}(L ⊗_{R} ^{e}M) = 0 for all J ⊆ R and for all e ∈ ℕ. (In case L = R/I, the above may be stated in terms of ∪_{e∈ℕ} Ass(M/I^{[q]}M) and H_{J}^{0}(M/I^{[q]}M).) Problem 5. Let R → S be a homomorphism of Noetherian rings of prime characteristic p and M a finitely generated R-module. (In this problem, we treat ^{e}M as an R-R-bimodule where r_{1} ·x·r_{2} =

r_{1}^{q}r_{2}x for any r_{1}, r_{2} ∈ R, x ∈ M. Also recall that #_{R}(^{e}M) = ℓ_{R}^{r}(Image(k ⊗_{R} ^{e}M ⟶^{ψ⊗1} E_{R}(k) ⊗_{R} ^{e}M))

where ψ : k → E_{R}(k) is any injective R-map and ℓ_{R}^{r}(−) denotes length as a right R-module.)

(1) For any R-module E, there is an isomorphism (E ⊗ ^{e}M) ⊗_{R} S ≅ (E ⊗_{R} S) ⊗_{S} ^{e}(M ⊗_{R} S). Moreover, the isomorphism is natural in the sense that for any R-linear map E_{1} → E_{2}, the following diagram commutes:

\begin{aligned}(E_{1} ⊗ ^{e}M) ⊗_{R} S & ⟶^{≅} & (E_{1} ⊗_{R} S) ⊗_{S} ^{e}(M ⊗_{R} S) \\ ↓ & & ↓ \\ (E_{2} ⊗ ^{e}M) ⊗_{R} S & ⟶^{≅} & (E_{2} ⊗_{R} S) ⊗_{S} ^{e}(M ⊗_{R} S) \end{aligned}

(2) Assume, furthermore, that (R, 𝔪, k) → (S, 𝔫, l) is a flat homomorphism of local rings such that 𝔪S = 𝔫. Show that (a) E_{R}(k) ⊗ S ≅ E_{S}(l) and (b) #_{R}(^{e}M) = #_{S}(^{e}(M ⊗_{R} S) for all e.