Math 615 (Winter 2005)  Homework Set #2  Due: 02/11/05 (Fri.)

Problem 1. Let (R, 𝔪) be a homomorphic image of a Noetherian Cohen-Macaulay local ring (S, 𝔫). Assume that R is equidimensional with dim(R) = d. For any system of parameters x_{1}, …, x_{d} of R, show that ((x_{1}, …, x_{i})_{R}^{∗} :_{R} x_{i+1}) = (x_{1}, …, x_{i})_{R}^{∗} for every i < d. (The case i = 0 may be interpreted as (0^{∗} :_{R} x_{1}) = 0^{∗}, the proof of which should be well incorporated into the general case.) Problem 2. Let R ⊆ S be an extension of Noetherian domains of prime characteristic p such that R is complete local and S is weakly F -regular. Moreover, assume that (IS) ∩ R = I for all ideals I of R (e.g. R is a direct summand of S as R-modules). Show R is Cohen-Macaulay. Problem 3. Let R ⊆ S be any integral extension of commutative rings of prime characteristic p in which R is Noetherian.

(1) Assume that S is module-finite over R. Then (IS)_{S}^{∗} ∩ R ⊆ I_{R}^{∗} for any ideal I of R.

(2) Show that (IS) ∩ R ⊆ I_{R}^{∗} for any ideal I of R. Problem 4. Let R be a Noetherian ring of prime characteristic p and N ⊊ M be finitely generated R-modules. Let Λ = \{L | N ⊆ L ⊆ M and \sqrt{Ann_{R}(M/L)} is a maximal ideal\}.

(1) Show that N = ∩_{L∈Λ}L. (This claim does not depend on characteristic p.)

(2) Suppose that J^{∗} = J for all ideals J of R such that \sqrt{J} are maximal ideals. Show that R is weakly F -regular, i.e. every ideal is tightly closed.

(3) Show that R is weakly F -regular if and only if R_{𝔪} is weakly F -regular for all maximal ideals

𝔪 of R. Problem 5. Let R be a commutative ring (not necessarily Noetherian or with characteristic p). Given an ideal I ⊆ R and x ∈ R, prove that x is in the integral closure of I if and only if x + P is in the integral closure of (I + P)/P ⊆ R/P for all P in min(R), the set of minimal primes of R. Problem 6. Let R be a Noetherian ring of prime characteristic p and N ⊆ M R-modules such that (N_{P})_{M_{P}}^{∗} = (N_{M}^{∗})_{P} for every P ∈ Ass(M/N_{M}^{∗}). Show that (W^{−1}N)_{W^{−1}M}^{∗} = W^{−1}(N_{M}^{∗}) for any multiplicatively closed set W ⊂ R.