Math 615 (Winter 2005)  Homework Set #1  Due: 01/28/05 (Fri.)

Problem 1. Let (R, 𝔪) be a 0-dimensional Noetherian (i.e. Artinian) local ring of prime characteristic p and M be a finitely generated R-module. Show that there exists e_{0} ∈ ℕ such that F^{e}(M) ≅ F^{e_{0}}(M) for all e ≥ e_{0}. (Actually, this is true as long as R is 0-dimensional Noetherian, the proof of which reduces to local case.) Problem 2. Let R be a Noetherian ring of prime characteristic p. It is known that I^{∗} ⊆ \sqrt{I} for every ideal I ⊆ R (by comparing tight closure with integral closure). Here we show this from definition of tight closure.

(1) If (R, 𝔪) is local (with maximal ideal 𝔪), show 𝔪^{∗} = 𝔪.

(2) Show P^{∗} = P for any prime ideal of R, which is not assumed to be local.

(3) Show I^{∗} ⊆ \sqrt{I} for every ideal I of R.

(4) Compute 0_{R}^{∗}. Here 0 refers to the zero ideal of R. Problem 3. Let k be a field of characteristic 2, S = k[X, Y] be a polynomial ring over k with indeterminates X, Y, and R = S/(X^{3})S. Let I = (X^{2})R ⊂ (X)R = J be ideals of R. (Then I ⊂ J are also modules over S under the natural ring homomorphism S → R.)

(1) Let F_{R}^{e}(−) be the Frobenius functor over R. Up to isomorphism, how many distinct Rmodules are there among 0, I, J, R, F_{R}(I), F_{R}(J), I_{R}^{[2]}, J_{R}^{[2]}, I_{J}^{[2]}? (Everything is considered as an R-module, including I as in the notation F_{R}(I), for example.) Group isomorphic R-modules together.

(2) Determine I_{R}^{∗} and I_{J}^{∗} over R. (Everything is considered as an R-module.)

(3) Determine I_{R}^{∗} and I_{J}^{∗} over S. (Everything is considered as an S-module, including R as in

I_{R}^{∗}.) Problem 4. Let R be a Noetherian ring of prime characteristic p. Suppose

G_{•} : 0 ⟶ G_{n} ⟶^{ϕ_{n}} G_{n−1} ⟶^{ϕ_{n−1}} ⋯ ⟶^{ϕ_{2}} G_{1} ⟶^{ϕ_{1}} G_{0}

is a complex with G_{i} free of finite rank for 0 ≤ i ≤ n. Apply Frobenius functor F^{e} to get F^{e}(G_{•}) for every e ∈ ℕ.

(1) If G_{•} is exact, show F^{e}(G_{•}) remains exact for every e ∈ ℕ.

(2) If ℓ_{R}(H_{i}(G_{•})) < ∞ for every 1 ≤ i ≤ n, show ℓ_{R}(H_{i}(F^{e}(G_{•}))) < ∞ for every 1 ≤ i ≤ n and for every e ∈ ℕ. (Here ℓ_{R}(−) represents the length of an R-module.)

Do (1) and (2) still hold if, instead, we assume G_{i} are finitely generated projective R-modules for 0 ≤ i ≤ n? Problem 5. Let R be a Noetherian ring of prime characteristic p and N ⊆ M be R-modules. Define N_{M}^{F} = {x ∈ M | x^{q} ∈ N_{M}^{[q]} ⊆ F^{e}(M) for some q = p^{e}}. (N_{M}^{F} is called the Frobenius closure of N in M.)

(1) Show that N_{M}^{F} ⊆ N_{M}^{∗}.

(2) Show that Frobenius closure commutes with localization, i.e. (W^{−1}N)_{W^{−1}M}^{F} = W^{−1}(N_{M}^{F}) for any multiplicatively closed set W ⊂ R.

(Assume R is Noetherian and M is a finitely generated R-module. Note that, for every ideal I of R, \sqrt{I} is defined to be \{r ∈ R | r^{n} ∈ I for some n ∈ ℕ\}. We may use the fact that depth_{I}(M) = depth\sqrt{I}(M) without proof. We may also use the fact without proof that F_{W^{−1}R}^{e}(W^{−1}M) ≅ W^{−1}F_{R}^{e}(M) for any multiplicatively closed set W ⊂ R although the proof of it is, actually, very short/easy. Finally, a polynomial ring over a field is regular.)