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Math 221 : Midterm Math 221 : Midterm Due online by 10am Oct. 28, or as hard copy in class by 10:10am Oct. 28 (This midterm is out of 50 points, with a 5 point extra credit problem.) 1. (6pts) In this problem we use primary ideal decomposition to prove Theorem (Krull’s intersection theorem). If A is a local noetherian ring with maximal ideal T i m, then i≥1 m = 0. (In class we did this with the additional assumptions that A is an integral domain and has Krull dimension 1. There are a number of different ways to prove this theorem in full generality, and you can find a different proof given in Eisenbud.) T i T Let m¥ = i≥1 m . Let mm¥ = k qk be a primary decomposition of the ideal mm¥. Prove that each qk contains m¥. Use this to show that mm¥ = m¥, and then deduce Krull’s intersection theorem. 2. Let A = C[x, y]/(y2 − x3). It follows from Problem Set 6 that A is noetherian of Krull dimension 1, but not a Dedekind domain; so it must be the case that A is not integrally closed in Frac(A) a) (2pts) Show this directly by showing that the element y/x 2 Frac(A) is integral over A but does not lie in A. b) (5pts) Show that there is an isomorphism C(t) =∼ Frac(A) which sends t 7! y/x. Show that this isomorphism identifies A with a proper subring of C[t], and that the integral closure of A in C(t) is precisely C[t]. 3. Let A be the ring defined in the previous problem. Using the isomorphism from part b), identify A with a subring of C[t]. For this problem, it will be more convenient to compute with A as a subring of C[t] rather than as a quotient of C[x, y], so please write your answers that way. Give examples, with proof, of the following: (a) (4pts) A fractional ideal of A which is not invertible. 1 (b) (4pts) A primary ideal of A which is not a prime power. (c) (4pts) Fractional ideals I, J, K of A such that IJ = IK but J 6= K. (Hint: in all cases you can examples where the ideals are m-primary for the the maximal ideal m = (x, y) ⊂ C[x, y]/(y2 − x3).) 4. (12pts) Let A be a Dedekind domain, with field of fractions K. Show that the following are equivalent for an A-module M: (i) M =∼ I as A-modules for some fractional ideal I of A. ∼ (ii) M is finitely generated, and for every prime ideal p of A, Mp = M ⊗A Ap is a ∼ free Ap-module of rank 1 (that is, Mp = Ap as Ap-modules). (iii) M is a finitely generated flat A-module and dimK(M ⊗A K) = 1. 5. Let A be an integral domain with field of fractions K, and let I and J be fractional ideals of A. a) (4pts) Suppose that A is Dedekind. Show that the natural morphism I ⊗A J ! IJ of A-modules induced by the multiplication map I × J ! IJ is an isomorphism. (Hint: use the previous problem.) (Extra credit (5pts): give a counterexample to this when A is the non-Dedekind domain from Problem 2.) b) (4pts) Suppose that I is invertible, with inverse I−1. Show that the natural mor- phism I−1 =∼ Hom(I, A) defined by sending an element x 2 I−1 to the multiplica- tion by x -map mx 2 HomA(I, A), is an isomorphism. c) (5pts) Suppose that A is Dedekind. Show that the set of isomorphism classes of A- modules satisfying any of the equivalent conditions of Problem 4 form an abelian group G, where multiplication is given by tensor product, and the inverse of M is given by HomA(M, A). Show that there is a surjective group homomorphism f from the group I(A) of fractional ideals of A defined in class to G, given by sending an ideal to its isomorphism class as a module. Show that the kernel of f is exactly the group of principal fractional ideals of A. Conclude that G is canonically isomorphic to the class group Cl(A) = I(A)/P(A) defined in class. 2.
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