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Mathematics 9607B Winter 2016 Problem Set 3 Due: Monday Mar.7, 2016. 1. Let A be a UFD, and let B be an integral domain, integral over A. Prove that, for every b ∈ B, there is a unique monic polynomial P ∈ A[w] of minimal degree, such that P (b) = 0. 2. Let X be an analytic subset of a manifold M , and let ξ ∈ X. We define the local ring of X at ξ as OX,ξ := OM,ξ /I(Xξ ) , where I(Xξ ) denotes the full ideal of the analytic germ Xξ . Prove the following: (a) The set H(X, ξ) of restrictions of holomorphic germs f ∈ OM,ξ to Xξ is a ring (more precisely, a C-algebra). (b) H(X, ξ) can be identified with OX,ξ (find an isomorphism of C-algebras). (c) OX,ξ is a local ring (find the unique maximal ideal). (d) The ring OX,ξ is an integral domain if and only if Xξ is an irreducible germ. (e) The ring OX,ξ need not be a UFD, even if X is locally principal and locally irreducible (give an example and justify). 3. Recall that the Krull dimension of a Noetherian local ring (R, m) is the maximal length d of a chain of prime ideals p0 p1 · · · pd = m in R. Let X be an analytic subset of a manifold M , and let ξ ∈ X. (a) Prove that the zero ideal (0) is prime in OX,ξ if and only if Xξ is an irreducible germ. (b) Suppose that Xξ is irreducible. Prove that the Krull dimension of OX,ξ equals dimξ X. 4. Recall that a Noetherian local ring (R, m) is called regular when the maximal ideal m can be generated by n elements, where n is the Krull dimension of R. Show that, if Xξ is smooth, then the local ring OX,ξ is regular. 5. Let X be a k-dimensional analytic subset of a manifold M , and let, for every d ∈ N, X (d) = {x ∈ X : dimx X = d} . Prove the following: (a) X (k) is an analytic subset of X (in particular, closed!). (b) For every d ∈ N, X (d) is analytic of pure dimension d. Moreover, X = X (k) ∪ X (k−1) ∪ · · · ∪ X (0) = X (k) ∪ X (k−1) ∪ · · · ∪ X (0) . [ (c) For every d = 1, . . . , k, X (i) is open in X. i<d (d) For every d = 1, . . . , k, X \ [ X (i) is analytic in X. i<d (d) 6. Let X and X be as above. Give, as explicit as possible, a characterization of X (d) (d = 0, . . . , k) in terms of components of regX. 7. Prove that the irreducible components of an analytic set X defined in Theorem 7.8 are precisely the maximal (with resp. to inclusion) irreducible analytic subsets of X. 1