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Unformatted text preview: HOMEWORK PROBLEMS FOR MATH 8320: 128 PETE L. CLARK Here are some homework problems for the 8320 course. Some of them are re peated from places in the lecture notes. Some of them are not specifically tied to any one topic. Some of them are easy, and some are quite difficult, etc. There are more problems than any one student will want to work out, but every student should work at least some of them and be prepared to present solutions to some of them. Unless mention is made to the contrary, k denotes an arbitrary field. Exercise 1 (Properties of the category of affine kalgebras): a) Show that any quo tient of an affine kalgebra is an affine kalgebra. b) Show that an integral affine kalgebra can be expressed as the quotient of a polynomial ring by a prime ideal. c) Show that the product of two (hence also any finite number) affine kalgebras is an affine kalgebra. d) Show that the tensor product, over k , of two affine kalgebras is an affine k algebra. 1 e) When is an infinite product of affine kalgebras an affine kalgebra? 2 f) Show by example that a ksubalgebra of an affine kalgebra need not be an affine kalgebra. Exercise 2: a) Give an example of a integral kalgebra (necessarily infinitely gener ated!) A for which the Krull dimension A is not equal to the transcendence degree of the fraction field K of A over k . b) Can one at least say either that the Krull dimension is always at most the transcendence degree or that the transcendence degree is always at most the Krull dimension? Exercise 3: We would like to say that an affine variety is the union of its ir reducible components, but at the moment we have to phrase everything in terms of the opposite category of kalgebras. What is a true statement about an affine kalgebra and its irrreducible components A/ p i (where p 1 ,..., p n are the minimal prime ideals) that would plausibly be the dual of this geometric statement?...
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 Fall '10
 Clark
 Math, Algebra

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