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Unformatted text preview: HOMEWORK PROBLEMS FOR MATH 8320: 1-28 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 k-algebras): a) Show that any quo- tient of an affine k-algebra is an affine k-algebra. b) Show that an integral affine k-algebra 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 k-algebras is an affine k-algebra. d) Show that the tensor product, over k , of two affine k-algebras is an affine k- algebra. 1 e) When is an infinite product of affine k-algebras an affine k-algebra? 2 f) Show by example that a k-subalgebra of an affine k-algebra need not be an affine k-algebra. Exercise 2: a) Give an example of a integral k-algebra (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 k-algebras. What is a true statement about an affine k-algebra 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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