Math 480, Spring 2013, Assignment 14

From cartan.math.umb.edu

Carefully define the following terms, then give one example and one non-example of each:

  1. Maximal ideal.
  2. Point.
  3. Prime ideal.
  4. Irreducible variety.
  5. Irreducible component (of some reducible variety).
  6. Associated prime (of an ideal).

For each operation on varieties, describe the corresponding operation on ideals, then describe an algorithm that computes this operation (modulo the problem of computing radicals):

  1. Union.
  2. Intersection.
  3. Zariski closure of the difference.

Solve the following problems:

  1. Let \(I, J,\) and \(K\) be ideals. Prove that \(\left(I + J\right)K = IK + JK.\) (Hint: each side is, by definition, the smallest ideal containing certain elements. Which elements?)
  2. Working in \(\mathbb{C}[x]\), compute the associated primes of \(\left\langle x^3-3x^2+2x\right\rangle\).
  3. Working in \(\mathbb{C}[x,y,u,v]\), compute the associated primes of \(\left\langle xu, xv, yu, yv\right\rangle\). Do you think the variety of this ideal has any singular points?

Questions:

  1. I was wondering about a couple of things. 1.) Would you be willing to briefly recap the notion of the Associated Primes and how they are calculated? I remember you saying there was an algorithm, and the textbook did as well, but for some reason I do not have it in my notes, and 2.) Will you be posting answers for this and last homework set so that we can make sure we have the right answers when we study for the final? --Robert.Moray (talk) 19:43, 15 May 2013 (EDT)
1) The associated primes of an ideal are the ideals corresponding to the irreducible components of the corresponding variety. There is an algorithm to compute them, but we haven't talked about it in this class (and even if we had I wouldn't recommend it for these problems). The thing to do in these problems is to understand the corresponding variety by elementary methods, then see whether it's reducible (i.e. whether you can write it as the union of two smaller varieties). If so, then repeat the same process on the smaller pieces, and so on until you've written the variety as the union of irreducible varieties. Then finds the ideals corresponding to the irreducible pieces. 2) If you or anyone else posts solutions on the wiki, then I will check them to make sure they're right. --Steven.Jackson (talk) 07:40, 16 May 2013 (EDT)
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