I.2.1 can be use to solve I.2.2 and I.2.3 giving the latter two height and volume 2. Similarly I.2.3 can be used to solve I.2.4 giving the it height 2 and volume 3. The rest of the problems have height 1.
- the hint says it all
- For (1) ==> (2) use I.2.1
- Here is (3) ==> (1). For any projective point p, some coordinate of p doesn't vanish, say . Then is not zero at p hence
- The first three parts are basic algebraic reasoning.
- For d, if is nonempty then is a proper homogeneous ideal and I.2.1 implies , the other inclusion follows because a field has no nonzero nilpotentns.
- For e, give parts a and b the same proof that is used in prop. I.1.2 works here.
- a) Say are algebraic sets such that then using I.2.3e etc so which shows .
- a similar argument works to show implies
- b) similar to the affine argument given in section 1.1
- c) follows from b) because (0) is a prime ideal.
- The first statement follows because a descending chain of closed sets corresponds to an ascending chain of radical ideals in a noetherian ring.
- The second statement follows form prop. I.1.5 ( for a noeth. top. sp. every closed can be expressed as finite union of irred. closed sets.)
You'll Want to use proposition I.1.5