This problem has Hartshorne Height 2; problem I.1.6 can help solve this one. No big commutative algebra results to apply, just work through the topology.


a) Use exercise I.1.6 and show if Z \subset Z' are distinct and closed in an arbitrary subset Y \subset X then  \overline{Z}, \ \overline{Z'} are distinct in X. In fact take a point of Z' that is not in Z and show this point is still not in the closure.

b) The hard part is to show there is an i such that \dim X \le \dim U_i. The idea is to take a chain or closed irreducible subsets in X, Z_0 \subset ... \subset Z_n and pick a U_i intersecting Z_0. Now show the chain still has the same length as a chain in U_i. Basically if this wasn't the case you can find two disjoint open subsets in an irreducible closed subset.

c) It can be done by constructing a simple two point topological space.

d) If Y \ne X then for every chain in Y, there is a strictly longer chain in X by taking X to be the last element of the chain.

e) Think: countably many finite dimensional noetherian topological spaces.

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