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This problem has Hartshorne Height 1.


  • a) This is a standard set of equivalences in commutative algebra.
  • b) Choose a cover with no finite subcover, and construct an infinite descending chain of closed subsets (the complement of some ascending chain of open sets)
  • c) For Y \subset X with X Noetherian and c \subset Y irreducible, use topology to show c = \bar{c} \cap Y so chains must stablize in Y
  • d)Proceed by contradiction, (i.e. there are infinitely many point) pick two distinct points and separate them by disjoint opens. Use the complement of the opens to write X = Z_1 \cup Z_2 where the Z_i closed. At least one must have infinitely points, and also Hausdorff and Noeth. Construct infinite descending chain...

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