# I.1.7

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This problem has Hartshorne Height 1.

### HAPPY

• 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...