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This problem has Harthshorne Height 1.


This problem has height 1.

HAPPY

  • The problem translates to showing if P \supset a is prime, then ht(P)\le r.
  • Recall  \sqrt{a} = \cap_i P_i for certain minimal primes, use the assumptions to show that each P_i is generated by at most r elements.
  • Use commutative algebra below to finish.

Commutative Algebra

  • If you have a Noetherian local ring A with maximal ideal m then ht\ m = \dim A \le the number of generators of m.

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