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I.1.8

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This problem has Harthshorne Height 1

HAPPY

Say Y = Z(I) and H = Z(f) then recall Y \cap H = Z(I,f). As Y is a variety, its associated ideal is prime. Use commutative algebra results to show  \dim Y = ht P_i + \dim B/P_i where B is the affine coordinate ring of Y and P_i are certain minimal primes.

Commutative Algebra

  • \sqrt I = \cap P where the intersections ranges over all primes P \supset I
  • Krull: If f is a nonzero divisor then the minimal primes containing (f) have height 1.
  •  P \subset k[x_1,...,x_n] = A a prime then ht P + \dim A/P = \dim A

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