# 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 $ht P + \dim A/P = \dim A$