I.1.9

216,076pages on
this wiki

Wikia is a free-to-use site that makes money from advertising. We have a modified experience for viewers using ad blockers Wikia is not accessible if you’ve made further modifications. Remove the custom ad blocker rule(s) and the page will load as expected.

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 $m$.