(Hauptidealsatz)
Let \(A\) be Noetherian and \(f \in A\) neither a unit nor a zero divisor. Every minimal prime over \((f)\) has height one. [@Har10a, Theorem I.1.11A]
(Consequences used on varieties)
For \(B\) a finitely generated domain over a field and \(\mfp \in \Spec B\), \[ \height \mfp + \krulldim (B/\mfp) = \krulldim B . \] [@Har10a, Theorem I.1.8A] A closed irreducible \(Y \subseteq \AA^n\) has \(\codim Y = 1\) exactly when \(Y = V(f)\) for a single irreducible nonconstant \(f \in k[x_1,\ldots,x_n]\). [@Har10a, Proposition I.1.13]
(Unique factorization via height-one primes)
A Noetherian domain \(A\) is a UFD if and only if every prime ideal of \(A\) of height one is principal. [@Har10a, Proposition I.1.12A]
The Hauptidealsatz says that one nonzerodivisor cuts codimension one; in affine space, height-one primes are principal because \(k[x_1,\ldots,x_n]\) is a UFD, so irreducible codimension-one subvarieties are hypersurfaces.
The additivity formula fails without finite generation over a field. Codimension does not bound the number of equations: the ideal of the image of \(t \mapsto (t^3,t^4,t^5)\) in \(\AA^3\) is a prime of height two that cannot be generated by two elements (P-AGHNOTLCI).
Dependencies
D-5LJUXDimension of a varietyuses