Let $F$ be a field and $R=F[X_1,X_2,\ldots,X_n]$ be the polynomial ring in $n$ variables over $F$ and $P$ be a prime ideal in $R$, I'm trying to prove that$$\operatorname{ht}P+\dim R/P=\dim R$$where $\dim$ is the Krull dimension of a ring.
Any ideas, thank you for your help.
Best Answer
$\mathrm{ht}(P) + \dim(R/P) \leq \dim(R)$ is trivial, but the converse is not so easy to prove directly. You can find a proof in several books on algebraic geometry, for example Qing Liu's Algebraic geometry and arithmetic curves, Proposition 2.5.23. Another reference is the CRing Project, Theorem ยง10.3.22. It holds more generally for every domain which is a finitely generated $F$-algebra (this generalization is important for the induction step).
For a more detailed discussion about the equation, see math.SE/49136.