Dimension of Polynomial and Power Series Rings

Understanding how Krull dimension behaves under polynomial and power series extensions is essential for computing dimensions in algebraic geometry and local algebra.

Polynomial Ring Dimension

Theorem6.9Dimension of Polynomial Rings

Let $R$ be a Noetherian ring. Then $\dim R[x] = \dim R + 1$ More generally, $\dim R[x_1, \ldots, x_n] = \dim R + n$ .

The proof has two parts. The inequality $\dim R[x] \geq \dim R + 1$ follows by extending any chain of primes in $R$ to a longer chain in $R[x]$ via $\mathfrak{p} \mapsto \mathfrak{p}[x] \subsetneq \mathfrak{p}[x] + (x)$ . The reverse inequality $\dim R[x] \leq \dim R + 1$ is more delicate and uses the going-up and going-down theorems for the integral extension $R \hookrightarrow R[x]$ .

ExampleAffine space

For a field $k$ , $\dim k[x_1, \ldots, x_n] = n$ . This recovers the geometric fact that affine $n$ -space $\mathbb{A}^n_k$ has dimension $n$ . The maximal chains of primes correspond to chains of irreducible subvarieties: $(0) \subsetneq (f_1) \subsetneq (f_1, f_2) \subsetneq \cdots \subsetneq (x_1, \ldots, x_n)$

Power Series Rings

Theorem6.10Dimension of Power Series Rings

For a Noetherian local ring $(R, \mathfrak{m})$ , $\dim R[[x]] = \dim R + 1$ More generally, $\dim R[[x_1, \ldots, x_n]] = \dim R + n$ .

ExampleFormal power series over a field

$\dim k[[x_1, \ldots, x_n]] = n$ . This ring is a regular local ring of dimension $n$ with maximal ideal $(x_1, \ldots, x_n)$ . It serves as the "model" local ring in dimension $n$ , and its properties often guide intuition for general local rings.

Fibers and Dimension

Theorem6.11Dimension Formula for Fibers

Let $\varphi : R \to S$ be a homomorphism of Noetherian rings, and let $\mathfrak{q}$ be a prime of $S$ lying over $\mathfrak{p} = \varphi^{-1}(\mathfrak{q})$ in $R$ . Then $\operatorname{ht}(\mathfrak{q}) \leq \operatorname{ht}(\mathfrak{p}) + \dim(S_\mathfrak{q} / \mathfrak{p} S_\mathfrak{q})$ If $\varphi$ is a flat homomorphism of finite type, equality holds.

RemarkGeometric fiber dimension

For a morphism $f : X \to Y$ of algebraic varieties, the fiber dimension formula says $\dim f^{-1}(y) \geq \dim X - \dim Y$ at every point. The locus where the fiber dimension jumps is a closed subset, and for flat morphisms, all fibers have the same dimension. This is the geometric content of the algebraic dimension formulas.