Prime Spectrum of a Commutative Ring

Overview

The prime spectrum provides a bridge from commutative algebra to algebraic geometry. It treats prime ideals as points and ring elements as functions on a space. This article concerns the underlying topological construction and its local rings; a full affine scheme additionally includes a structure sheaf.

Prerequisites

Use commutative unital rings, prime ideals, localization, and elementary topology. A prime ideal is proper and contains one factor whenever it contains a product. Equivalently, quotienting by it gives an integral domain. Maximal ideals are prime, but the converse need not hold.

Definition and conventions

For a commutative ring $R$,

Give it the Zariski topology whose closed sets are

where $I$ ranges over ideals of $R$. The distinguished open set $D(f)$ consists of primes not containing $f$. The local ring at $\mathfrak p$ is $R_{\mathfrak p}$.

The ID here is qualified as algebraic-prime-spectrum: it does not denote the spectrum of a linear operator or a topological spectrum.

Intuition

The condition $f\in\mathfrak p$ says that the function represented by $f$ vanishes at that prime point. Localization allows division by functions not vanishing there. Prime ideals include generic points that describe an entire irreducible region, not just ordinary closed points.

Examples and non-examples

In $\operatorname{Spec}\mathbb Z$, the points are $(0)$ and $(p)$ for positive primes $p$. The point $(0)$ is dense, since every prime contains the zero ideal. The point $(p)$ is closed, since it is maximal. The open set $D(6)$ omits $(2)$ and $(3)$ but contains $(0)$ and all other prime points.

For an algebraically closed field $k$, the maximal ideals of $k[x]$ are $(x-a)$, with $a\in k$, and $(0)$ is a generic point. Restricting to maximal ideals would remove this generic point and lose part of the spectral topology.

The ideal $(6)\subset\mathbb Z$ is not a point of the spectrum: $2\cdot3$ belongs to it although neither factor does. Also, the spectrum is usually not Hausdorff; a dense nonclosed singleton already demonstrates the failure in $\operatorname{Spec}\mathbb Z$.

Important properties

The formulas

verify the finite closed-set axioms; arbitrary intersections are handled by sums of ideals. The union formula uses primality, which is why arbitrary ideals cannot serve as points in this construction.

A ring homomorphism $R\to S$ induces a continuous map $\operatorname{Spec}S\to\operatorname{Spec}R$ by contraction. Distinguished opens identify with spectra of localizations. Nilpotent elements lie in every prime, so the underlying space cannot alone detect nilpotent structure.

Connections and further reading

The structure sheaf restores the local algebra that topology loses, including nilpotents. The spectrum is therefore an entry point to affine schemes, not a replacement for their structure sheaves. Read Milne, Section 14, The spectrum of a ring, for the topology and localization correspondence.