\( \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\uHom}{\,\underline{\!Hom\!}\,} \DeclareMathOperator{\Mor}{Mor} \DeclareMathOperator{\Map}{Map} \DeclareMathOperator{\map}{map} \DeclareMathOperator*{\colim}{colim} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\Id}{Id} \DeclareMathOperator{\Ob}{Ob} \DeclareMathOperator{\comp}{comp} \DeclareMathOperator{\Fun}{Fun} \DeclareMathOperator{\true}{true} \DeclareMathOperator{\Sub}{Sub} \DeclareMathOperator{\Lan}{Lan} \DeclareMathOperator{\Ran}{Ran} \DeclareMathOperator{\PSh}{PSh} \DeclareMathOperator{\Sh}{Sh} \DeclareMathOperator{\Sp}{Sp} \DeclareMathOperator{\Glue}{Glue} \DeclareMathOperator{\Res}{Res} \DeclareMathOperator{\End}{End} \DeclareMathOperator{\oneim}{1im} \DeclareMathOperator{\twoim}{2im} \DeclareMathOperator{\charr}{char} \DeclareMathOperator{\Spec}{Spec} \newcommand{\ProFinSet}{\mathrm{ProFinSet}} \newcommand{\sSet}{\mathrm{sSet}} \newcommand{\Top}{\mathrm{Top}} \newcommand{\deltacat}{\boldsymbol{\Delta}} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Set}{\mathrm{Set}} \newcommand{\Ring}{\mathrm{Ring}} \newcommand{\CatMon}{\mathrm{CatMon}} \newcommand{\Cof}{\mathrm{Cof}} \newcommand{\Fib}{\mathrm{Fib}} \newcommand{\Frm}{\mathrm{Frm}} \newcommand{\Loc}{\mathrm{Loc}}\)

Geometric Construction of the Specturm of a Ring

Classically the spectrum \( \Spec R \) of a commutative ring \( R \) is constructed as a locally ringed space (the points of the underlying topological space are prime ideals). There are two related problems with this construction. First, when appealing to our geometric intuition there seems to be no reason why the points of the topological space should be the prime ideals of \(R \). Second, in more general settings like a topos the classical construction doesn’t work (a ring may have no prime ideals, hence the underlying topological space would be empty). We give a geometric and constructive construction of the spectrum of a commutative ring that solves both problems and is equivalent to the classical construction in the classical setting.

First we give a list of criteria of adequacy for the spectrum of a commutative ring. Let \( R \) be a commutative ring. Then the spectrum of \(R \) has to satisfy:

  1. It should be a space such that the elements of \(R \) are functions on it and its geometry is completely determined by the functions on it.
  2. For every \(f,g \in R \) it should contain a closed subspace “\(\{ f = g \} \)”. Since \(R \) is a ring, equivalently it should contain an open subspace \(\{ f \neq 0\} \).
  3. Coming from classical algebraic geometry where all functions map into a field, for every local function \(f \) without roots there should exists a function \(1/f \).

To meet the first criteria we take the approach of constructing the spectrum of \(R \) as a sheaf of rings. Denote the underlying category by \(L(R) \) and the functor to the category of rings by \(\mathcal O _R \). Since \(L(R) \) is the category of open subspaces, we assume that \(L(R) \) is a locale (or equivalently a frame), ie a poset with finite meets, all joins, an initial object and a terminal object (think of the open subset of a topological space). The second criteria tells us that for every \(f \in R \) there should be an object \(\{ f \neq 0 \} \) in \(L(R) \). But what are the relations between the objects? The third criteria says that \(\mathcal O _R(\{ f \neq 0 \}) = R[1/f] \). But \(R \) induces canonical maps between localizations, ie by the universal property there may exist a canonical map \(R[1/f] \rightarrow R[1/g] \) for \(f,g \in R \). Therefore we add a map \(\{ g \neq 0 \} \rightarrow \{ f \neq 0 \} \) in this case. This identifies \(\{ f \neq 0 \} \) and \(\{ g \neq 0 \} \) iff \(R[1/f] \) and \(R[1/g] \) are canonically isomorphic and we get a poset with finite meets. The following is aſ formal definition of this poset.

Let \( f \in R \). The basic open \( D(f) \) is the set

\begin{equation*} \{ g \in R \mid g \text{ divides a power of } f \} \end{equation*}

(this is exactly the set of elements that get inverted in the localization \( R[1/f] \)). The poset (with finite meets) of basic opens is

\begin{equation*} D(R) := \{ D(f) \mid f \in R \} \end{equation*}

where \(D(f) \le D(g) \) if \(D(g) \subseteq D(f) \) and \(D(f) \wedge D(g) = D(fg) \).

To construct a locale we additionally need to know when a family \(\{ f_i \neq 0 \} _{i \in I} \) covers \(\{ g \neq 0 \} \). This is the case iff \(\{ f_i \neq 0 \} \le \{ g \neq 0 \} \) and every function on \(\{ g \neq 0 \} \) is a gluing of compatible functions on \(\{ f_i \neq 0 \} _{i \in I} \). Hence exactly if \(R[1/g] \) is canonically isomorphic to \(\lim _{i \in I} R[1/f_i] \) (the diagram consists of maps \(R[1/f_i] \rightarrow R[1/f_if_j] \)). This defines a coverage on \(D(R) \) and hence \(D(R) \) is a posite. Then \(D(R) \) generates a free locale by looking at the ideals of \(D(R) \) (if we think of topological spaces, \(D(R) \) plays the role of a base that generates a topology).

An ideal in \(D(R) \) is a subset \(I \subseteq D(R) \) such that

  1. if \(U \le V \) and \(V \in I \), then \(U \in I \),
  2. if \((U_i) _{i \in I} \) are in \(I \) and cover \(V \), then \(V \in I \).

Now we can define \(L(R) \).

Let \(L(R) \) be the locale of ideals in \(D(R) \) with

  1. \(I \le J \) if \(I \subseteq J \),
  2. \(I \wedge J = I \cap J \),
  3. \(\bigvee _{i \in I} I_i = \bigcap \{ I \subseteq D(R) \text{ ideal} \mid \bigcup I_i \subseteq I \} \).

Finally we can define \(\mathcal O _R \).

Let \(\mathcal O _R \colon L(R) \rightarrow \mathrm{Rings} \) be the functor given by

\begin{equation*} \mathcal O _R(I) := \lim _{D(f) \le I} R[1/f]. \end{equation*}

This is exactly the classical construction of the structure sheaf.

We can compare our construction to the classical construction. First we note that \(L(R) \) is canonically isomorphic to the locale of radical ideals.

Let \(L'(R) \) be the set of radical ideals of \(R \). Then \(L'(R) \) is a locale with \(I \le J \) if \(I \subseteq J \), \(I \wedge J = I \cap J \) and \(\bigvee _{i \in I} I_i = \sqrt{ \bigoplus _{i \in I} I_i} \). Furthermore \(L(R) \rightarrow L'(R), I \mapsto \bigvee _{D(f) \le I} \sqrt{ (f) }\) and \(L'(R) \rightarrow L(R), I \mapsto \{ D(f) \mid \sqrt{(f)} \subseteq I \} \) are inverses of each other.

The construction of \(L'(R) \) is easier on a technical level but is harder to justify geometrically (but it can be done, a radical ideal \(\sqrt{(f)} \) represents \(\{ f = 0 \} \)). See “Stone Spaces”, V.3 by Johnstone. To recover the classical construction we have to look at the topological space associated to \(L(R) \). It is given by the set of points of \(L(R) \) equipped with the canonical topology (see nlab). A point of a locale \(L \) is a completely prime filter on \(L \).

Let \(X' \) be the topological spaces associated to \(L'(R) \). Then \(\Spec R \rightarrow X', \mathfrak p \mapsto \{ I \in L'(R) \mid I \subseteq \mathfrak p \} \) and \(X' \rightarrow \Spec R, P \mapsto \sum _{I \in P} I \) are inverse homeomorphisms.

In a constructive setting the ring \( R \) may have no prime ideals, then the locale \( L(R) \) has no points but is still a non-trivial locale.

Author: Frederik Gebert

Created: 2025-02-10 Mon 21:45