A locally ringed locale is a pair \((X, \mathcal{O}_X)\) where:

1. \(X\) is a locale (a frame)

2. \(\mathcal{O}_X\) is a sheaf of rings on \(X\)

3. For every open \(u \in X\), the stalk \(\mathcal{O}_{X,\bar{u}}\) is a local ring (has a unique maximal ideal)

For a sheaf \(\mathcal{F}\) on a locale \(X\) and an element \(u \in X\), the stalk is defined as:

where \(\downarrow u := \{ v \in X : v \leq u\} \) is the principal order filter at \(u\).

An affine scheme is a locally ringed locale of the form \((\mathrm{Spec} R, \mathcal{O}_{\mathrm{Spec} R})\) for some commutative ring \(R\), where:

1. \(\mathrm{Spec} R = \mathrm{Rad}(R)\) is the Zariski locale

2. \(\mathcal{O}_{\mathrm{Spec} R}\) is the structure sheaf constructed in Definition 38

A morphism of affine schemes from \(\mathrm{Spec} R\) to \(\mathrm{Spec} S\) is a morphism of locally ringed locales, i.e., a pair \((f_{\# }, f^{\sharp })\) where:

1. \(f_{\# }: \mathrm{Spec} R \to \mathrm{Spec} S\) is a locale morphism (i.e., a frame homomorphism \(f_{\# }^*: \mathrm{Rad}(S) \to \mathrm{Rad}(R)\))

2. \(f^{\sharp }: \mathcal{O}_S \to f_{\# *} \mathcal{O}_R\) is a morphism of sheaves of rings respecting the local ring structure

A scheme is a locally ringed locale \((X, \mathcal{O}_X)\) such that \(X\) has an open cover \(\{ u_i : i \in I\} \) (where \(\bigvee _i u_i = \top \)) with the property that:

1. The restriction \((u_i, \mathcal{O}_X|_{u_i})\) to each open is isomorphic to an affine scheme \(\mathrm{Spec} R_i\)

2. The transition functions between affine pieces are given by ring homomorphisms