Let AAA be a Boolean ring. For each φ:A→F2\varphi : A \to \mathbb{F}_2φ:A→F2 we can define φ↦kerφ\varphi \mapsto \operatorname{ker} \varphiφ↦kerφ these maps form a bijection between hom-set Hom(A,F2)\text{Hom}(A, \mathbb{F}_2)Hom(A,F2) and SpecA\operatorname{Spec} ASpecA. Proof [local-0] If φ:R→S\varphi : R \to Sφ:R→S is surjective, then S≃R/kerφS \simeq R / \operatorname{ker}\varphiS≃R/kerφ (Chapter 0 III. Corollary 3.10.). Therefore, for each x∈SpecAx \in \operatorname{Spec} Ax∈SpecA we have A/x≃A/kerφA / x \simeq A / \operatorname{ker} \varphiA/x≃A/kerφ Remind that A/x=F2A / x = \mathbb{F}_2A/x=F2. hence we choose φ\varphiφ. Inversely, we choose kerφ∈SpecA\operatorname{ker} \varphi \in \operatorname{Spec} Akerφ∈SpecA.
If φ:R→S\varphi : R \to Sφ:R→S is surjective, then S≃R/kerφS \simeq R / \operatorname{ker}\varphiS≃R/kerφ (Chapter 0 III. Corollary 3.10.). Therefore, for each x∈SpecAx \in \operatorname{Spec} Ax∈SpecA we have A/x≃A/kerφA / x \simeq A / \operatorname{ker} \varphiA/x≃A/kerφ Remind that A/x=F2A / x = \mathbb{F}_2A/x=F2. hence we choose φ\varphiφ. Inversely, we choose kerφ∈SpecA\operatorname{ker} \varphi \in \operatorname{Spec} Akerφ∈SpecA.