Affine blue scheme - Knowlpedia

Definition

An affine blue scheme is a locally blueprinted space isomorphic to

\operatorname{Spec}B

for some blueprint $B$ , where the spectrum carries its Zariski topology and structure sheaf.

Contravariantly, blueprint morphisms induce morphisms of affine blue schemes:

\operatorname{Hom}_{\mathrm{Blpr}}(B,C) \;\cong\; \operatorname{Hom}_{\mathrm{BSch}}(\operatorname{Spec}C,\operatorname{Spec}B).

Role in blue geometry

Affine blue schemes are the local models for blue schemes. Principal opens $U_h\subseteq\operatorname{Spec}B$ are affine and correspond to blueprint localizations $B[h^{-1}]$ , so the usual scheme-theoretic gluing pattern survives.

Semiring schemes and ordinary affine schemes enter through the canonical embedding of commutative semirings into blueprints. The resulting base-extension functors carry extra hypotheses and should not be read as identifying all blue schemes with classical schemes.

References

Oliver Lorscheid, The geometry of blueprints, Part I, §3.