Spectrum of a blueprint

Core idea

For a blueprint $B$ , its spectrum

\operatorname{Spec}B

is the set of prime $k$ -ideals of $B$ . It has the Zariski topology whose basic opens are

U_h=\{\mathfrak p\mid h\notin\mathfrak p\}

for $h\in B$ , together with a structure sheaf $\mathcal O_{\operatorname{Spec}B}$ locally modeled on blueprint localizations.

This is the prime- $k$ -ideal spectrum used in the original theory of blue schemes. Other blueprint geometries also use congruence spectra, which need not have the same points.

Basic affine calculation

The structure sheaf satisfies

\Gamma(U_h,\mathcal O_{\operatorname{Spec}B})\simeq B[h^{-1}]

for basic opens under the standard blueprint-spectrum construction. In particular, $\operatorname{Spec}B$ is the affine blue scheme represented by $B$ .

Convention warning

“Spectrum of a blueprint” is not a unique phrase across every later variant of blueprint geometry. This knowl records Lorscheid's prime- $k$ -ideal spectrum underlying blue schemes. Ordered blue schemes use the corresponding ordered-blueprint theory.

References

Oliver Lorscheid, The geometry of blueprints, Part I, §§2–3.