Localization of a blueprint

Core idea

Let $B=A/\!/\mathcal R$ be a blueprint and let $S\subseteq A$ be multiplicatively closed. The localization $S^{-1}B$ has underlying monoid $S^{-1}A$ , and its pre-addition is generated by the images of the relations in $\mathcal R$ .

It is characterized by the usual universal property: a blueprint morphism $f:B\to C$ factors uniquely through $S^{-1}B$ exactly when every $f(s)$ , $s\in S$ , is a unit of $C^\bullet$ .

Fractions and relations

Elements are represented by fractions $a/s$ . An additive relation between fractions holds when, after multiplying by a suitable common denominator, it is induced by a relation of $B$ . This simultaneously localizes the multiplicative monoid and transports the formal additive structure.

For $h\in B$ , one writes $B[h^{-1}]$ for localization at $\{1,h,h^2,\ldots\}$ . These principal localizations define the basic open charts used in blueprint spectra.

References

Oliver Lorscheid, The geometry of blueprints, Part I, §1.9.