Blueprint - Knowlpedia

Definition

A blueprint $B=A/\!/\mathcal R$ consists of a commutative monoid with zero $A$ together with a pre-addition $\mathcal R$ on $A$ .

A morphism $f:A/\!/\mathcal R\to A'/\!/\mathcal R'$ is a multiplicative map preserving $0$ and $1$ whose termwise extension sends every relation in $\mathcal R$ to one in $\mathcal R'$ .

What the notation records

The monoid $A$ is the multiplicative skeleton $B^\bullet$ . The relations $\mathcal R$ say which formal sums are to count as equal without requiring every sum to be represented by an element of $A$ . The semiring completion is

B^+=\mathbb N[A]/\mathcal R

and the canonical map $B^\bullet\to B^+$ remembers which elements are designated as monomials.

Familiar structures inside blueprints

There are fully faithful embeddings of commutative semirings and commutative monoids with zero into blueprints, but the two constructions impose different additive relations.

Blueprints support localization, spectra, and blue schemes. The canonical embedding into ordered blueprints regards every additive equality as a pair of opposite inequalities.

References

Oliver Lorscheid, The geometry of blueprints, Part I: Algebraic background and scheme theory, §1.