Semiring completion of a blueprint

Core idea

Let $B=A/\!/\mathcal R$ be a blueprint. Its semiring completion is the commutative semiring

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

Thus two formal sums define the same element of $B^+$ exactly when they are related by the pre-addition $\mathcal R$ .

The map

B^\bullet=A\longrightarrow B^+

sends each element of the multiplicative monoid to its one-term sum. Properness of $\mathcal R$ makes this map injective, and its image generates $B^+$ as a semiring. The pair consisting of $B^+$ and the distinguished multiplicative subset $B^\bullet$ retains the blueprint data; the semiring $B^+$ alone may not.

Functoriality

A blueprint morphism $f:B\to C$ extends termwise to formal sums and therefore induces a semiring homomorphism

f^+:B^+\longrightarrow C^+.

Consequently $B\mapsto B^+$ is a functor from blueprints to commutative semirings.

References

Oliver Lorscheid, The geometry of blueprints, Part I: Algebraic background and scheme theory, §1.2.