Idempotentization of Affine Schemes and Sheaves

In this article, we introduce the idempotentization process, which bears some philosophical and mathematical similarities with modern analytification and tropicalization. Idempotentization associates to any affine scheme an idempotent version of itself with respect to a fixed covering by distinguished affine open subschemes. Once this cover is fixed, we can functorially associate a Zariski sheaf of rings or modules to a sheaf of idempotent semiring or a sheaf of idempotent semimodules. We show that idempotentization is independent of the chosen cover and in the Noetherian case, the idempotentization of the structure sheaf recovers the global sections. Underlying our formalism is a combinatorial reflection of lattices of subobjects of ordered-theoretic objects seen as lattices coming from commutative algebra. This has topological consequences for the semiring of subtractive ideals of a commutative semiring $S$: On one hand, it is a topological retract of the semiring of congruence relations of $S$ for the coarse lower topology. On the other hand, it is a topological retract of the semiring of ideals of $S$ for the coarse upper topology.