Lattices, Spectral Spaces, and Closure Operations on Idempotent Semirings

Abstract

A classical theorem of Hochster provides purely topological characterization of prime spectra of commutative rings. In this paper, we first prove an analogous statement for idempotent semirings, showing that for a spectral space X, we can construct an idempotent semiring A in such a way that the saturated prime spectrum of A is homeomorphic to X. We further provide examples of spectral spaces arising from sets of congruence relations of semirings. In particular, we prove that the space of valuations and the space of prime congruences on an idempotent semiring A are spectral, and there is a natural bijection of sets between two. We then develop several aspects of commutative algebra of semirings. We mainly focus on the notion of closure operations for semirings, and provide several examples. In particular, we introduce an integral closure operation and a Frobenius closure operation for idempotent semirings.

Vector bundles on tropical schemes

“…One may then expect that some properties of ideals of A (more generally modules over A) to be captured purely in terms of certain structures of this associated idempotent semiring. We refer the interested readers to [JRT20] for detailed treatment of this line of thought along with an idempotent semiring analogue of Hochster's theorem on spectral spaces [Hoc69]. To this end, we first prove the following.…”

“…In this section, we briefly recall basic definitions and properties of monoid schemes and semiring schemes (also tropical schemes as a special case) which play a key role in the later sections. Most of the material in this section can be found in [GG16], [Jun17a], [JMT19], [JRT20].…”

“…More precisely, the closed sets in Spec s S are of the form V (I) = {p ∈ Spec s S | I ⊆ p}. For more details on this topology, see [JRT20,§3].…”

Proto-exact categories of modules over semirings and hyperrings

“…In particular, for each subset S of M, there exists a smallest saturated submodule of M containing S. We call this the saturation closure (or simply the saturation) of S, denoted by S or S . The following is proved in [JRT20] for ideals, but the same proof works for submodules. We include the proof for completeness.…”

“…For more details in relation to tropical geometry, we refer the readers to [JMT20]. This correspondence was also employed in [JRT20] to enrich Hochster's theorem on spectral spaces in [Hoc69] by showing that a topological space X is spectral if and only if X is the "saturated prime spectrum" of an idempotent semiring. In fact, in [JRT20], with S. Ray, the first and the third authors proved that the category of spectral spaces is equivalent to a certain subcategory of the category of idempotent semirings.…”