Commutative $\nu$-algebra and supertropical algebraic geometry

TL;DR

This paper develops a foundational framework for supertropical algebraic geometry using commutative $ u$-algebras, introducing new congruences and spectra to establish a scheme theory analogous to classical algebraic geometry.

Contribution

It introduces $ u$-algebras, $rak{q}$-congruences, and their spectra, creating a systematic foundation for supertropical algebraic geometry and extending scheme theory to this setting.

Findings

Defined $rak{g}$-prime, $rak{g}$-radical, and maximal $rak{q}$-congruences.

Established spectra of $rak{g}$-prime congruences as the underlying spaces for schemes.

Connected supertropical algebraic geometry with classical scheme theory via Grothendieck's approach.

Abstract

This paper lays out a foundation for a theory of supertropical algebraic geometry, relying on commutative $\nu$-algebra. To this end, the paper introduces $\mathfrak{q}$-congruences, carried over $\nu$-semirings, whose distinguished ghost and tangible clusters allow both quotienting and localization. Utilizing these clusters, $\mathfrak{g}$-prime, $\mathfrak{g}$-radical, and maximal $\mathfrak{q}$-congruences are naturally defined, satisfying the classical relations among analogous ideals. Thus, a foundation of systematic theory of commutative $\nu$-algebra is laid. In this framework, the underlying spaces for a theoretic construction of schemes are spectra of $\mathfrak{g}$-prime congruences, over which the correspondences between $\mathfrak{q}$-congruences and varieties emerge directly. Thereby, scheme theory within supertropical algebraic geometry follows the Grothendieck approach, and is applicable to polyhedral geometry.