# Commutative ν-Algebra and Supertropical Algebraic Geometry

Research output: Working paper

## 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.
Original language English ArXiv Submitted - 23 Jan 2019

### Publication series

Name arXiv

• math.AC
• math.AG