### 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 |
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Publisher | ArXiv |

Publication status | Submitted - 23 Jan 2019 |

### Publication series

Name | arXiv |
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### Keywords

- math.AC
- math.AG

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## Cite this

Izhakian, Z. (2019).

*Commutative ν-Algebra and Supertropical Algebraic Geometry*. (arXiv). ArXiv. http://arxiv.org/abs/1901.08032v1