### Abstract

Original language | English |
---|---|

Journal | arXiv |

Publication status | Submitted - 23 Jan 2019 |

### Fingerprint

### Keywords

- math.AC
- math.AG

### Cite this

**Commutative ν-Algebra and Supertropical Algebraic Geometry.** / Izhakian, Zur.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Commutative ν-Algebra and Supertropical Algebraic Geometry

AU - Izhakian, Zur

N1 - 83 pages

PY - 2019/1/23

Y1 - 2019/1/23

N2 - 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.

AB - 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.

KW - math.AC

KW - math.AG

M3 - Article

JO - arXiv

JF - arXiv

ER -