A geometric interpretation of Krull dimensions of $\mathbf{T}$-algebras

Abstract

We investigate Krull dimensions of semirings and semifields dealt in tropical geometry. For a congruence $C$ on a tropical Laurent polynomial semiring $\mathbf{T}[X_1^{\pm },\dots ,X_n^{\pm }]$, a finite subset $T$ of $C$ is called a finite congruence tropical basis of $C$ if the congruence variety $\mathbf{V}(T)$ associated with $T$ coincides with $\mathbf{V}(C)$. For $C$ proper, we prove that the Krull dimension of the quotient semiring $\mathbf{T}[X_1^{\pm },\dots ,X_n^{\pm }]/C$ coincides with the maximum of the dimension of $\mathbf{V}(C)$ as a polyhedral complex plus one and that of $\mathbf{V}(C_{\mathbf{B}})$ when both $C$ and $C_{\mathbf{B}}$ have finite congruence tropical bases, respectively. Here $C_{\mathbf{B}}$ is the congruence on $\mathbf{T}[X_1^{\pm },\dots ,X_n^{\pm }]$ generated by $\{(f_{\mathbf{B}},g_{\mathbf{B}})|(f,g)\in C\}$ and $f_{\mathbf{B}}$ is defined as the tropical Laurent polynomial obtained from $f$ by replacing the coefficients of all non $-\infty$ terms of $f$ with the real number zero. With this fact, we also show that rational function semifields of tropical curves that do not consist of only one point have Krull dimension two.

30 pages