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

Abstract

We investigate Krull dimensions of semirings and semifields dealt in tropical geometry. For a congruence $C$ on a tropical Laurent polynomial semiring $\boldsymbol{T}[X_1^{\pm}, \ldots, X_n^{\pm}]$, a finite subset $T$ of $C$ is called a finite congruence tropical basis of $C$ if the congruence variety $\boldsymbol{V}(T)$ associated with $T$ coincides with $\boldsymbol{V}(C)$. For $C$ proper, we prove that the Krull dimension of the quotient semiring $\boldsymbol{T}[X_1^{\pm}, \ldots, X_n^{\pm}] / C$ coincides with the maximum of the dimension of $\boldsymbol{V}(C)$ as a polyhedral complex plus one and that of $\boldsymbol{V}(C_{\boldsymbol{B}})$ when both $C$ and $C_{\boldsymbol{B}}$ have finite congruence tropical bases, respectively. Here $C_{\boldsymbol{B}}$ is the congruence on $\boldsymbol{T}[X_1^{\pm}, \ldots, X_n^{\pm}]$ generated by $\{ (f_{\boldsymbol{B}}, g_{\boldsymbol{B}}) \,|\, (f, g) \in C \}$ and $f_{\boldsymbol{B}}$ 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.