𝘛代数のクルル次元の幾何学的解釈

Abstract

Recently many researches to construct algebraic foundation for tropical geometry appear. One of them is Joó-Mincheva's research on Krull dimensions of 𝘉-algebras, where 𝘉 is the Boolean semifield. They revealed that the tropical Laurent polynomial semiring 𝘛[[+-]𝘟₁, ..., [+-]𝘟ₙ] in 𝓃-variables has Krull dimension n plus one. In this talk, with their technique, we explain that the Krull dimension of 𝘛[[+-]𝘟₁, ..., [+-]𝘟ₙ]/𝘊 for a congruence 𝘊 having a finite congruence tropical basis and nonempty congruenc variety 𝘝(𝘊) is max{dim 𝘝(𝘊) + 1, dim V(𝘊[𝘉])} if 𝘊[𝘉] also has a finite congruence tropical basis. Here 𝘊[𝘉] is the congruence obtained from 𝘊 with the 𝘉-algebra homomorphism 𝘛→𝘉. This is a joint work with Yasuhito Nakajima.