On Bands and Limit Theorems in Tropical Geometry

We review the basic theory of bands and band schemes introduced by Baker-Jin-Lorscheid, which is an algebraic framework for tropicalization, analytification, and $\mathbb{F}_1$-geometry. For an affine scheme $X$ over a non-Archimedean valued field $k$, one can associate to every affine embedding $\iota$ of $X$ a naturally defined affine band scheme $Y_\iota$ whose rational points over the tropical band $\mathbb{T}$ recover the tropicalization $Trop(X,\iota)$. We prove that $X$ is the limit of the $Y_\iota$ in the category of band schemes, thereby obtaining a scheme-theoretic enhancement of Payne's limit theorem. By taking $\mathbb{T}$-rational points, this recovers Payne's theorem for affine tropicalizations from the perspective of band scheme theory and the same method provides an analogous result in the real tropical setting.