On Bands and Limit Theorems in Tropical Geometry

Abstract

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 ${\hat{I}}^1$ of $X$ a naturally defined affine band scheme $Y_{\hat{I}}^1$ whose rational points over the tropical band $\mathbb{T}$ recover the tropicalization $Trop(X,{\hat{I}}^1)$. We prove that $X$ is the limit of the $Y_{\hat{I}}^1$ 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.

13 pages