Picard groups for tropical toric schemes

Abstract

From any monoid scheme $X$ (also known as an $\mathbb{F}_1$-scheme) one can pass to a semiring scheme (a generalization of a tropical scheme) $X_S$ by scalar extension to an idempotent semifield $S$. We prove that for a given irreducible monoid scheme $X$ (satisfying some mild conditions) and an idempotent semifield $S$, the Picard group $Pic(X)$ of $X$ is stable under scalar extension to $S$ (and in fact to any field $K$). In other words, we show that the groups $Pic(X)$ and $Pic(X_S)$ (and $Pic(X_K)$) are isomorphic. In particular, if $X_\mathbb{C}$ is a toric variety, then $Pic(X)$ is the same as the Picard group of the associated tropical scheme. The Picard groups can be computed by considering the correct sheaf cohomology groups. We also define the group $CaCl(X_S)$ of Cartier divisors modulo principal Cartier divisors for a cancellative semiring scheme $X_S$ and prove that $CaCl(X_S)$ is isomorphic to $Pic(X_S)$.