Picard groups for tropical toric schemes (Q2326786)

Given a variety over a valued field, one can tropicalize it and construct a combinatorial object. Originally, these tropical varieties are polyhedral complexes which inherit topology from \(\mathbb{R}^n\). \textit{J. Giansiracusa} and \textit{N. Giansiracusa} [Duke Math. J. 165, No. 18, 3379--3433 (2016; Zbl 1409.14100)] combined \(\mathbb{F}_1\)-geometry with the notion of bend loci to equip tropical varieties with scheme structure, and therefore, obtained tropical schemes. This paper investigates Picard groups of tropical schemes as a first step towards building scheme-theoretic tropical divisor theory which can lead to finding a scheme-theoretic tropical Riemann-Roch theorem. For monoid \(M\) one can pass from monoid scheme \(X = \text{Spec}\, M\) to scheme \(X_K = \text{Spec}\, K[M]\) by scalar extension to field \(K\). \textit{J. Flores} and \textit{C. Weibel} [J. Algebra 415, 247--263 (2014; Zbl 1314.14003)] show Picard groups \(\text{Pic} (X)\) and \(\text{Pic} (X_K)\) are isomorphic. The current paper proves this isomorphism in tropical setting: for irreducible monoid scheme \(X\) and idempotent semifield \(S\) Picard groups \(\text{Pic} (X)\) and \(\text{Pic} (X_S)\) are both isomorphic to certain sheaf cohomology groups, and hence, are isomorphic. They also construct the group \(\text{CaCl}\, (X_S)\) of Cartier divisors modulo principal Cartier divisors and show that \(\text{CaCl}\, (X_S)\) is isomorphic to \(\text{Pic} (X_S)\).