2017/09/10 by Jaiung Jun, Jun, Jaiung, Kalina Mincheva +3
Computer Science · Mathematics · Physics and Astronomy · #12K10(secondary) #14C22(primary) #14T05(primary) #16Y60(secondary) #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Nonlinear Waves and Solitons #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1709.03130
openalex publication_date 2017/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
From any monoid scheme X (also known as an \mathbbF1-scheme) one can pass to a semiring scheme (a generalization of a tropical scheme) XS 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(XS) (and Pic(XK)) are isomorphic. In particular, if X_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(XS) of Cartier divisors modulo principal Cartier divisors for a cancellative semiring scheme XS and prove that CaCl(XS) is isomorphic to Pic(XS).