Quasicoherent sheaves on projective schemes over F_1

TL;DR

This paper develops the theory of quasicoherent sheaves on projective monoid schemes over F_1, establishing foundational results and classifying coherent sheaves on P^1 with novel combinatorial structures.

Contribution

It introduces a framework for quasicoherent sheaves on projective monoid schemes over F_1 and classifies coherent sheaves on P^1 using combinatorial methods.

Findings

Every quasicoherent sheaf can be constructed from a graded A--set.

High twists of coherent sheaves are finitely generated by global sections.

Coherent sheaves have finite global sections.

Abstract

Given a graded monoid A with 1, one can construct a projective monoid scheme MProj(A) analogous to Proj(R) of a graded ring R. This paper is concerned with the study of quasicoherent sheaves (of pointed sets) on MProj(A), and we prove several basic results regarding these. We show that: 1.) Every quasicoherent sheaf F on MProj(A) can be constructed from a graded A--set in analogy with the construction of quasicoherent sheaves on Proj(R) from graded R--modules. 2.) High enough twists of coherent sheaves are generated by finitely many global sections, hence that every coherent sheaf is a quotient of a locally free sheaf. 3.) Coherent sheaves have finite spaces of global sections. The last part of the paper is devoted to classifying coherent sheaves on P^1 in terms of certain directed graphs and gluing data. The classification of these over F_1 is shown to be much richer and combinatorially interesting than in the case of ordinary P^1, and several new phenomena emerge.