Quasicoherent sheaves on projective schemes over F_1

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 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 from graded R-modules

2. if F is coherent on MProj(A), then F(n) is globally generated for large enough n, and consequently, that F is a quotient of a finite direct sum of invertible sheaves

3. if F is coherent on MProj(A), then gamma(MProj(A)) is finitely generated over A0 (and hence a finite set if A0 = {0, 1}).

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.