Quasicoherent sheaves on projective schemes over $\Bbb F_1$

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 Proj(R)
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 Γ(MProj(A), F) 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 P1 in terms of
certain directed graphs and gluing data. The classification of these over F1 is shown
to be much richer and combinatorially interesting than in the case of ordinary P1,
and several new phenomena emerge.