Quasicoherent sheaves on projective schemes over F_1 (1602.05233v1)
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 P1 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 P1, and several new phenomena emerge.