Quasicoherent sheaves on projective schemes over F_1
Files
Accepted manuscript
Date
2017-07
Authors
Lorscheid, Oliver
Szczesny, Matt
Version
OA Version
Citation
Oliver Lorscheid, Matt Szczesny. June 2018. "Quasicoherent sheaves on projective schemes over." Journal of Pure and Applied Algebra, Volume 222, Issue 6, pp. 1337-1354
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.