31,459 research outputs found
Quasicoherent sheaves on projective schemes over 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 (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.Comment: arXiv admin note: text overlap with arXiv:1009.357
A Generalization of Kochen-Specker Sets Relates Quantum Coloring to Entanglement-Assisted Channel Capacity
We introduce two generalizations of Kochen-Specker (KS) sets: projective KS
sets and generalized KS sets. We then use projective KS sets to characterize
all graphs for which the chromatic number is strictly larger than the quantum
chromatic number. Here, the quantum chromatic number is defined via a nonlocal
game based on graph coloring. We further show that from any graph with
separation between these two quantities, one can construct a classical channel
for which entanglement assistance increases the one-shot zero-error capacity.
As an example, we exhibit a new family of classical channels with an
exponential increase.Comment: 16 page
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