2018/11/21 by Belmans, Pieter, Presotto, Dennis, Bergh, Michel Van den
#Algebraic Geometry (math.AG) #FOS: Mathematics #Quantum Algebra (math.QA) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1811.08810
Recently the Euler forms on numerical Grothendieck groups of rank 4 whose properties mimick that of the Euler form of a smooth projective surface have been classified. This classification depends on a natural number m, and suggests the existence of noncommutative surfaces which up to that point had not been considered for m≥ 2. These have been constructed for m=2 using noncommutative ℙ1-bundles, and for all m≥ 2 by a different construction using maximal orders on Blxℙ2. In this article we compare the constructions for m=2, i.e. we compare the categories arising from half-ruled del Pezzo quaternion orders on \mathbbF1 with noncommutative ℙ1-bundles on ℙ1. This can be seen as a noncommutative instance of the classical isomorphism \mathbbF1\congBlxℙ2.