Weighted noncommutative regular projective curves

作者：Kussin Dirk^*

来源：Journal of Noncommutative Geometry, 2016, 10(4): 1465-1540.

DOI：10.4171/JNCG/264

• Noncommutative regular projective curve
• noncommutative function field
• Auslander-Reiten translation
• Picard-shift
• ghost group
• maximal order over a scheme
• ramification
• Witt curve
• noncommutative elliptic curve
• Klein bottle
• Fourier-Mukai partner
• weighted curve
• orbifold Euler characteristic
• noncommutative orbifold
• tubular curve

Let H be a noncommutative regular projective curve over a perfect field k. We study global and local properties of the Auslander-Reiten translation tau and give an explicit description of the complete local rings, with the involvement of tau. We introduce the tau-multiplicity e(tau) (x), the order of tau as a functor restricted to the tube concentrated in x. We obtain a local-global principle for the (global) skewness s(H), defined as the square root of the dimension of the skew-) field over its centre. In the case of genus zero we show how the ghost group, that is, the group of automorphisms of H which fix all objects, is determined by the points x with e(tau) (x) >1. Based on work of Witt we describe the noncommutative regular (smooth) projective curves over the real numbers; those with s(H) = 2 we call Witt curves. In particular, we study noncommutative elliptic curves, and present an elliptic Witt curve which is a noncommutative Fourier-Mukai partner of the Klein bottle. If H is weighted, our main result will be formulae for the orbifold Euler characteristic, involving the weights and the tau-multiplicities. As an application we will classify the noncommutative 2-orbifolds of nonnegative Euler characteristic, that is, the real elliptic, domestic and tubular curves. Throughout, many explicit examples are discussed.

• 出版日期2016