JournalsjncgVol. 10, No. 4pp. 1465–1540

Weighted noncommutative regular projective curves

  • Dirk Kussin

    University of Paderborn, Germany
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Let H\mathcal H be a noncommutative regular projective curve over a perfect field kk. We study global and local properties of the Auslander–Reiten translation τ\tau and give an explicit description of the complete local rings, with the involvement of τ\tau. We introduce the τ\tau-multiplicity eτ(x)e_{\tau}(x), the order of τ\tau as a functor restricted to the tube concentrated in xx. We obtain a local-global principle for the (global) skewness s(H)s(\mathcal H), defined as the square root of the dimension of the function (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\mathcal H which fix all objects, is determined by the points xx with eτ(x)>1e_{\tau}(x) > 1. Based on work of Witt we describe the noncommutative regular (smooth) projective curves over the real numbers; those with s(H)=2s(\mathcal 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\mathcal H is weighted, our main result will be formulae for the orbifold Euler characteristic, involving the weights and the τ\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.

Cite this article

Dirk Kussin, Weighted noncommutative regular projective curves. J. Noncommut. Geom. 10 (2016), no. 4, pp. 1465–1540

DOI 10.4171/JNCG/264