Tilting on non-commutative rational projective curves

2009/05/31 by Igor Burban, Yuriy Drozd · 1 citation
Mathematics · #math.AG #math.RT #msc:14F05 #msc:14H60 #msc:16G99

paper · pdf

published as Math. Ann. 351 (2011), 665-709 · 40 pages; misprints fixed in the new version

arxiv created 2011/04/10 · arxiv updated 2012/01/24

Abstract

In this article we introduce a new class of non-commutative projective curves and show that in certain cases the derived category of coherent sheaves on them has a tilting complex. In particular, we prove that the right bounded derived category of coherent sheaves on a reduced rational projective curve with only nodes and cusps as singularities, can be fully faithfully embedded into the right bounded derived category of the finite dimensional representations of a certain finite dimensional algebra of global dimension two. As an application of our approach we show the dimension of the bounded derived category of coherent sheaves on a rational projective curve with only nodal or cuspidal singularities is at most two. In the case of the Kodaira cycles of projective lines, the corresponding tilted algebras belong to a well-known class of gentle algebras. We work out in details the tilting equivalence in the case of the plane nodal cubic.

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