Singular Derived Categories of Q-factorial terminalizations and Maximal Modification Algebras

Details Singular Derived Categories of Q-factorial terminalizations and Maximal Modification Algebras Singular Derived Categories of Q-factorial terminalizations and Maximal Modification Algebras

2011-08-23

arxiv.org/abs/1108.4518v1

Mathematics

Algebraic Geometry

Preliminary version. 34 pages

Scientific paper

Let X be a Gorenstein normal 3-fold satisfying (ELF) with local rings which are at worst isolated hypersurface (e.g. terminal) singularities. By using the singular derived category D_{sg}(X), we give a necessary and sufficient categorical condition for X to be Q-factorial. By passing to the idempotent completion we obtain a necessary and sufficient condition for X to be complete locally Q-factorial. We then relate this information to maximal modification algebras(=MMAs), introduced in [IW10], by showing that if a modifying algebra A is derived equivalent to X as above, then X is Q-factorial if and only if A is a MMA. This shows that MMAs categorify Q-factorial terminalizations in the same way that noncommutative crepant resolutions (=NCCRs) categorify 3-fold crepant resolutions. We then apply these results to the case of cA_n singularities, generalizing some results of Burban-Iyama-Keller-Reiten [BIKR] and Dao-Huneke [DH].

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