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

Abstract

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) and its idempotent completion, we give necessary and sufficient categorical conditions for X to be Q-factorial and complete locally Q-factorial respectively. We then relate this information to maximal modification algebras(=MMAs), introduced in [IW10], by showing that if an algebra A is derived equivalent to X as above, then X is Q-factorial if and only if A is an MMA. Thus all rings derived equivalent to Q-factorial terminalizations in dimension three are MMAs. As an application, we extend some of the algebraic results in Burban-Iyama-Keller-Reiten [BIKR] and Dao-Huneke [DH] using geometric arguments.

Very minor changes, 24 pages

References in corpus (2)

• Calabi-Yau algebras
• Relative singularity category of a non-commutative resolution of singularities

Cited by in corpus (8)

• Noncommutative deformations and flops
• Lectures on Noncommutative Resolutions
• Reduction of triangulated categories and Maximal Modification Algebras for cA_n singularities
• Singularity categories via the derived quotient
• New realizations of modular forms in Calabi-Yau threefolds arising from ${\varphi }^4$ theory
• A new equivalence between singularity categories of commutative algebras
• Fourier-Mukai partners of singular genus one curves
• Derived equivalences and Kodaira fibers