Singular derived categories of Q-factorial terminalizations and maximal modification algebras

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 [20], 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 [6] and [14] using geometric arguments. (c) 2014 Elsevier Inc. All rights reserved.