Singular derived categories of QQ-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 Dsg(X)Dsg(X) and its idempotent completion Dsg(X)¯, we give necessary and sufficient categorical conditions for X   to be QQ-factorial and complete locally QQ-factorial respectively. We then relate this information to maximal modification algebras (= MMAs), introduced in [20], by showing that if an algebra Λ is derived equivalent to X as above, then X   is QQ-factorial if and only if Λ   is an MMA. Thus all rings derived equivalent to QQ-factorial terminalizations in dimension three are MMAs. As an application, we extend some of the algebraic results in [6] and [14] using geometric arguments.