Categorical Resolutions of Irrational Singularities

Abstract

We show that the derived category of any singularity over a field of characteristic 0 can be embedded fully and faithfully into a smooth triangulated category which has a semiorthogonal decomposition with components equivalent to derived categories of smooth varieties. This provides a categorical resolution of the singularity.

Categorical crepant resolutions for quotient singularities

“…The functor π * always commutes with arbitrary direct sums whereas the functor π * commutes with arbitrary direct sums because it is right adjoint to π * and T is compactly generated. Finally, the inclusion I refer to [12] for interesting comments about this nice result and for a proof of it. In this note, we will focus on a very special type of categorical resolutions, the so-called categorical crepant resolutions of singularities.…”

“…I will not discuss the details of this definition and rather refer to [12] where the theory is developed with great care. In [14], the notion of categorical resolution was defined for the bounded derived category of coherent sheaves on X .…”

“…But T is the completion of an orthogonal component of D b (X ), so that T is also cocomplete, compactly generated and smooth ( [12], for instance).…”

Derived noncommutative schemes, geometric realizations, and finite dimensional algebras

“…The lower triangular DG category C " A v T B is not necessarily pretriangulated even if the components A and B are pretriangulated. To make this operation well-defined on the class of pretriangulated categories, we introduce gluing of pretriangulated categories (see [T2,KL,Ef,O6]…”

Exceptional collections in surface-like categories

“…Note that each of these categories can be realized as a semiorthogonal component of the derived category of a smooth projective variety [Or2], see also [BR2]. More generally, one can ask when a gluing [KL,Or1,Or3] of two triangulated categories is surface-like.…”