Abstract
We discuss some results on categorical crepant resolutions for varieties with quotient singularities. Namely, we prove that under appropriate hypotheses, the derived category of a smooth Deligne–Mumford stack is a strongly crepant non-commutative resolution of singularities of its coarse moduli space.
Highlights
The ramification formula for proper birational morphisms between smooth varieties implies that X = X
Kuznetsov has given a definition of categorical crepant resolution of singularities which seems very well-fit to deal with the issue of minimal categorical resolutions of singularities
The definition of strongly crepant categorical resolution will be given in Sect. 2 of this paper
Summary
I remind some basic facts about categorical crepant resolution of singularities and we exhibit some classical examples as they appear in [14]. A categorical resolution of X is a smooth cocomplete compactly generated category T with a pair of adjoint functors: π∗: T → D(X ) π ∗: D(X ) → T , such that π∗π ∗ i d,. In [14], the notion of categorical resolution was defined for the bounded derived category of coherent sheaves on X. The left adjoint to π∗ is only well defined on Dperf (X ), which happens to be not convenient This is one reason which explains the use of unbounded derived categories in the above definition. The existence of weakly crepant resolutions of singularities has been proved in a quite general context (see [2,3]).
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