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Non-commutative crepant resolutions. (English) Zbl 1082.14005
Laudal, Olav Arnfinn (ed.) et al., The legacy of Niels Henrik Abel. Papers from the Abel bicentennial conference, University of Oslo, Oslo, Norway, June 3–8, 2002. Berlin: Springer (ISBN 3-540-43826-2/hbk). 749-770 (2004).
For a Gorenstein algebra \(R\) over an algebraically closed field \(k\), there is the notion of a crepant resolution of singularities of \(X=\text{Spec}(R)\). While such resolutions do not exist in general, and if they do, they are not unique, there is a conjecture by Bondal and Orlov that the bounded derived category of coherent sheaves on a crepant resolution is unique up to equivalence.
In this paper, the author introduces a noncommutative version of the notion of crepant resolution. This is the algebra \(A\) of endomorphisms of a reflexive \(R\)–module which is homologically homogeneous. In case that \(A\) has finite global dimension it is sufficient that \(A\) is a maximal Cohen–Macaulay \(R\)–module. The main part of the paper is devoted to showing that in the case that \(R\) is three dimensional and has terminal singularities, existence of a noncommutative crepant resolution is equivalent to existence of a crepant resolution in the usual sense. Moreover, the two resolutions are shown to be derived equivalent, thus verifying the Bondal–Orlov conjecture in this special case.
Existence of a noncommutative crepant resolution is proved for two more classes of examples, namely cones over del Pezzo surfaces and invariants of a one dimensional torus acting linearly on a polynomial ring.
For the entire collection see [Zbl 1047.00019].
Reviewer: Andreas Cap (Wien)

MSC:
14A22 Noncommutative algebraic geometry
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
16S38 Rings arising from noncommutative algebraic geometry
16S50 Endomorphism rings; matrix rings
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