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].

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)