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Remarks on non-commutative crepant resolutions of complete intersections. (English) Zbl 1192.13011
Let \(R\) be a Gorenstein local normal domain. If \(M\) is a reflexive \(R\)-module such that \(A=\operatorname{Hom}_R(M,M)\) is maximal Cohen-Macaulay \(R\)-module of finite global dimension then \(A\) is called a non-commutative crepant resolution (shortly NCCR). M. van den Bergh [The legacy of Niels Henrik Abel. Papers from the Abel bicentennial conference, University of Oslo, Oslo, Norway, June 3–8, 2002. Berlin: Springer. 749–770 (2004; Zbl 1082.14005)] showed for dimension \(3\) isolated terminal singularities that the existence of a NCCR is equivalent with the existence of a desingularization \(f:Y\rightarrow X=\text{Spec}\;R\) such that \(f^*\omega_X=\omega_Y\).
Let \(R\) be a local hypersurface satisfying condition \((R_2)\) and such that \({\hat R}\cong S/(f)\) for an equicharacteristic or unramified local ring \(S\), \(f\) being a regular element of \(S\). The main results of the paper are the followings:
i) \(R\) admits no NCCR if either \(\dim \;R=3\) and \(R\) is \({\mathbb Q}\)-factorial, or \(R\) has isolated singularity and \(\dim\;R>3\) is even.
ii) \(R\) admits a NCCR (a suggestion how to build it is given) if there exists an indecomposable maximal Cohen-Macaulay \(R\)-module which is not Tor-rigid.

MSC:
13C14 Cohen-Macaulay modules
13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.)
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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