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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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