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Let $R$ be normal, local ring of dimension at least $2$. Let $M$ be a reflexive $R$-module and let $A=Hom_R(M,M)$. Suppose $A$ has finite global dimension. Then one can view $A$ as a weak non-commutative desingularization of $R$ (note that, a) there is a natural map $R\to A$ and b) in the commutative case, finite global dimension implies regularity, hence the name).

This concept imitates Van den Bergh's definition of non-commutative crepant resolution (NCCR). His definition arises from a proof of dimension $3$ case of Bondal-Orlov conjecture. This is a long story, but an excellent account of the reasons behind the definition can be found in Section 4 of this paper. For existence of NCCR in some high dimensions case, check out this.

Now, non-commutative crepant resolution does not always exist (the above papers proves the equivalence, in some cases, with existence of projective crepant resolutions, which exists rarely in high dimensions). What we do know from Hironaka, in characteristic $0$ at least, is that resolution of singularity exist. So:

Question: Does weak non-commutative desingularizations of $R$ (as specified in the first paragraph) always exist?

Some discussions (I am a beginner in this, so feel free to correct me):

1) Why weak? By Morita equivalence, to ensures the desingularization is an isomorphism on the regular locus one needs $M$ to be free on that locus of $R$.

2) If one requires extra conditions (like $M$ being an generator-cogenerator) then there are examples when such desingularization does not exist (in some paper by Iyama which I forgot the name). I have not seen an example in the generality above.

3) There are positive results in diemension $0,1$, but I care about normal rings, so we start in dimension $2$.

4) Some people like Van den Bergh or Lieven le Bruyn probably know. May be they are even on MO!

I would appreciate even heuristic reasons for one way or another.

EDIT: The question is now resolved, by Lieven's answer below (as expected (:). I will provide a little bit more details in case someone is interested: Van den Bergh and Stafford proved that, in characteristic $0$, if $A$ is a non-commutative crepant resolution, then $R$ has rational singularity. The definition of NCCR is stronger, but if, for example, $R$ is Gorenstein of dimension $2$, it coincides with my version. So a counterexample is something like $R=k[x,y,z]/(x^3+y^3+z^3)$, which is a non-rational hypersurface.

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Let $R$ be normal, local ring of dimension at least $2$. Let $M$ be a reflexive $R$-module and let $A=Hom_R(M,M)$. Suppose $A$ has finite global dimension. Then one can view $A$ as a weak non-commutative desingularization of $R$ (note that, a) there is a natural map $R\to A$ and b) in the commutative case, finite global dimension implies regularity, hence the name).

This concept imitates Van den Bergh's definition of non-commutative crepant resolution (NCCR). His definition arises from a proof of dimension $3$ case of Bondal-Orlov conjecture. This is a long story, but an excellent account of the reasons behind the definition can be found in Section 4 of this paper. For existence of NCCR in some high dimensions case, check out this.

Now, non-commutative crepant resolution does not always exist (the above papers proves the equivalence, in some cases, with existence of projective crepant resolutions, which exists rarely in high dimensions). What we do know from Hironaka, in characteristic $0$ at least, is that resolution of singularity exist. So:

Question: Does weak non-commutative desingularizations of $R$ (as specified in the first paragraph) always exist?

Some discussions (I am a beginner in this, so feel free to correct me):

1) Why weak? From the point of By Morita equivalence, to ensures the desingularization is an isomorphism on the regular locus one needs $M$ to be free on that locus of $R$.

2) If one requires extra conditions (like $M$ being an generator-cogenerator) then there are examples when such desingularization does not exist (in some paper by Iyama which I forgot the name). I have not seen an example in the generality above.

3) There are positive results in diemension $0,1$, but I care about normal rings, so we start in dimension $2$.

4) Some people like Van den Bergh or Lieven le Bruyn probably know. May be they are even on MO!

I would appreciate even heuristic reasons for one way or another.

EDIT: The question is now resolved, by Lieven's answer below (as expected (:). I will provide a little bit more details in case someone is interested: Van den Bergh and Stafford proved that, in characteristic $0$, if $A$ is a non-commutative crepant resolution, then $R$ has rational singularity. The definition of NCCR is stronger, but if $R$ is Gorenstein of dimension $2$, it coincides with my version. So a counterexample is something like $R=k[x,y,z]/(x^3+y^3+z^3)$, which is a non-rational hypersurface.

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Let $R$ be normal, local ring of dimension at least $2$. Let $M$ be a reflexive $R$-module and let $A=Hom_R(M,M)$. Suppose $A$ has finite global dimension. Then one can view $A$ as a weak non-commutative desingularization of $R$ (note that, a) there is a natural map $R\to A$ and b) in the commutative case, finite global dimension implies regularity, hence the name).

This concept imitates Van den Bergh's definition of non-commutative crepant resolution (NCCR). His definition arises from a proof of dimension $3$ case of Bondal-Orlov conjecture. This is a long story, but an excellent account of the reasons behind the definition can be found in Section 4 of this paper. For existence of NCCR in some high dimensions case, check out this.

Now, non-commutative crepant resolution does not always exist (the above papers proves the equivalence, in some cases, with existence of projective crepant resolutions, which exists rarely in high dimensions). What we do know from Hironaka, in characteristic $0$ at least, is that resolution of singularity exist. So:

Question: Does weak non-commutative desingularizations of $R$ (as specified in the first paragraph) always exist?

Some discussions (I am a beginner in this, so feel free to correct me):

1) While Why weak? From the point of Morita equivalence, to ensures the desingularization is an isomorphism on the regular locus one needs $M$ to be free on that locus of $R$.

2) If one requires extra conditions (like $M$ being an generator-cogenerator) then there are examples when such desingularization does not exist (in some paper by Iyama which I forgot the name). I have not seen an example in the generality above.

3) There are positive results in diemension $0,1$, but I care about normal rings, so we start in dimension $2$.

4) Some people like Van den Bergh or Lieven le Bruyn probably know. May be they are even on MO!

I would appreciate even heuristic reasons for one way or another.

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