Let $R$ be normal, local ring of dimension at least $2$. Let $M$ be a [reflexive](https://mathoverflow.net/questions/7490/differences-between-reflexives-and-projectives-modules/7588#7588) $R$-module and let $A=Hom\_R(M,M)$. Suppose $A$ has [finite global dimension](https://en.wikipedia.org/wiki/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](https://doi.org/10.1215/S0012-7094-04-12231-6). This is a long story, but an excellent account of the reasons behind the definition can be found in Section 4 of this [paper](https://arxiv.org/abs/math/0211064). For existence of NCCR in some high dimensions case, check out [this](https://arxiv.org/abs/0911.2659).


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](https://arxiv.org/abs/math/0612032) 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.



Existence of non-commutative desingularizations
There are already counter-examples in dimension 2. If you take a 2-dml non-rational singularity, then there cannot exist a non-commutative resolution in your sense. In fact, any 2-dml nc-resolution your sense is also an nc-resolution in Michel's sense and so must have rational singularities by a [result of Toby Stafford and Michel](https://arxiv.org/abs/math/0612032).


In dimension two you can resort to ancient stuff such as the book 'Graded orders' by Fred, Michel and me, an online scanned version can be found [here](http://matrix.uantwerpen.be/lieven.lebruyn/LeBruyn1988f.pdf) ([Wayback Machine](https://web.archive.org/web/20201112163640/http://matrix.uantwerpen.be/lieven.lebruyn/LeBruyn1988f.pdf)). Lemma IV.2.3 says that a tame order (such as your End(M)) of global dim 2 is 'moderated regular' (defn IV.1.4) and hence a tame order of finite representation type which are classified further in the book to yield rational central singularities. That lemma (and dfn IV.1.1 and thm IV.1.2) also shows that your and Michel's dfns coincide in dim 2.


Further, I didn't understand your remark about an Iyama counterexample in the case M is projective? In that case End(M) is Azumaya, so if it has finite gldim, then the center has to be regular too.


