Abstract. In this paper we study endomorphism rings of finite global dimension over not necessarily normal commutative rings. These objects have recently attracted attention as noncommutative (crepant) resolutions, or NC(C)Rs, of singularities. We propose a notion of a NCCR over any commutative ring that appears weaker but generalizes all previous notions. Our results yield strong necessary and sufficient conditions for the existence of such objects in many cases of interest. We also give new examples of NCRs of curve singularities, regular local rings and normal crossing singularities. Moreover, we introduce and study the global spectrum of a ring R, that is, the set of all possible finite global dimensions of endomorphism rings of MCM R-modules. Finally, we use a variety of methods to compute global dimension for many endomorphism rings.

We construct a noncommutative desingularization of the discriminant of a finite reflection group G as a quotient of the skew group ring A = S * G. If G is generated by order two reflections, then this quotient identifies with the endomorphism ring of the reflection arrangement A(G) viewed as a module over the coordinate ring S G /(∆) of the discriminant of G. This yields, in particular, a correspondence between the nontrivial irreducible representations of G to certain maximal Cohen-Macaulay modules over the coordinate ring S G /(∆). These maximal Cohen-Macaulay modules are precisely the nonisomorphic direct summands of the coordinate ring of the reflection arrangement A(G) viewed as a module over S G /(∆). We identify some of the corresponding matrix factorizations, namely the so-called logarithmic co-residues of the discriminant.In order to show that A is an endomorphism ring, we first view A as a CM-module over the (noncommutative) ring T * H and will use the functor i * : Mod(T * H) − → Mod(R/(∆)) , coming from a standard recollement. For this part we will need that G is a true reflection group, that is, generated by reflections of order 2. Then clearly H ∼ = µ 2 . In order to use the 1 Most of our results also hold if the characteristic of the field K does not divide the order |G| of the group G. However, in order to facilitate the presentation, we restrict to K = C.which gives us that the map S J − → S decomposes into components of the form Hom KG (U, S) ⊗ K U J − → Hom KG (U ⊗ det, S) ⊗ K U ⊗ det for each irreducible representation U of G.

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