We situate the noncrossing partitions associated with a finite Coxeter group within the context of the representation theory of quivers. We describe Reading's bijection between noncrossing partitions and clusters in this context, and show that it extends to the extended Dynkin case. Our setup also yields a new proof that the noncrossing partitions associated with a finite Coxeter group form a lattice. We also prove some new results within the theory of quiver representations. We show that the finitely generated, exact abelian, and extension-closed subcategories of the representations of a quiver Q without oriented cycles are in natural bijection with the cluster tilting objects in the associated cluster category. We also show that these subcategories are exactly the finitely generated categories that can be obtained as the semistable objects with respect to some stability condition.
We consider the class of singular double coverings X → P 3 ramified in the degeneration locus D of a family of 2-dimensional quadrics. These are precisely the quartic double solids constructed by Artin and Mumford as examples of unirational but nonrational conic bundles. With such quartic surface D one can associate an Enriques surface S which is the factor of the blowup of D by a natural involution acting without fixed points (such Enriques surfaces are known as nodal Enriques surfaces or Reye congruences). We show that the nontrivial part of the derived category of coherent sheaves on this Enriques surface S is equivalent to the nontrivial part of the derived category of a minimal resolution of singularities of X.
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 define and study canonical singularities of orders over surfaces. These are non-commutative analogues of Kleinian singularities that arise naturally in the minimal model program for orders over surfaces D. Chan and C. Ingalls, Invent. Math. 161 (2005) 427-452. We classify canonical singularities of orders using their minimal resolutions (which we define). We describe them explicitly as invariant rings for the action of a finite group on a full matrix algebra over a regular local ring. We also prove that they are Gorenstein, describe their Auslander-Reiten quivers, and note a simple version of the McKay correspondence.
Abstract. In this paper, we study the semi-stable subcategories of the category of representations of a Euclidean quiver, and the possible intersections of these subcategories. Contrary to the Dynkin case, we find out that the intersection of semi-stable subcategories may not be semi-stable. However, only a finite number of exceptions occur, and we give a description of these subcategories. Moreover, one can attach a simplicial fan in Q n to any acyclic quiver Q, and this simplicial fan allows one to completely determine the canonical presentation of any element in Z n . This fan has a nice description in the Dynkin and Euclidean cases: it is described using an arrangement of convex codimension-one subsets of Q n , each such subset being indexed by a real Schur root or a set of quasi-simple objects. This fan also characterizes when two different stability conditions give rise to the same semi-stable subcategory.
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.
Infinitesimal deformations of maximal orders over smooth algebraic surfaces are studied. We classify which maximal orders may admit deformations that are not finite over their centres. The classification is in terms of the algebraic surface that corresponds to the centre of the order and the ramification divisor of the order.
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