Exceptional Modules for Tubular Canonical Algebras

Let Λ be a tubular canonical algebra of quiver type over a field. We show that each exceptional Λ-module can be exhibited by matrices involving as coefficients 0, 1 and –1 if Λ is of type (3,3,3), (2,4,4) or (2,3,6) and by matrices involving as coefficients 0, 1, –1, λ, –λ and λ–1 if Λ is of type (2,2,2,2) and defined by a parameter λ. Presented by Claus M. Ringel.     

Schofield Induction for Sheaves on Weighted Projective Lines

We show that each exceptional vector bundle on a weighted projective line in the sense of Geigle and Lenzing can be obtained by Schofield induction from exceptional sheaves of rank one and zero. This relates to results of Ringel concerning modules over finite dimensional k-algebras over an arbitrary field.     

Prüfer Sheaves and Generic Sheaves Over the Weighted Projective Lines and Elliptic Curves

In the present paper, we introduce the concepts of Prüfer sheaves and adic sheaves over a weighted projective line of genus one or an elliptic curve, show that Prüfer sheaves and adic sheaves can characterize the category of coherent sheaves. Moreover, we describe the relationship between Prüfer sheaves and generic sheaves, and provide two methods to construct generic sheaves by using coherent sheaves and Prüfer sheaves.     

Generalized parabolic sheaves on an integral projective curve

We extend the notion of a parabolic vector bundle on a smooth curve to define generalized parabolic sheaves (GPS) on any integral projective curve X. We construct the moduli spacesM(X) of GPS of certain type onX. IfX is obtained by blowing up finitely many nodes inY then we show that there is a surjective birational morphism from M(X) to M (Y). In particular, we get partial desingularisations of the moduli of torsion-free sheaves on a nodal curveY.     

Weighted Projective Lines as Fine Moduli Spaces of Quiver Representations

We describe weighted projective lines in the sense of Geigle and Lenzing by a moduli problem on the canonical algebra of Ringel. We then go on to study generators of the derived categories of coherent sheaves on the total spaces of their canonical bundles, and show that they are rarely tilting. We also give a moduli construction for these total spaces for weighted projective lines with three orbifold points.     

Coherent Sheaves on P1: their Families and their Deformations Related to a Behavioral Approach to Singular Systems

Recently, Lomadze, Ravi, Rosenthal and Schumacher gave a natural and geometrically meaningful one-to-one correspondence between abstract linar behaviors and coherent sheaves on P ¹. Motivated by their result, here we give a complete picture of the deformation theory of coherent sheaves on P ¹. We use our results to study deformations over the field R obtaining a connectedness result for the real locus of the algebraic parameter spaces.     

On the derived category of the classical Godeaux surface

We construct an exceptional sequence of length 11 on the classical Godeaux surface X$X$ which is the Z/5Z$\mathbb{Z}/5\mathbb{Z}$-quotient of the Fermat quintic surface in P³${\mathbb{P}}^3$. This is the maximal possible length of such a sequence on this surface which has Grothendieck group Z¹¹⊕Z/5Z${\mathbb{Z}}^{11}\oplus \mathbb{Z}/5\mathbb{Z}$. In particular, the result answers Kuznetsov’s Nonvanishing Conjecture, which concerns Hochschild homology of an admissible subcategory, in the negative. The sequence carries a symmetry when interpreted in terms of the root lattice of the simple Lie algebra of type _E8$E_8$. We also produce explicit nonzero objects in the (right) orthogonal to the exceptional sequence.     

Configurations in abelian categories. II. Ringel–Hall algebras

This is the second in a series on configurations in an abelian category . Given a finite poset (I,), an (I,)-configuration (σ,ι,π) is a finite collection of objects σ(J) and morphisms ι(J,K) or in satisfying some axioms, where J,KI. Configurations describe how an object X in decomposes into subobjects.The first paper defined configurations and studied moduli spaces of (I,)-configurations in , using the theory of Artin stacks. It showed well-behaved moduli stacks of objects and configurations in exist when is the abelian category coh(P) of coherent sheaves on a projective scheme P, or mod- of representations of a quiver Q.Write for the vector space of -valued constructible functions on the stack . Motivated by the idea of Ringel–Hall algebras, we define an associative multiplication * on using pushforwards and pullbacks along 1-morphisms between configuration moduli stacks, so that is a -algebra. We also study representations of , the Lie subalgebra of functions supported on indecomposables, and other algebraic structures on .Then we generalize all these ideas to stack functions , a universal generalization of constructible functions, containing more information. When Extⁱ(X,Y)=0 for all and i>1, or when for P a Calabi–Yau 3-fold, we construct (Lie) algebra morphisms from stack algebras to explicit algebras, which will be important in the sequels on invariants counting τ-semistable objects in .     

Exceptional collections of line bundles on the Beauville surface

We construct quasi-phantom admissible subcategories in the derived category of coherent sheaves on the Beauville surface S$S$. These quasi-phantoms subcategories appear as right orthogonals to subcategories generated by exceptional collections of maximal possible length 4 on S$S$. We prove that there are exactly 6 exceptional collections consisting of line bundles (up to a twist) and these collections are spires of two helices.     

Explicit Models for Perverse Sheaves,II

In this paper we show that any p-perverse sheaf on an arbitrary stratified topological space (p is a perversity function) is functorially determined by a system of usual sheaves on the open sets U _r (r≥0) and certain gluing data, where U _r is the union of strata of perversity ≤r. Both authors are partially supported by BFM2001-3207 and MTM2004-07203-C02-01 and FEDER.     

Double Hall algebras and derived equivalences

We show that the reduced Drinfeld double of the Ringel-Hall algebra of a hereditary category is invariant under derived equivalences. By associating an explicit isomorphism to a given derived equivalence, we also extend the results of Burban and Schiffmann [5] and [006], Sevenhant and Van den Bergh (1999) [25], and Xiao and Yang (2001) [31].     

Generalized logic and the representation of rings

Following the progression towards weaker logics, a number of authors have considered the notion of a sheaf over a quantale or, equivalently, a quantale valued set. In this paper, we use ideas from enriched category theory to motivate the definition of a quantic sheaf. Given a localic subquantale of Q, a quantic sheaf over Q gives a sheaf in the usual sense. As an application, we derive a series of sheaf representations for commutative rings including the familiar Pierce representation.     

Abelianness of the “missing part” from a sheaf category to a module category

This paper investigates the structure of the"missing part"from the category of coherent sheaves over a weighted projective line of weight type(2,2,n)to the category of finitely generated right modules on the associated canonical algebra.By constructing a t-structure in the stable category of the vector bundle category,we show that the"missing part"is equivalent to the heart of the t-structure,hence it is abelian.Moreover,it is equivalent to the category of finitely generated modules on the path algebra of type An-1.