Yuriy Drozd

Vector Bundles and Torsion Free Sheaves on Degenerations of Elliptic Curves

In this paper we give a survey about the classification of vector bundles and torsion free sheaves on degenerations of elliptic curves. Coherent sheaves on singular curves of arithmetic genus one can be studied using the technique of matrix problems or via Fourier-Mukai transforms, both methods are discussed here. Moreover, we include new proofs of some classical results about vector bundles on elliptic curves.

TILTING ON NON-COMMUTATIVE RATIONAL PROJECTIVE CURVES

In this article we introduce a new class of non-commutative projective curves and show that in certain cases the derived category of coherent sheaves on them has a tilting complex. In particular, we prove that the right bounded derived category of coherent sheaves on a reduced rational projective curve with only nodes and cusps as singularities, can be fully faithfully embedded into the right bounded derived category of the finite dimensional representations of a certain finite dimensional algebra of global dimension two. As an application of our approach we show that the dimension of the bounded derived category of coherent sheaves on a rational projective curve with only nodal or cuspidal singularities is at most two. In the case of the Kodaira cycles of projective lines, the corresponding tilted algebras belong to a well-known class of gentle algebras. We work out in details the tilting equivalence in the case of the Weierstrass nodal curve zy2 = x3 + x2z.

Stable vector bundles over cuspidal cubics

We give a complete classification of stable vector bundles over a cuspidal cubic and calculate their cohomologies. The technique of matrix problems is used, similar to [2, 3].

Maximal Cohen–Macaulay Modules Over Non–Isolated Surface Singularities and Matrix Problems

In this article we develop a new method to deal with maximal Cohen-Macaulay modules over non-isolated surface singularities. In particular, we give a negative answer on an old question of Schreyer about surface singularities with only countably many indecomposable maximal Cohen-Macaulay modules. Next, we prove that the degenerate cusp singularities have tame Cohen-Macaulay representation type. Our approach is illustrated on the case of

Maximal Cohen–Macaulay modules over surface singularities

k\llbracket x,y,z\rrbracket/(xyz)

MATRIX PROBLEMS AND STABLE HOMOTOPY TYPES OF POLYHEDRA

as well as several other rings. This study of maximal Cohen-Macaulay modules over non-isolated singularities leads to a new class of problems of linear algebra, which we call representations of decorated bunches of chains. We prove that these matrix problems have tame representation type and describe the underlying canonical forms.

Simple vector bundles on plane degenerations of an elliptic curve

This is a survey article about properties of Cohen-Macaulay modules over surface singularities. We discuss properties of the Macaulayfication functor, reflexive modules over simple, quotient and minimally elliptic singularities, geometric and algebraic McKay Corre- spondence. Finally, we describe matrix factorizations corresponding to indecomposable Cohen- Macaulay modules over the non-isolated singularities A1 and D1.For a finite dimensional algebra A of finite global dimension the bounded derived category of finite dimensional A-modules admits Auslander- Reiten triangles such that the Auslander-Reiten translation τ is an equivalence. On the level of the Grothendieck group τ induces the Coxeter transformation �A. More generally this extends to a homologically finite triangulated category T admitting Serre duality. In both cases the Coxeter polynomial, that is, the characteristic polynomial of the Coxeter transformation yields an important homological invariant of A or T. Spectral analysis is the study of this interplay, it often reveals unexpected links between apparently different subjects. This paper gives a summary on spectral techniques and studies the links to singularity theory. In particular, it offers a contribution to the categorifica- tion of the Milnor lattice through triangulated categories which are naturally attached to a weighted projective line.We review the definition of a Calabi-Yau triangulated category and survey examples coming from the representation theory of quivers and finite-dimensional algebras. Our main motivation comes from the links between quiver representations and Fomin-Zelevinsky’s cluster algebras. Mathematics Subject Classification (2000). Primary 18E30; Secondary 16D90, 18G10.The singular cochain complex of a topological space is a classical object. It is a Differential Graded algebra which has been studied intensively with a range of methods, not least within rational homotopy theory. More recently, the tools of Auslander-Reiten theory have also been applied to the singular cochain complex. One of the highlights is that by these methods, each Poincare duality space gives rise to a Calabi-Yau category. This paper is a review of the theory.An introduction to moduli spaces of representations of quivers is given, and results on their global geometric properties are surveyed. In particular, the geometric approach to the problem of classification of quiver representations is motivated, and the construction of moduli spaces is reviewed. Topological, arithmetic and algebraic methods for the study of moduli spaces are discussed.We use the representation theory of preprojective algebras to construct and study certain cluster algebras related to semisimple algebraic groups.We recall several results in Auslander-Reiten theory for finite-dimensional algebras over fields and orders over complete local rings. Then we introduce

On Cubic Functors

n

On genera of polyhedra

-cluster tilting subcategories and higher theory of almost split sequences and Auslander algebras there. Several examples are explained.

On Hom-spaces of tame algebras

This is a survey of the results on stable homotopy types of polyhedra of small dimensions, mainly obtained by H.-J. Baues and the author [3, 5, 6]. The proofs are based on the technique of matrix problems (bimodule categories).