Publications (19)8.52 Total impact
 [Show abstract] [Hide abstract]
ABSTRACT: Working over an algebraically closed field k of any characteristic, we determine the matrix factorizations for the — suitably graded — triangle singularities f = x^a+y^b+ z^c of domestic type, that is, we assume that (a, b, c) are integers at least two, satisfying 1/a+ 1/b+ 1/c > 1. Using work by KussinLenzingMeltzer, this is achieved by determining projective covers in the Frobenius category of vector bundles on the weighted projective line of weight type (a, b, c). Equivalently, in a representationtheoretic context, we can work in the mesh category of Z ̃∆ ove rk, where ̃∆ is the extended Dynkin diagram, corresponding to the Dynkin diagram ∆ = [a, b, c]. Our work is related to, but in methods and results different from, the determination of matrix factorizations for the Zgraded simple singularities by KajiuraSaitoTakahashi. In particular, we obtain symmetric matrix factorizations whose entries are scalar multiples of monomials, with scalars taken from {0,±1}.  [Show abstract] [Hide abstract]
ABSTRACT: We give a description of matrix bimodules parametrizing all indecomposable homogeneous Λmodules with a fixed integral slope over a tubular canonical algebra Λ, for all possible integers (Theorem 4.1). An important role in the first step of this description (Theorem 2.4) is played by the translation of the shift functor for coherent sheaves over the associated weighted projective line to the language of Λmodules (Theorem 3.2).  [Show abstract] [Hide abstract]
ABSTRACT: We give an algorithmic description of matrix bimodules parametrizing all indecomposable homogeneous Λmodules with a fixed slope q over a tubular canonical algebra Λ, for all possible slopes q (Main Theorem 3.3). A crucial role in this description is played by universal extensions of bimodules and their nice properties (Theorems 3.1 and 3.2).  [Show abstract] [Hide abstract]
ABSTRACT: We show a surprising link between singularity theory and the invariant subspace problem of nilpotent operators as recently studied by C. M. Ringel and M. Schmidmeier, a problem with a longstanding history going back to G. Birkhoff. The link is established via weighted projective lines and (stable) categories of vector bundles on those. The setup yields a new approach to attack the subspace problem. In particular, we deduce the main results of Ringel and Schmidmeier for nilpotency degree p from properties of the category of vector bundles on the weighted projective line of weight type (2,3,p), obtained by Serre construction from the triangle singularity x^2+y^3+z^p. For p=6 the RingelSchmidmeier classification is thus covered by the classification of vector bundles for tubular type (2,3,6), and then is closely related to Atiyah's classification of vector bundles on a smooth elliptic curve. Returning to the general case, we establish that the stable categories associated to vector bundles or invariant subspaces of nilpotent operators may be naturally identified as triangulated categories. They satisfy Serre duality and also have tilting objects whose endomorphism rings play a role in singularity theory. In fact, we thus obtain a whole sequence of triangulated (fractional) CalabiYau categories, indexed by p, which naturally form an ADEchain.  [Show abstract] [Hide abstract]
ABSTRACT: 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 kalgebras over an arbitrary field.  [Show abstract] [Hide abstract]
ABSTRACT: We investigate the triangle singularity $f=x^a+y^b+z^c$, or $S=k[x,y,z]/(f)$, attached to a weighted projective line $X$ given by the weight triple $(a,b,c)$. We investigate the stable category of vector bundles on $X$ obtained from the vector bundles by factoring out all line bundles. This category is triangulated and has Serre duality. It is, moreover, naturally equivalent to the stable category of graded maximal CohenMacaulay modules over $S$ (or matrix factorizations of $f$), and then by results of Buchweitz and Orlov to the graded singularity category of $f$. We show that this category is fractional CalabiYau with a CYdimension that is a function of the Euler characteristic of $X$. We show the existence of a tilting object which has the shape of an $(a1)(b1)(c1)$cuboid. Particular attention is given to the weight types $(2,a,b)$, yielding an explanation of HappelSeidel symmetry for a class of important Nakayama algebras. In particular, the weight sequence $(2,3,p)$ corresponds to an ADEchain, the $E_n$chain, extrapolating the exceptional Dynkin cases $E_6$, $E_7$ and $E_8$ to a whole sequence of triangulated categories.  [Show abstract] [Hide abstract]
ABSTRACT: Let Λ be a tubular canonical algebra of quiver type. We describe an algorithm, which for numerical data computes all regular exceptional Λmodules, or more generally all indecomposable modules in exceptional tubes. The input for the algorithm is a quadruple consisting of the slope, the number of the tube, the quasisocle and the quasilength, the output are explicit matrices for the module with the data above.  [Show abstract] [Hide abstract]
ABSTRACT: We study the category C(X, Y) generated by an exceptional pair (X, Y) in a hereditary category H. If r = dimkHom(X, Y) ≥ 1 we show that there are exactly 3 possible types for C(X, Y), all derived equivalent to the category of finite dimensional modules mod(Hr) over the rKronecker algebra Hr. In general C(X, Y) will not be equivalent to a module category. More specifically, if H is the category of coherent sheaves over a weighted projective line , then C(X, Y) is equivalent to the category of coherent sheaves on the projective line 1 or to mod(Hr) and, if is wild, then every r ≥ 1 can occur in this way.  [Show abstract] [Hide abstract]
ABSTRACT: We describe explicitly all indecomposable modules of rank 6 over a domestic canonical algebra of quiver type over a field k of arbitrary characteristic. Together with the results given in [5] this yields an explicit description of all preprojective and preinjective indecomposable modules (and of all indecomposable modules if k is algebraically closed) for a domestic canonical algebra of quiver type. In particular for those algebras each preprojective and each preinjective indecomposable module can be represented by matrices whose coefficients are 0 and 1.  [Show abstract] [Hide abstract]
ABSTRACT: We show that–up to precisely one–each exceptional module over a domestic canonical algebra of quiver type over a field k can be represented by matrices whose entries are just 0 and 1. In the case we calculate the matrices of these representations explicitly.  [Show abstract] [Hide abstract]
ABSTRACT: A hyperelliptic algebra Λ is a canonical algebra in the sense of Ringel of type (2,2,⋯,2). Using universal extensions we give an explicit description of all but finitely many omnipresent exceptional modules of minimal rank over those algebras. All these modules are exhibited by matrices involving as coefficients the parameters appearing in the definition of Λ as a bound quiver.  [Show abstract] [Hide abstract]
ABSTRACT: Let Λ be a tubular canonical algebra of quiver type over a field. We show that each exceptional Λmodule 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.  [Show abstract] [Hide abstract]
ABSTRACT: We discuss the problem of classification of indecomposable representations for extended Dynkin quivers of type Ẽ 8, with a fixed orientation. We describe a method for an explicit determination of all indecomposable preprojective and preinjective representations for those quivers over an arbitrary field and for all indecomposable representations in case the field is algebraically closed. This method uses tilting theory and results about indecomposable modules for a canonical algebra of type (5; 3; 2) obtained by Kussin and Meltzer and by Komoda and Meltzer. Using these techniques we calculate all series of preprojective indecomposable representations of rank 6. The same method has been used by Kussin and Meltzer to determine indecomposable representations for extended Dynkin quivers of type D̃ n and Ẽ 6. Moreover, our techniques can be applied to calculate indecomposable representations of extended Dynkin quivers of type Ẽ 7. The indecomposable representations for extended Dynkin quivers of type Ã n are known.  [Show abstract] [Hide abstract]
ABSTRACT: We show that the operation of the braid group on the set of complete ex ceptional sequences in the category of coherent sheaves on an exceptional curve X over afi eldk is transitive. As a consequence the list of endomorphism skewfields of the inde composable direct summands of a tilting complex is a derived invariant. Furthermore, we apply the result in order to establish a bijection (which is compatible with the Ktheory) between the sets of translation classes of exceptional objects in the derived categories of two derivedcanonical algebras with the same Cartan matrix, but which are defined over possibly distinct fields.  [Show abstract] [Hide abstract]
ABSTRACT: We investigate complete exceptional sequences E in the derived category of finitedimensional modules over a canonical algebra, equivalently in the derived category of coherent sheaves on a weighted projective line, and the associated Cartan matrices C(E). As a consequence of the transitivity of the braid group action on such sequences we show that a given Cartan matrix has at most finitely many realizations by an exceptional sequence E, up to an automorphism and a multitranslation . Moreover, we determine a bound on the number of such realizations. Our results imply that a derived canonical algebra A is determined by its Cartan matrix up to isomorphism if and only if the Hochschild cohomology of A vanishes in nonzero degree, a condition satisfied if A is representationfinite.  [Show abstract] [Hide abstract]
ABSTRACT: We show that up to a translation each automorphism of the derived category DX of coherent sheaves on a weighted projective line X, equivalently of the derived category DA of finite dimensional modules over a derived canonical algebra A, is composed of tubular mutations and automorphisms of X. In the case of genus one this implies that the automorphism group is a semidirect product of the braid group on three strands by a finite group. Moreover we prove that most automorphisms lift from the Grothendieck group to the derived category. 


Publication Stats
218  Citations  
8.52  Total Impact Points  
Top Journals
Institutions

20062014

University of Szczecin
 Institute of Mathematics
Stettin, West Pomeranian Voivodeship, Poland


2002

Universität Paderborn
 Department of Mathematics
Paderborn, North RhineWestphalia, Germany


2000

Technische Universität Chemnitz
 Department of Mathematics
Chemnitz, Saxony, Germany
