[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 Kussin-Lenzing-Meltzer, this is achieved by determining projective covers in the Frobenius category of vec-tor bundles on the weighted projective line of weight type (a, b, c). Equivalently, in a representation-theoretic 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 Z-graded simple singularities by Kajiura-Saito-Takahashi. In particular, we obtain symmetric matrix factorizations whose entries are scalar mul-tiples of monomials, with scalars taken from {0,±1}.
[Show abstract][Hide abstract] ABSTRACT: We give a description of matrix bimodules parametrizing all indecomposable homogeneous Λ-modules 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 Λ-modules (Theorem 3.2).
Algebras and Representation Theory 02/2014; 17(1). DOI:10.1007/s10468-012-9386-7 · 0.54 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We give an algorithmic description of matrix bimodules parametrizing all indecomposable homogeneous Λ-modules 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).
Algebras and Representation Theory 02/2014; 17(1). DOI:10.1007/s10468-013-9430-2 · 0.54 Impact Factor
[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 Ringel-Schmidmeier 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) Calabi-Yau
categories, indexed by p, which naturally form an ADE-chain.
Journal für die reine und angewandte Mathematik (Crelles Journal) 12/2013; 685:33-71. DOI:10.1515/crelle-2012-0014 · 1.43 Impact Factor
[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 k-algebras over an arbitrary field.
Communications in Algebra 05/2013; 41(6). DOI:10.1080/00927872.2011.642042 · 0.39 Impact Factor
[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 Cohen-Macaulay 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 Calabi-Yau with a CY-dimension that is a function of the Euler characteristic of $X$. We show the existence of a tilting object which has the shape of an $(a-1)(b-1)(c-1)$-cuboid. Particular attention is given to the weight types $(2,a,b)$, yielding an explanation of Happel-Seidel symmetry for a class of important Nakayama algebras. In particular, the weight sequence $(2,3,p)$ corresponds to an ADE-chain, the $E_n$-chain, extrapolating the exceptional Dynkin cases $E_6$, $E_7$ and $E_8$ to a whole sequence of triangulated categories.
Advances in Mathematics 04/2013; 237:194-251. DOI:10.1016/j.aim.2013.01.006 · 1.29 Impact Factor
[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 Λ-modules, 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
quasi-socle and the quasi-length, the output are explicit matrices
for the module with the data above.
Journal of Algebra 09/2010; 323(10):2710-2734. DOI:10.1016/j.jalgebra.2009.12.027 · 0.60 Impact Factor
[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 r-Kronecker 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.
Communications in Algebra 07/2009; 37(8):2547-2556. DOI:10.1080/00927870802174140 · 0.39 Impact Factor
[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 de-scription of all preprojective and preinjective indecomposable modules (and of all indecomposable modules if k is algebraically closed) for a domestic canon-ical 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.
Journal of Pure and Applied Algebra 11/2007; 211(2-211):471-483. DOI:10.1016/j.jpaa.2007.02.001 · 0.47 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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.
Algebras and Representation Theory 09/2007; 10(5):481-496. DOI:10.1007/s10468-007-9067-0 · 0.54 Impact Factor
[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 skew-fields 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 K-theory) between the sets of translation classes of exceptional objects in the derived categories of two derived-canonical algebras with the same Cartan matrix, but which are defined over possibly distinct fields.
Archiv der Mathematik 11/2002; 79(5):335-344. DOI:10.1007/PL00012455 · 0.39 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We investigate complete exceptional sequences E in the derived category of finite-dimensional 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 multi-translation . 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 representation-finite.
Algebras and Representation Theory 01/2002; 5(2):201-209. DOI:10.1023/A:1015646412663 · 0.54 Impact Factor
[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, equiv-alently 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 semi-direct 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.
Communications in Algebra 01/2000; 28(4):1685-1700. DOI:10.1080/00927870008826922 · 0.39 Impact Factor
Representation theory of algebras (Cocoyoc, 1994); CMS Conf. Proc. 18, Edited by Raymundo Bautista, Roberto Martínez-Villa and José Antonio de La Peña, 01/1996: chapter Tilting sheaves and concealed-canonical algebras: pages 455-473; American Mathematical Society, Providence, RI..
Representations of algebras (Ottawa, ON, 1992), Edited by Vlastimil Dlab and Helmut Lenzing, 07/1993: chapter Sheaves on a weighted projective line of genus one and representations of a tubular algebra: pages 313-337; American Mathematical Society, CMS Conf. Proc., 14..