Helmut Lenzing

Perpendicular Categories with Applications to Representations and Sheaves

This paper is concerned with the omnipresence of the formation of the subcategories right (left) perpendicular to a subcategory of objects in an abelian category. We encounter these subcategories in various contexts: • • the formation of quotient categories with respect to localizing subcategories (cf. Section 2); • • the deletion of vertices and shrinking of arrows (see [37]) in the representation theory of finite dimensional algebras (cf. Section 5); • • the comparison of the representation theories of different extended Dynkin quivers (cf. Section 10); • • the theory of tilting (cf. Sections 4 and 6); • • the study of homological epimorphisms of rings (cf. Section 4); • • the passage from graded modules to coherent sheaves on a possibly weighted projective variety or scheme (cf. Section 7 and [21]); • • the study of (maximal) Cohen-Macaulay modules over surface singularities (cf. Sections); • • the comparison of weighted projective lines for different weight sequences (cf. Section 9); • • the formation of affine and local algebras attached to path algebras of extended Dynkin quivers, canonical algebras, and weighted projective lines (cf. Section 11 and [21] and the concept of universal localization in [40]). Formation of the perpendicular category has many aspects in common with localization and allows one to dispose of localization techniques in situations not accessible to any of the classical concepts of localization. This applies in particular to applications in the domain of finite dimensional algebras and their representations. Several applications of the methods presented in this paper are already in existence, partly published, or appearing in print in the near future (see, for instance, [40, 39, 4, 45, 26, 49, 46]) and have show the versatility of the notion of a perpendicular category. It seems that (right) perpendicular categories first appeared—as the subcategories of so-called closed objects—in the process of the formation of the quotient cateogory of an abelian category with respect to a localizing Serre subcategory (see [18, 47, 34]). Another natural occurrence is encountered in Commutative Algebra, forming the possibly infinitely generated modules of depth ⩾2 (cf. Section 7). The concept and some of the central applications were first presented in a talk given by the first author at the Honnef meeting in January 1985. We also note that the perfectly matching nomination “perpendicular category” was coined by A. Schofield, who discovered independently the usefulness of this concept in dealing with hereditary algebras (see [39], cf. also Section 7). The authors further acknowledge the support of the Deutsche Forschungsgemeinschaft (SPP “Darstellungstheorie von endlichen Gruppen und endlichdimensionalen Algebren”). Throughout this paper rings are associative with unit and modules are unitary right modules. Mod(R) (respectively mod(R)) denotes the category of all (respectively all finitely presented) right R-modules.

CONCEALED-CANONICAL ALGEBRAS AND SEPARATING TUBULAR FAMILIES

We characterise those concealed-canonical algebras which arise as endomorphism rings of tilting modules, all of whose indecomposable summands have strictly positive rank, as those artin algebras whose module categories have a separating exact subcategory (that is, a separating tubular family of standard tubes). This paper develops further the technique of shift automorphisms which arises from the tubular structure. It is related to the characterisation of hereditary noetherian categories with a tilting object as the categories of coherent sheaves on a weighted projective line.

Nilpotentoperators and weighted projective lines

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 Atiyahs 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.

The cluster category of a canonical algebra

We study the cluster category of a canonical algebra A in terms of the hereditary category of coherent sheaves over the corresponding weighted projective line X. As an application we determine the automorphism group of the cluster category and show that the cluster-tilting objects form a cluster structure in the sense of Buan, Iyama, Reiten and Scott. The tilting graph of the sheaf category always coincides with the tilting or exchange graph of the cluster category. We show that this graph is connected if the Euler characteristic of X is non-negative, or equivalently, if A is of tame (domestic or tubular) representation type.

The automorphism group of the derived category for a weighted projective line

We show that up to a translation each automorphism of the derived category D b X of coherent sheaves on a weighted projective line X, equiv-alently of the derived category D b A 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.

One-point extensions and derived equivalence

Abstract Work of the first author with de la Pena [M. Barot, J.A. de la Pena, Proc. Amer. Math. Soc. 127 (1999) 647–655], concerned with the class of algebras derived equivalent to a tubular algebra, raised the question whether a derived equivalence between two algebras can be extended to one-point extensions. The present paper yields a positive answer.

Hereditary noetherian categories with a tilting complex

We are characterizing the categories of coherent sheaves on a weighted projective line as the small hereditary noetherian categories without projectives and admitting a tilting complex. The paper is related to recent work with de la Pena (Math. Z., to appear) characterizing finite dimensional algebras with a sincere separating tubular family, and further gives a partial answer to a question of Happel, Reiten, Smalo (Mem. Amer. Math. Soc. 120 (1996), no. 575) regarding the characterization of hereditary categories with a tilting object. A characterization of weighted projective lines We are going to characterize the categories coh(X) of coherent sheaves on a weighted projective line X [3, 4]. Theorem 1. Let k be an algebraically closed field. For a small connected abelian k-category H with finite dimensional morphism and extension spaces the following assertions are equivalent: (i) H is equivalent to the category of coherent sheaves on a weighted projective line. (ii) Each object of H is noetherian. H is hereditary, has no non-zero projectives, and admits a tilting complex. (iii) Each object of H is noetherian, moreover (a) There exists an equivalence τ : H → H (Auslander-Reiten translation) such that Serre duality DExt(A,B) ∼= Hom(B, τA) holds functorially in A,B ∈ H. (b) The Grothendieck group K0(H) is finitely generated free, and the Euler form 〈−,−〉 : K0(H) × K0(H) → Z given on classes of objects of H by 〈[X ], [Y ]〉 = dimk Hom(X,Y )−dimk Ext(X,Y ) is non-degenerate of determinant ±1. (c) H has an object without self-extensions which is not of finite length. Received by the editors February 9, 1995. 1991 Mathematics Subject Classification. Primary 14G14, 16G20; Secondary 18F20, 18E30.

Wild Canonical Algebras and Rings of Automorphic Forms

This paper aims to show that the modern representation theory of finite dimensional algebras, usually judged to be of fairly recent origin, has strong roots within classical mathematics. We do justify this claim by exploring the links between tame hereditary algebras ∑ and simple surface singularities (corresponding to the invariant theory of binary polyhedral groups) wild canonical algebras Λ and automorphic forms, respectively.

A homological approach to representations of algebras II: tame hereditary algebras

Abstract We determine the pure global dimension of finite dimensional hereditary or radical-squared zero algebras over algebraically closed fields. The results are applied to algebras of dimension four and to the incidence algebras of critical ordered sets studied by Loupias. We further prove that the path algebra of an oriented cycle shares with Dedekind domains the Kulikov property (submodules of pure-projective modules are pure-projective).

Homological Dimension and representation type of algebras under base field extension.

We prove that for finite-dimensional associative algebras the finitistic global dimension in the sense of H. Bass and also the self-injective dimension are preserved under arbitrary extensions of the base field. Further, the global dimension, and as a consequence also finite representation type, are preserved under base field extensions which are separable in the sense of S. MacLane. The results are derived with the aid of ultraproduct-techniques and are applied to the model theory of finite dimensional algebras.