Otto Kerner

Self-injective algebras of wild tilted type

Abstract We develop the representation theory of self-injective algebras which admit Galois coverings by the repetitive algebras of tilted algebras of wild type. Moreover, we exhibit differences between this class of algebras and wild blocks of group algebras.

Exceptional components of wild hereditary algebras

This paper studies regular components of wild hereditary algebras. It is clear that the existence of extremely many maps in the infinite radical of the module category causes algebras to be wild. Whereas the dimension of the Horn-spaces of maps from indecomposable preprojective modules or to indecomposable preinjective modules can be calculated by linear methods via the Euler-bilinearform ( , ), these methods give only poor information if we pass to regular modules. So it is natural to study as a first step maps between regular modules which are in the same regular component V, and we see in (4.7) that this is a way to get information about the maps between arbitrary regular modules. There is another, less obvious reason for the study of single components: If A and B are connected wild hereditary algebras, then by [9], via tilting modules, there can be constructed bijections between the regular components of the Auslander-Reiten quivers T(A) and T(B). In order to find finally properties of these bijections, we have to study properties of components. Proposition 6.3 points in that direction. If G? is a regular component in T(A), with A wild hereditary, we denote by the quasi-rank rk(%‘) of V the smallest integer N such that rad(X, r”x) # 0 for all n 2 N and all XE %. We call a component V exceptional, if there is an indecomposable (quasi-simple) module XE %? such that Hom(X, YX) # 0 but Hom(X, z r + ‘1) = 0 for some r > 0. In Section 4 we show that there are only finitely many exceptional components in T(A) and that the existence of those components has strict consequences for the algebra A, that is, for its ordinary quiver 9(A). In Section 5 we present the numerical invariants of the most important exceptional components. In the last section we give applications to tilting theory. The applications are chosen under the aspect that the notion of the quasi-rank of a regular component might be a pendant to the period of a tube in the tame case. 184 0021-8693/92

Preprojective components of wild tilted algebras

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Constructing tilting modules

If A is a finite dimensional, connected, hereditary wild k-algebra, k algebraically closed and T a tilting module without preinjective direct summands, then the preprojective componentP of the tilted algebra B=EndA (T) is the preprojective component of a concealed wild factoralgebra C of B. Our first result is, that the growth number ρ(C) of C is always bigger or equal to the growth number ρ(A). Moreover the growth number ρ(C) can be arbitrarily large; more precise: if A has at least 3 simple modules and N is any positive integer, then there exists a natural number n>N such that C is the Kronecker-algebraKn, that is the path-algebra of the quiver (n arrows).

Finiteness of the strong global dimension of radical square zero algebras

We investigate the structure of (infinite dimensional) tilting modules over hereditary artin algebras. For connected algebras of infinite representation type with Grothendieck group of rank n, we prove that for each 0 < i < n - 1, there is an infinite dimensional tilting module T i with exactly i pairwise non-isomorphic indecomposable finite dimensional direct summands. We also show that any stone is a direct summand in a tilting module. In the final section, we give explicit constructions of infinite dimensional tilting modules over iterated one-point extensions.

Regular stones of wild hereditary algebras

The strong global dimension of a finite dimensional algebra A is the maximum of the width of indecomposable bounded differential complexes of finite dimensional projective A-modules. We prove that the strong global dimension of a finite dimensional radical square zero algebra A over an algebraically closed field is finite if and only if A is piecewise hereditary. Moreover, we discuss results concerning the finiteness of the strong global dimension of algebras and the related problem on the density of the push-down functors associated to the canonical Galois coverings of the trivial extensions of algebras by their repetitive algebras.

Auslander–Reiten Components Containing Cones

Abstract If A is a finite-dimensional wild hereditary algebra we study the classof regular components containing stones of quasi-lenght two, where a stone is a brick without self-extensions. We show that there are only finitely many non-sincere components of this type and, provided the algebra has elementary modules with self-extensions, no sincere component with stones of quasi-length two.

On socle-projective categories and tilting modules

Let A be a locally finite Abelian R-category with Auslander–Reiten sequences and with Auslander–Reiten quiver Γ(A). We give a criterion for Auslander–Reiten components to contain a cone and apply this result to various categories.

Stichwörter A-Z

Socle-projective categories seem to be a link between the representation theory of modules over finite dimensional algebras or artin algebras and other topics in representation theory: The the category of lattices over generalised) ackstrom orders can be described in terms of the category of socle-projective modules over a hereditary algebra, see for example the survey paper [16]. If S is a finite poset, the category of S-spaces of the poset S is nothing else than the category of socle-projective modules of the incidence algebra k(S*) of the enlarged poset S U {w}, see [18, 5.11.] Results of Konig on tame and wild generalised Biickstrom orders (in the language of socle-projective modules over hereditary algebras) show that the structure of the socle-projective category over a hereditary algebra A also in the representation-infinite case is extremely parallel to the structure of the torsion-free class F(T) C A - mod of a tilting module: In the case of a tame generalised Backstrom order in [12] he got the s...

Stable components of wild tilted algebras

Sind X und Y Mengen und F ⊂ X × Y eine ° Relation zwischen X und Y, die die beiden folgenden Bedingungen erfullt (I) zu jedem x ∈ X gibt es ein y ∈ Y, so das (x, y) ∈ F gilt. (Linksvollstandigkeit der Relation F) (II) fur alle (x, y), (x’, y’) ∈ F gilt: aus x = x’ folgt y = y’ (Rechtseindeutigkeit der Relation F) so nennt man das Tripel f:=(X, Y, F) eine Abbildung von X nach Y; man nennt die Relation F den Graphen von f, X den Definitionsbereich (Quelle) von f und Y den Wertebereich (Ziel, Bildbereich) von f. Fur jedes x ∈ X bezeichnet man das eindeutig bestimmte y ∈ Y, so das (x, y) ∈ F gilt, meist mit f(x); die Zuordnung, die zu jedem x ∈ X das Element f(x) eindeutig bestimmt, wird durch die Symbole x ↦ f(x) beschrieben und die Zuordnungsvorschrift (Abbildungsvorschrift) von f genannt. Man beachte: eine Abbildung ist allein durch ihre Abbildungsvorschrift nicht vollstandig beschrieben.