Abstract. Let Λ be an artin algebra or, more generally, a locally bounded associative algebra, and S(Λ) the category of all embeddings (A ⊆ B) where B is a finitely generated Λ-module and A is a submodule of B. Then S(Λ) is an exact Krull-Schmidt category which has Auslander-Reiten sequences. In this manuscript we show that the Auslander-Reiten translation in S(Λ) can be computed within mod Λ by using our construction of minimal monomorphisms. If in addition Λ is uniserial, then any indecomposable nonprojective object in S(Λ) is invariant under the sixth power of the Auslander-Reiten translation.
Let be a commutative local uniserial ring with radical factor field k. We consider the category S( ) of embeddings of all possible submodules of finitely generated -modules. In case =Z/ p n , where p is a prime, the problem of classifying the objects in S( ), up to isomorphism, has been posed by Garrett Birkhoff in 1934. In this paper we assume that has Loewy length at least seven. We show that S( ) is controlled k-wild with a single control object I ∈ S( ). It follows that each finite dimensional k-algebra can be realized as a quotient End(X)/End(X) I of the endomorphism ring of some object X ∈ S( ) modulo the ideal End(X) I of all maps which factor through a finite direct sum of copies of I .
We study geometric properties of varieties associated with invariant subspaces of nilpotent operators. There are algebraic groups acting on these varieties. We give dimensions of orbits of these actions. Moreover, a combinatorial characterization of the partial order given by degenerations is described.
We present a sum-product formula for the classical Hall polynomial which is based on tableaux that have been introduced by T. Klein in 1969. In the formula, each summand corresponds to a Klein tableau, while the product is taken over the cardinalities of automorphism groups of short exact sequences which are derived from the tableau. For each such sequence, one can read off from the tableau the summands in an indecomposable decomposition, and the size of their homomorphism and automorphism groups. Klein tableaux are refinements of LittlewoodRichardson tableaux in the sense that each entry ≥ 2 carries a subscript r. We describe module theoretic and categorical properties shared by short exact sequences which have the same symbol r in a given row in their Klein tableau. Moreover, we determine the interval in the Auslander-Reiten quiver in which the indecomposable sequences of p n -bounded groups which carry such a symbol occur.
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