Algebraic Modules and the Auslander–Reiten Quiver
نویسنده
چکیده
Recall that an algebraic module is aKG-module that satisfies a polynomial with integer coefficients, with addition and multiplication given by direct sum and tensor product. In this article we prove that non-periodic algebraic modules are very rare, and that if the complexity of an algebraic module is at least 3, then it is the only algebraic module on its component of the (stable) Auslander–Reiten quiver. We include a strong conjecture on the relationship between periodicity and algebraicity.
منابع مشابه
Shapes of Connected Components of the Auslansder-Reiten Quivers of Artin Algebras
The aim of these notes is to report some new developments on the problem of describing all possible shapes of the connected components of the Auslander-Reiten quiver ΓA of an artin algebra A. The problem is interesting since the shapes of these components carry some important information of the module category of A. For instance the algebra A is hereditary if and only if ΓA has a connected comp...
متن کاملA pr 2 00 3 THE AUSLANDER - REITEN QUIVER OF A POINCARÉ DUALITY SPACE
In a previous paper, Auslander-Reiten triangles and quivers were introduced into algebraic topology. This paper shows that over a Poincaré duality space, each component of the Auslander-Reiten quiver is isomorphic to ZA∞.
متن کاملAuslander-reiten Components Containing Modules with Bounded Betti Numbers
Let R be a connected selfinjective Artin algebra, and M an indecomposable nonprojective R-module with bounded Betti numbers lying in a regular component of the Auslander-Reiten quiver of R. We prove that the Auslander-Reiten sequence ending at M has at most two indecomposable summands in the middle term. Furthermore we show that the component of the Auslander-Reiten quiver containing M is eithe...
متن کاملAuslander-reiten Triangles and Quivers over Topological Spaces
In this paper, Auslander-Reiten triangles are introduced into algebraic topology, and it is proved that their existence characterizes Poincaré duality spaces. Invariants in the form of quivers are also introduced, and Auslander-Reiten triangles and quivers over spheres are computed. The quiver over the d-dimensional sphere turns out to consist of d − 1 components, each isomorphic to ZA∞. So qui...
متن کاملAuslander-Reiten theory in a Krull-Schmidt category
We first introduce the notion of an Auslander-Reiten sequence in a Krull-Schmidt category. This unifies the notion of an almost split sequence in an abelian category and that of an Auslander-Reiten triangle in a triangulated category. We then define the Auslander-Reiten quiver of a Krull-Schmidt category and describe the shapes of its semi-stable components. The main result generalizes those fo...
متن کامل