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Register a new userRepresentations of equipped graphs: Auslander-Reiten theory
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Authors
William Crawley-Boevey
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Representations of equipped graphs were introduced by Gelfand and Ponomarev; they are similar to representation of quivers, but one does not need to choose an orientation of the graph. In a previous article we have shown that, as in Kacs Theorem for quivers, the dimension vectors of indecomposable representations are exactly the positive roots for the graph. In this article we begin by surveying that work, and then we go on to discuss Auslander-Reiten theory for equipped graphs, and give examples of Auslander-Reiten quivers.
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Our main theorem classifies the Auslander-Reiten triangles according to properties of the morphisms involved. As a consequence, we are able to compute the mapping cone of an irreducible morphism. We finish by showing a technique for constructing the connecting component of the derived category of any tilted algebra. In particular we obtain a technique for constructing the derived category of any tilted algebra of finite representation type.
Auslander-Reiten conjecture, which says that an Artin algebra does not have any non-projective generator with vanishing self-extensions in all positive degrees, is shown to be invariant under certain singular equivalences induced by adjoint pairs, which occur often in matrix algebras, recollements and change of rings. Accordingly, several reduction methods are established to study this conjecture.
For a finitely generated module $ M $ over a commutative Noetherian ring $R$, we settle the Auslander-Reiten conjecture when at least one of ${rm Hom}_R(M,R)$ and ${rm Hom}_R(M,M)$ has finite injective dimension. A number of new characterizations of Gorenstein local rings are also obtained in terms of vanishing of certain Ext and finite injective dimension of Hom.
Let $mathbf{k}$ be an algebraically closed field, let $Lambda$ be a finite dimensional $mathbf{k}$-algebra, and let $widehat{Lambda}$ be the repetitive algebra of $Lambda$. For the stable category of finitely generated left $widehat{Lambda}$-modules $widehat{Lambda}$-underline{mod}, we show that the irreducible morphisms fall into three canonical forms: (i) all the component morphisms are split monomorphisms; (ii) all of them are split epimorphisms; (iii) there is exactly one irreducible component. We next use this fact in order to describe the shape of the Auslander-Reiten triangles in $widehat{Lambda}$-underline{mod}. We use the fact (and prove) that every Auslander-Reiten triangle in $widehat{Lambda}$-underline{mod} is induced from an Auslander-Reiten sequence of finitely generated left $widehat{Lambda}$-modules.
Given a graph E we define E-algebraic branching systems, show their existence and how they induce representations of the associated Leavitt path algebra. We also give sufficient conditions to guarantee faithfulness of the representations associated to E-algebraic branching systems and to guarantee equivalence of a given representation (or a restriction of it) to a representation arising from an E-algebraic branching system.
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