Submodule categories of wild representation type
By Claus Michael Ringel, Markus Schmidmeier. 2004
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Abstract Let Λ be a commutative local uniserial ring with radical factor field k . We consider the category S ( Λ ) of embeddings of all possible submodules of finitely generated Λ -modules. In case Λ = Z / 〈 p n 〉 , where p is a prime, the problem of classifying the objects in S ( Λ ) , up to isomorphism, has been posed by Garrett Birkhoff in 1934. In this paper we assume that Λ has Loewy length at least seven. We show that S ( Λ ) is controlled k -wild with a single control object I ∈ S ( Λ ) . 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 ( Λ ) modulo the ideal End ( X ) I of all maps which factor through a finite direct sum of copies of I .
View Paper: https://www.semanticscholar.org/paper/e5a4dd988298cc5490f116d8d9ac02e13e3759bb
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Citations in the corpus
- Representation types of the category of subprojective representations of a finite poset over K(t)/(t m ) and a solution of a Birkhoff type problem (2007, background): “The only solution we know up to now follows from the results obtained recently by Ringel and Schmidmeier in [27,28].”
- Invariant subspaces of nilpotent linear operators, I (2006, methodology): “ociated with 8+1+1 = 10 = 2(6−1) points. 7. Three Remarks about Wild Submodule Categories. In the cases where n≥7, the categories S(en) and S(n) have wild representation type. In fact, it is shown in [10] that for any commutative local uniserial ring Λ of Loewy length 7 the category S(Λ) is controll”