The Auslander-Reiten translation in submodule categories

By Claus Michael Ringel, Markus Schmidmeier. 2005

104 citations recorded by the source corpus.

Let A be an artin algebra or, more generally, a locally bounded associative algebra, and S(Λ) the category of all embeddings (A C B) where B is a finitely generated A-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 A by using our construction of minimal monomorphisms. If in addition A is uniserial, then any indecomposable nonprojective object in S(Λ) is invariant under the sixth power of the Auslander-Reiten translation.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ffd790f2abda6001cbe2745664c0dd2552cd3930

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Citations in the corpus

On algebras of finite Cohen-Macaulay type (2011, methodology): “Since Subpmod-Λq is contravariantly finite in mod-T2pΛq, see [65], the first three conditions are equivalent by Theorem 3.”
Nilpotent operators and weighted projective lines (2010, background): “We advise the reader in this context to check the paper [21] for the ubiquitous appearance of the number 6.”
Invariant subspaces of nilpotent linear operators, I (2006, methodology): “[ 11 ], Proposition 4.1, the isomorphism class of Mimo(˜ a) in S(n) does not depend on the choice for ˜ nor on the choice for e.”
Auslander–Reiten translations in monomorphism categories (2011, methodology): “idmeier’s work on S2(A) to Sn(A). As in the case of n= 2, Sn(A) has Auslander-Reiten sequences, and τS of Sn(A) can be formulated in the same form as above: these can be achieved by using the idea in [RS2]. For selﬁnjective algebras, Sections 3 and 4 of this paper contain new considerations. In orde”
Monic representations and Gorenstein-projective modules (2011, background): “r a monic Tn(A)-module, exactly means that each φi is injective, 1 ≤i ≤n −1. This kind of monic Tn(A)-modules have arisen from diﬀerent questions and in diﬀerent terminologies, for examples in [RS1], [RS2], [S], [LZ], [KLM], [C], [Z], [IKM]. 2.5. There is another similar but diﬀerent notion. Let A =”
The Stable Monomorphism Category of a Frobenius category (2009, background): “In this paper, we will call the triangulated category S(p̃) the stable category of Ringel-Schmidmeier.”
Monomorphism categories, cotilting theory, and Gorenstein-projective modules (2011, methodology): “ation types of C(n,k[x]/hxti), have been given by D. Simson [S] (see also [SW]). C. M. Ringel and M. Schmidmeier ([RS1] - [RS3]) have intensively studied the monomorphism category S2(A). According to [RS2], Sn(A) is a functorially ﬁnite subcategory in Tn(A)-mod; and hence Sn(A) has Auslander-Reiten ”