Classification of discrete derived categories

Bobiński, Grzegorz; Geiß, Christof; Skowroński, Andrzej

doi:10.2478/bf02475948

Cited by 61 publications

(106 citation statements)

References 16 publications

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“…An object of K b (projA) is indecomposable if and only it is isomorphic to one of the following complexes [11]. The objects in the second and third families are exceptional with endomorphism algebra k, the projective P 1 is exceptional with endomorphism algebra k[x]/x 2 , while the other objects in the first and fourth families are not exceptional.…”

Section: Proposition 33 Suppose That K Is a Field And A B C Are Fmentioning

confidence: 99%

Ladders and simplicity of derived module categories

et al. 2017

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Abstract. Recollements of derived module categories are investigated, using a new technique, ladders of recollements, which are maximal mutation sequences. The position in the ladder is shown to control whether a recollement restricts from unbounded to another level of derived category. Ladders also turn out to control derived simplicity on all levels. An algebra is derived simple if its derived category cannot be deconstructed, that is, if it is not the middle term of a non-trivial recollement whose outer terms are again derived categories of algebras. Derived simplicity on each level is characterised in terms of heights of ladders.These results are complemented by providing new classes of examples of derived simple rings, in particular indecomposable commutative rings, as well as by a finite dimensionsal counterexample to the Jordan-Hölder problem for derived module categories. Moreover, recollements are used to compute homological and K-theoretic invariants.

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Section: Proposition 33 Suppose That K Is a Field And A B C Are Fmentioning

confidence: 99%

Ladders and simplicity of derived module categories

et al. 2017

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“…A study of derived categories (in particular derived equivalences) in the representation theory of algebras was initiated by papers of Happel [28, Presented by Henning Krause. 29] and motivated by tilting theory, and is now an important direction of research (see for example [3,10,13,15,16,18,19,21,25,30,32,34,35,37,38]). …”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Derived Equivalence Classification of the Gentle Two-Cycle Algebras

Bobiński

2017

Algebr Represent Theor

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“…This has only been achieved for the finite dimensional algebras having discrete derived categories (in the sense of [14]). A derived equivalence classification of these very special gentle algebras was given in [4].…”

Section: Vol 85 2005mentioning

confidence: 99%

Cartan determinants for gentle algebras

Holm

2005

Arch. Math.

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The determinant of the Cartan matrix of a finite dimensional algebra is an invariant of the derived category and can be very helpful for derived equivalence classifications. In this paper we determine the determinants of the Cartan matrices for all gentle algebras. This is a class of algebras of tame representation type which occurs naturally in various places in representation theory. The definition of these algebras is of a purely combinatorial nature, and so are our formulae for the Cartan determinants.

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Classification of discrete derived categories

Cited by 61 publications

References 16 publications

Ladders and simplicity of derived module categories

Ladders and simplicity of derived module categories

Derived Equivalence Classification of the Gentle Two-Cycle Algebras

Cartan determinants for gentle algebras

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