Strictly stratified algebras

Ágoston, István; Dlab, Vlastimil; Lukács, Eugene

doi:10.1515/9783110805697.17

Cited by 33 publications

(104 citation statements)

References 0 publications

Supporting

Mentioning

102

Contrasting

Order By: Relevance

“…Triangulated categories. Let C be a triangulated k-category with shift functor [1]. An object X of C is exceptional if Hom C (X, X[n]) = 0 unless n = 0.…”

Section: Derived Categories and Recollementsmentioning

confidence: 99%

“…Let C be a triangulated k-category with shift functor [1]. A t-structure on C is a pair of full subcategories (C ≤0 , C ≥0 ) which are closed under isomorphisms and which satisfy the following conditions…”

Section: Derived Categories and Recollementsmentioning

confidence: 99%

“…Notice that D b (modA) then coincides with the full subcategory D f l (A) of D(ModA) consisting of complexes of A-modules whose total cohomology has finite length over k. The shift functor will be denoted by [1]. Let K b (projA) denote the homotopy category of bounded complexes of finitely generated projective A-modules.…”

Section: Derived Categoriesmentioning

confidence: 99%

“…A is a monomial algebra of wild representation type (see table W in [22]). Its opposite algebra is a standardly stratified algebra (in the sense of [1]; that is, its regular representation has a filtration by standard modules, using the order 1 < 2).…”

Section: Corollary 73 the Derived Jordan-hölder Theorem Holds True mentioning

confidence: 99%

See 3 more Smart Citations

Ladders and simplicity of derived module categories

et al. 2017

View full text Add to dashboard Cite

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.

show abstract

“…Triangulated categories. Let C be a triangulated k-category with shift functor [1]. An object X of C is exceptional if Hom C (X, X[n]) = 0 unless n = 0.…”

Section: Derived Categories and Recollementsmentioning

confidence: 99%

Section: Derived Categories and Recollementsmentioning

confidence: 99%

Section: Derived Categoriesmentioning

confidence: 99%

Section: Corollary 73 the Derived Jordan-hölder Theorem Holds True mentioning

confidence: 99%

See 2 more Smart Citations

Ladders and simplicity of derived module categories

et al. 2017

View full text Add to dashboard Cite

show abstract

“…We suppose for the moment that R is left artinian and gl dim R < oo. The first method was initiated by Zacharia in [12] and was used in [4], [9] and [1]. Here the idea is to find e 2 = e e R so that det C(R) = det C(eRe) and gl dim eRe < oo.…”

Section: Introductionmentioning

confidence: 99%

The Cartan determinant and generalizations of quasihereditary rings

Burgess

Fuller

1998

Proceedings of the Edinburgh Mathematical Society

View full text Add to dashboard Cite

The Cartan determinant conjecture for left artinian rings was verified for quasihereditary rings showing detC(R) = detC(U//), where / is a projective ideal generated by a primitive idempotent. This article identifies classes of rings generalizing the quasihereditary ones, first by relaxing the "projective" condition on heredity ideals. These rings, called left fc-hereditary are all of finite global dimension. Next a class of rings is defined which includes left serial rings of finite global dimension, quasihereditary and left 1-hereditary rings, but also rings of infinite global dimension. For such rings, the Cartan determinant conjecture is true, as is its converse. This is shown by matrix reduction. Examples compare and contrast these rings with other known families and a recipe is given for constructing them.

show abstract