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Decomposability criterion for linear sheaves

Marcos Jardim, Vitor Silva (2012)

Open Mathematics

We establish a decomposability criterion for linear sheaves on ℙn. Applying it to instanton bundles, we show, in particular, that every rank 2n instanton bundle of charge 1 on ℙn is decomposable. Moreover, we provide an example of an indecomposable instanton bundle of rank 2n − 1 and charge 1, thus showing that our criterion is sharp.

Deformations of bimodule problems

Christof Geiß (1996)

Fundamenta Mathematicae

We prove that deformations of tame Krull-Schmidt bimodule problems with trivial differential are again tame. Here we understand deformations via the structure constants of the projective realizations which may be considered as elements of a suitable variety. We also present some applications to the representation theory of vector space categories which are a special case of the above bimodule problems.

Deformed mesh algebras of Dynkin type ℂₙ

Jerzy Białkowski, Karin Erdmann, Andrzej Skowroński (2012)

Colloquium Mathematicae

In our recent paper (J. Algebra 345 (2011)) we prove that the deformed preprojective algebras of generalized Dynkin type ₙ (in the sense of our earlier work in Trans. Amer Math. Soc. 359 (2007)) are exactly (up to isomorphism) the stable Auslander algebras of simple plane singularities of Dynkin type 2 n . In this article we complete the picture by showing that the deformed mesh algebras of Dynkin type ℂₙ are isomorphic to the canonical mesh algebras of type ℂₙ, and hence to the stable Auslander algebras...

Derived dimension via τ -tilting theory

Yingying Zhang (2021)

Czechoslovak Mathematical Journal

Comparing the bounded derived categories of an algebra and of the endomorphism algebra of a given support τ -tilting module, we find a relation between the derived dimensions of an algebra and of the endomorphism algebra of a given τ -tilting module.

Derived endo-discrete artin algebras

Raymundo Bautista (2006)

Colloquium Mathematicae

Let Λ be an artin algebra. We prove that for each sequence ( h i ) i ∈ ℤ of non-negative integers there are only a finite number of isomorphism classes of indecomposables X ∈ b ( Λ ) , the bounded derived category of Λ, with l e n g t h E ( X ) H i ( X ) = h i for all i ∈ ℤ and E(X) the endomorphism ring of X in b ( Λ ) if and only if b ( M o d Λ ) , the bounded derived category of the category M o d Λ of all left Λ-modules, has no generic objects in the sense of [4].

Derived equivalence classification of weakly symmetric algebras of domestic type

Rafał Bocian, Andrzej Skowroński (2016)

Colloquium Mathematicae

We complete the derived equivalence classification of all weakly symmetric algebras of domestic type over an algebraically closed field, by solving the problem of distinguishing standard and nonstandard algebras up to stable equivalence, and hence derived equivalence. As a consequence, a complete stable equivalence classification of weakly symmetric algebras of domestic type is obtained.

Derived equivalences between generalized matrix algebras

QingHua Chen, HongJin Liu (2020)

Czechoslovak Mathematical Journal

We construct derived equivalences between generalized matrix algebras. We record several corollaries. In particular, we show that the n -replicated algebras of two derived equivalent, finite-dimensional algebras are also derived equivalent.

Dichte Ringe*

Günther Haugner, Wolfgang Zimmermann (1974)

Mathematische Annalen

Differentiation and splitting for lattices over orders

Wolfgang Rump (2001)

Colloquium Mathematicae

We extend our module-theoretic approach to Zavadskiĭ’s differentiation techniques in representation theory. Let R be a complete discrete valuation domain with quotient field K, and Λ an R-order in a finite-dimensional K-algebra. For a hereditary monomorphism u: P ↪ I of Λ-lattices we have an equivalence of quotient categories ∂ ̃ u : Λ - l a t / [ ℋ ] ⭇ δ u Λ - l a t / [ B ] which generalizes Zavadskiĭ’s algorithms for posets and tiled orders, and Simson’s reduction algorithm for vector space categories. In this article we replace u by a more...

Directing components for quasitilted algebras

Flávio Coelho (1999)

Colloquium Mathematicae

We show here that a directing component of the Auslander-Reiten quiver of a quasitilted algebra is either postprojective or preinjective or a connecting component.

Distinguishing derived equivalence classes using the second Hochschild cohomology group

Deena Al-Kadi (2010)

Colloquium Mathematicae

We study the second Hochschild cohomology group of the preprojective algebra of type D₄ over an algebraically closed field K of characteristic 2. We also calculate the second Hochschild cohomology group of a non-standard algebra which arises as a socle deformation of this preprojective algebra and so show that the two algebras are not derived equivalent. This answers a question raised by Holm and Skowroński.

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