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Endomorphism rings of maximal rigid objects in cluster tubes

Dagfinn F. Vatne (2011)

Colloquium Mathematicae

We describe the endomorphism rings of maximal rigid objects in the cluster categories of tubes. Moreover, we show that they are gentle and have Gorenstein dimension 1. We analyse their representation theory and prove that they are of finite type. Finally, we study the relationship between the module category and the cluster tube via the Hom-functor.

Exceptional collections on isotropic Grassmannians

Alexander Kuznetsov, Alexander Polishchuk (2016)

Journal of the European Mathematical Society

We introduce a new construction of exceptional objects in the derived category of coherent sheaves on a compact homogeneous space of a semisimple algebraic group and show that it produces exceptional collections of the length equal to the rank of the Grothendieck group on homogeneous spaces of all classical groups.

Explicit cogenerators for the homotopy category of projective modules over a ring

Amnon Neeman (2011)

Annales scientifiques de l'École Normale Supérieure

Let R be a ring. In two previous articles [12, 14] we studied the homotopy category 𝐊 ( R - Proj ) of projective R -modules. We produced a set of generators for this category, proved that the category is 1 -compactly generated for any ring R , and showed that it need not always be compactly generated, but is for sufficiently nice R . We furthermore analyzed the inclusion j ! : 𝐊 ( R - Proj ) 𝐊 ( R - Flat ) and the orthogonal subcategory 𝒮 = 𝐊 ( R - Proj ) . And we even showed that the inclusion 𝒮 𝐊 ( R - Flat ) has a right adjoint; this forces some natural map to be an equivalence...

Explicit models for perserse sheaves.

Félix Gudiel Rodríguez, Luis Narváez Macarro (2003)

Revista Matemática Iberoamericana

We consider categories of generalized perverse sheaves, with relaxed constructibility conditions, by means of the process of gluing t-structures and we exhibit explicit abelian categories defined in terms of standard sheaves categories which are equivalent to the former ones. In particular , we are able to realize perverse sheaves categories as non full abelian subcategories of the usual bounded complexes of sheaves categories. Our methods use induction on perversities. In this paper, we restrict...

Ext-algebras and derived equivalences

Dag Madsen (2006)

Colloquium Mathematicae

Using derived categories, we develop an alternative approach to defining Koszulness for positively graded algebras where the degree zero part is not necessarily semisimple.

Extensions of covariantly finite subcategories revisited

Jing He (2019)

Czechoslovak Mathematical Journal

Extriangulated categories were introduced by Nakaoka and Palu by extracting the similarities between exact categories and triangulated categories. A notion of homotopy cartesian square in an extriangulated category is defined in this article. We prove that in an extriangulated category with enough projective objects, the extension subcategory of two covariantly finite subcategories is covariantly finite. As an application, we give a simultaneous generalization of a result of X. W. Chen (2009) and...

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