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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 ℵ 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...

Finite presentation and purity in categories σ[M]

Mike Prest, Robert Wisbauer (2004)

Colloquium Mathematicae

For any module M over an associative ring R, let σ[M] denote the smallest Grothendieck subcategory of Mod-R containing M. If σ[M] is locally finitely presented the notions of purity and pure injectivity are defined in σ[M]. In this paper the relationship between these notions and the corresponding notions defined in Mod-R is investigated, and the connection between the resulting Ziegler spectra is discussed. An example is given of an M such that σ[M] does not contain any non-zero finitely presented...

Generalized canonical algebras and standard stable tubes

Andrzej Skowroński (2001)

Colloquium Mathematicae

We introduce a new wide class of finite-dimensional algebras which admit families of standard stable tubes (in the sense of Ringel [17]). In particular, we prove that there are many algebras of arbitrary nonzero (finite or infinite) global dimension whose Auslander-Reiten quivers admit faithful standard stable tubes.

Gorenstein star modules and Gorenstein tilting modules

Peiyu Zhang (2021)

Czechoslovak Mathematical Journal

We introduce the notion of Gorenstein star modules and obtain some properties and a characterization of them. We mainly give the relationship between n -Gorenstein star modules and n -Gorenstein tilting modules, see L. Yan, W. Li, B. Ouyang (2016), and a new characterization of n -Gorenstein tilting modules.

Localization and colocalization in tilting torsion theory for coalgebras

Yuan Li, Hailou Yao (2021)

Czechoslovak Mathematical Journal

Tilting theory plays an important role in the representation theory of coalgebras. This paper seeks how to apply the theory of localization and colocalization to tilting torsion theory in the category of comodules. In order to better understand the process, we give the (co)localization for morphisms, (pre)covers and special precovers. For that reason, we investigate the (co)localization in tilting torsion theory for coalgebras.

On flat covers in varieties

David Kruml (2008)

Commentationes Mathematicae Universitatis Carolinae

Flat covers do not exist in all varieties. We give a necessary condition for the existence of flat covers and some examples of varieties where not all algebras have flat covers.

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