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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...

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.

Extending modules relative to a torsion theory

Semra Doğruöz (2008)

Czechoslovak Mathematical Journal

An R -module M is said to be an extending module if every closed submodule of M is a direct summand. In this paper we introduce and investigate the concept of a type 2 τ -extending module, where τ is a hereditary torsion theory on Mod - R . An R -module M is called type 2 τ -extending if every type 2 τ -closed submodule of M is a direct summand of M . If τ I is the torsion theory on Mod - R corresponding to an idempotent ideal I of R and M is a type 2 τ I -extending R -module, then the question of whether or not M / M I is...

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...

Galois coverings and splitting properties of the ideal generated by halflines

Piotr Dowbor (2004)

Colloquium Mathematicae

Given a locally bounded k-category R and a group G ⊆ A u t k ( R ) acting freely on R we study the properties of the ideal generated by a class of indecomposable locally finite-dimensional modules called halflines (Theorem 3.3). They are applied to prove that under certain circumstances the Galois covering reduction to stabilizers, for the Galois covering F: R → R/G, is strictly full (Theorems 1.5 and 4.2).

Generic representations of orthogonal groups: projective functors in the category ℱ q u a d

Christine Vespa (2008)

Fundamenta Mathematicae

We continue the study of the category of functors ℱ q u a d , associated to ₂-vector spaces equipped with a nondegenerate quadratic form, initiated in J. Pure Appl. Algebra 212 (2008) and Algebr. Geom. Topology 7 (2007). We define a filtration of the standard projective objects in ℱ q u a d ; this refines to give a decomposition into indecomposable factors of the first two standard projective objects in ℱ q u a d : P H ₀ and P H ₁ . As an application of these two decompositions, we give a complete description of the polynomial functors...

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.

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