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For a tensor‐triangulated category and any regular cardinal α$\alpha$, we study the frame of α$\alpha$‐localizing tensor ideals and its associated space of points. For a well‐generated category and its frame of localizing tensor ideals, we provide conditions such that the associated space is obtained by refining the topology of the corresponding space for the triangulated subcategory of α$\alpha$‐compact objects. This is illustrated by several known examples for α=ℵ0$\alpha =\aleph _0$, and new spaces arise for α>ℵ0$\alpha >\aleph _0$. [ABSTRACT FROM AUTHOR]

A notion of stratification is introduced for any compactly generated triangulated category ⊤ endowed with an action of a graded‐commutative noetherian ring R. The utility of this notion is demonstrated by establishing diverse consequences that follow if ⊤ is stratified by R. Among them are a classification of the localizing subcategories of ⊤ in terms of subsets of the set of prime ideals in R; a classification of the thick subcategories of the subcategory of compact objects in ⊤; and results on the support of the graded R‐module of morphisms Hom*⊤(C, D) leading to analogs of the tensor product theorem for support varieties of modular representation of groups. [ABSTRACT FROM AUTHOR]

A classical theorem of Zimmermann describes the relation between almost split sequences in the category of finitely presented modules and those in the category of all modules over some fixed ring. An analogue of Auslander–Reiten triangles in triangulated categories is proved in this paper. This is used to explain the relation between different existence results for Auslander–Reiten triangles, which are based either on Brown's representability theorem, or on the existence of Serre functors. 2000 Mathematics Subject Classification 18E30, 16G70 (primary), 14F05, 55U35 (secondary). [ABSTRACT FROM PUBLISHER]

Let Λ and Γ be finite dimensional algebras. It is shown that any stable equivalence f:mod¯Λ→mod¯Γ between the categories of finitely generated modules induces a bijection M ↦ Mf between the sets of isomorphism classes of generic modules over Λ and Γ such that the endolength of Mf is bounded by the endolength of M up to a scalar which depends only on f. Using Crawley-Boevey's characterization of tame representation type in terms of generic modules, one obtains as a consequence a new proof for the fact that a stable equivalence preserves tameness. This proof also shows that polynomial growth is preserved. 2000 Mathematics Subject Classification 16G60, 16D90. [ABSTRACT FROM AUTHOR]