Relative theory in subcategories

Abstract

Abstract. We generalize the relative (co)tilting theory of Auslander-Solberg in the category mod Λ of finitely generated left modules over an artin algebra Λ to certain subcategories of mod Λ. We then use the theory (relative (co)tilting theory in subcategories) to generalize one of the main result of Marcos et al. [Comm. Algebra 33 (2005)].Introduction. Let Λ be an artin algebra, and let mod Λ denote the category of finitely generated left Λ-modules. Auslander and Solberg [9, 10] developed a relative (co)tilting theory in mod Λ which is a generalization of standard (co)tilting theory [3], [12], [14], [23]. In this paper we develop a relative (co)tilting theory in extension-closed functorially finite subcategories of mod Λ.Let T be an ordinary tilting module over Λ. Then the module DT , where D is the usual duality between left and right modules, is a cotilting module

Split-By-Nilpotent Extensions Algebras and Stratifying Systems

“…Definition 2.4. [3,28] Let Γ and Λ be algebras, and let I be a two-sided ideal of Γ. It is said that Γ is a split-by-nilpotent extension of Λ by I, if I ⊆ rad (Γ) and there is an exact sequence of abelian groups…”

“…Stratifying systems where introduced in [12,20,21,27,31] and developed in [16,23,24,17,26] with some applications, for example, in [10,11,13,14,15,19,22,28].…”