Auslander-type conditions and cotorsion theory

We study the properties of rings satisfying Auslander-type conditions. If an artin algebra Λ satisfies the Auslander condition (that is, Λ is an ∞-Gorenstein artin algebra), then we construct two kinds of subcategories which form functorially finite cotorsion theories. Noetherian rings satisfying ‘Auslander-type conditions’ on self-injective resolutions can be regarded as certain non-commutative analogs of commutative Gorenstein rings. Such conditions, especially dominant dimension and the n-Gorenstein condition, play a crucial role in representation theory and non-commutative algebraic geometry (e.g. [A], [AR2,3], [B], [C], [EHIS], [FGR], [FI], [HN], [IS], [I3,4], [M], [R], [S], [T], [W]). They are also interesting from the viewpoint of some mysterious homological conjectures, e.g. the finitistic dimension conjecture, Nakayama conjecture, and so on. It is therefore important to understand non-commutative ‘regular’ or ‘Gorenstein’ rings though it is still far from realized even for the case of finite dimensional algebras. Recently, several authors (e.g. [Hu1,2,3], [I1,2,4]) study some Auslander-type conditions, e.g. the quasi n-Gorenstein condition, the (l, n)-condition, and so on. This paper is devoted to enlarge our knowledge of the homological behavior of non-commutative rings. Especially we introduce Auslander-type conditions Gn(k) and gn(k) and study their properties. Throughout this paper, let Λ be a noetherian ring (unless stated otherwise) and modΛ the category of finitely generated left Λ-modules. We denote by modΛ the stable category of Λ [AB]. Put ( ) := HomΛ( ,Λ) and En := Ext n Λ( ,Λ) : modΛ → modΛ op for n ≥ 0, Tn := Tr ◦Ω n−1 : modΛ → modΛ for n > 0, where Ω : modΛ → modΛ is the syzygy functor and Tr : modΛ → modΛ is the transpose functor [AB]. Let grade X := inf{i ≥ 0 | EiX 6= 0} the grade, The first author was partially supported by Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20060284002) and NSF of Jiangsu Province of China (Grant No. BK2005207).