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nrepresentation infinite algebras
"... From the viewpoint of higher dimensional AuslanderReiten theory, we introduce a new class of finite dimensional algebras of global dimension n, which we call nrepresentation infinite. They are a certain analog of representation infinite hereditary algebras, and we study three important classes o ..."
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From the viewpoint of higher dimensional AuslanderReiten theory, we introduce a new class of finite dimensional algebras of global dimension n, which we call nrepresentation infinite. They are a certain analog of representation infinite hereditary algebras, and we study three important classes of modules: npreprojective, npreinjective and nregular modules. We observe that their homological behaviour is quite interesting. For instance they provide first examples of algebras having infinite Ext 1orthogonal families of modules. Moreover we give general constructions of nrepresentation infinite algebras. Applying Minamoto’s theory on Fano algebras in noncommutative algebraic geometry, we describe the category of nregular modules in terms of the corresponding preprojective algebra. Then we introduce nrepresentation tame algebras, and show that the category of nregular modules decomposes into the categories of finite dimensional modules over localizations of the preprojective algebra. This generalizes the classical description of regular modules over tame hereditary algebras. As an application, we show that the representation dimension of an nrepresentation tame algebra is at least n+2.