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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.
AMPLENESS OF TWOSIDED TILTING COMPLEXES AND FANO ALGEBRAS
"... Abstract. From the view point of noncommutative algebraic geometry (NCAG), a twosided tilting complex is an analog of a line bundle. In this paper we define the notion of ampleness for twosided tilting complexes over finite dimensional algebras of finite global dimension, and prove its basic prope ..."
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Abstract. From the view point of noncommutative algebraic geometry (NCAG), a twosided tilting complex is an analog of a line bundle. In this paper we define the notion of ampleness for twosided tilting complexes over finite dimensional algebras of finite global dimension, and prove its basic properties, which justify the name ”ampleness”. From the view point of NCAG, Serre functors are considered to be shifted canonical bundles. A finite dimensional algebra A of finite global dimension is called Fano if the shifted Serre functor A∗[¡d] is antiample. Some classes of algebras studies before are Fano. We show by an example that the property of A∗[¡d] from the view point of NCAG captures some representation theoretic property of the algebra A. From our view point, we give a structure theorem of ASregular algebras. ASregular algebras are defined to extract a good homological property of polynomial algebras. Our theorem shows that ASregular algebra is polynomial algebra in some sense. 1.