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Cluster tilting for higher Auslander algebras
, 2008
"... The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representationfinite algebras and Auslander algebras. The nAuslanderReiten translation functor τn plays an important role in the study of ncluster tilting subcategories. We study the category M ..."
Abstract

Cited by 30 (9 self)
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The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representationfinite algebras and Auslander algebras. The nAuslanderReiten translation functor τn plays an important role in the study of ncluster tilting subcategories. We study the category Mn of preinjectivelike modules obtained by applying τn to injective modules repeatedly. We call a finite dimensional algebra Λ ncomplete if Mn = add M for an ncluster tilting object M. Our main result asserts that the endomorphism algebra EndΛ(M) is (n + 1)complete. This gives an inductive construction of ncomplete algebras. For example, any representationfinite hereditary algebra Λ (1) is 1complete. Hence the Auslander algebra Λ (2) of Λ (1) is 2complete. Moreover, for any n ≥ 1, we have an ncomplete algebra Λ (n) which has an ncluster tilting object M (n) such that Λ (n+1) = End Λ (n)(M (n)). We give the presentation of Λ (n) by a quiver with relations. We apply our results to construct ncluster tilting subcategories of derived categories of ncomplete algebras.