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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 ..."
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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.
nStrongly Gorenstein Projective, Injective and Flat Modules
, 2009
"... In this paper, we study the relation between mstrongly Gorenstein projective (resp. injective) modules and nstrongly Gorenstein projective (resp. injective) modules whenever m = n, and the homological behavior of nstrongly Gorenstein projective (resp. injective) modules. We introduce the notion ..."
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In this paper, we study the relation between mstrongly Gorenstein projective (resp. injective) modules and nstrongly Gorenstein projective (resp. injective) modules whenever m = n, and the homological behavior of nstrongly Gorenstein projective (resp. injective) modules. We introduce the notion of nstrongly Gorenstein flat modules. Then we study the homological behavior of nstrongly Gorenstein flat modules, and the relation between nstrongly Gorenstein flat modules and nstrongly Gorenstein projective (resp. injective) modules.
Trivial Maximal 1Orthogonal Subcategories For Auslander’s 1Gorenstein Algebras
, 2009
"... Let Λ be an Auslander’s 1Gorenstein Artinian algebra with global dimension 2. If Λ admits a trivial maximal 1orthogonal subcategory of mod Λ, then for any indecomposable module M ∈ mod Λ, we have that the projective dimension of M is equal to 1 if and only if so is its injective dimension and th ..."
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Let Λ be an Auslander’s 1Gorenstein Artinian algebra with global dimension 2. If Λ admits a trivial maximal 1orthogonal subcategory of mod Λ, then for any indecomposable module M ∈ mod Λ, we have that the projective dimension of M is equal to 1 if and only if so is its injective dimension and that M is injective if the projective dimension of M is equal to 2, which implies that Λ is almost hereditary.
nREPRESENTATIONFINITE ALGEBRAS AND FRACTIONALLY CALABIYAU ALGEBRAS
, 2009
"... In this short paper, we study nrepresentationfinite algebras from the viewpoint of fractionally CalabiYau algebras. We shall show that all nrepresentationfinite algebras are twisted fractionally CalabiYau. We also show that twisted n(ℓ−1)CalabiYau algebras of global dimension n are ℓ nrepr ..."
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In this short paper, we study nrepresentationfinite algebras from the viewpoint of fractionally CalabiYau algebras. We shall show that all nrepresentationfinite algebras are twisted fractionally CalabiYau. We also show that twisted n(ℓ−1)CalabiYau algebras of global dimension n are ℓ nrepresentationfinite for any ℓ> 0. As an application, we give a construction of nrepresentationfinite algebras using the tensor product.