127 research outputs found
Cluster tilting for higher Auslander algebras
The concept of cluster tilting gives a higher analogue of classical Auslander
correspondence between representation-finite algebras and Auslander algebras.
The -Auslander-Reiten translation functor plays an important role
in the study of -cluster tilting subcategories. We study the category
\MM_n of preinjective-like modules obtained by applying to injective
modules repeatedly. We call a finite dimensional algebra
\emph{-complete} if \MM_n=\add M for an -cluster tilting object .
Our main result asserts that the endomorphism algebra \End_\Lambda(M) is
-complete. This gives an inductive construction of -complete
algebras. For example, any representation-finite hereditary algebra
is 1-complete. Hence the Auslander algebra of
is 2-complete. Moreover, for any , we have an
-complete algebra which has an -cluster tilting object
such that \Lambda^{(n+1)}=\End_{\Lambda^{(n)}}(M^{(n)}). We give
the presentation of by a quiver with relations. We apply our
results to construct -cluster tilting subcategories of derived categories of
-complete algebras.Comment: 42 pages. Typos are correcte
Higher dimensional Auslander-Reiten theory on maximal orthogonal subcategories
Auslander-Reiten theory is fundamental to study categories which appear in
representation theory, for example, modules over artin algebras, Cohen-Macaulay
modules over Cohen-Macaulay rings, lattices over orders, and coherent sheaves
on projective curves. In these Auslander-Reiten theories, the number `2' is
quite symbolic. For one thing, almost split sequences give minimal projective
resolutions of simple functors of projective dimension `2'. For another,
Cohen-Macaulay rings of Krull-dimension `2' provide us with one of the most
beautiful situation in representation theory, which is closely related to
McKay's observation on simple singularities. In this sense, usual
Auslander-Reiten theory should be `2-dimensional' theory, and it be natural to
find a setting for higher dimensional Auslander-Reiten theory from the
viewpoint of representation theory and non-commutative algebraic geometry.
We introduce maximal -orthogonal subcategories as a natural domain of
higher dimensional Auslander-Reiten theory which should be
`-dimensional'. We show that the -Auslander-Reiten translation
functor and the -Auslander-Reiten duality can be defined quite naturally for
such categories. Using them, we show that our categories have {\it -almost
split sequences}, which give minimal projective resolutions of simple objects
of projective dimension `' in functor categories. We show that an
invariant subring (of Krull-dimension `') corresponding to a finite
subgroup of has a natural maximal -orthogonal
subcategory. We give a classification of all maximal 1-orthogonal subcategories
for representation-finite selfinjective algebras and representation-finite
Gorenstein orders of classical type.Comment: 25 pages. Final Version. To appear in Adv. Mat
Tilting Cohen-Macaulay representations
This is a survey on recent developments in Cohen-Macaulay representations via
tilting and cluster tilting theory. We explain triangle equivalences between
the singularity categories of Gorenstein rings and the derived (or cluster)
categories of finite dimensional algebras.Comment: To appear in the ICM 2018 proceeding
Auslander correspondence
We study Auslander correspondence from the viewpoint of higher dimensional
Auslander-Reiten theory on maximal orthogonal subcategories. We give
homological characterizations of Auslander algebras, especially an answer to a
question of M. Artin. They are also closely related to Auslander's
representation dimension of artin algebras and Van den Bergh's non-commutative
crepant resolutions of Gorenstein singularities.Comment: 29 pages. Final Version. To appear in Adv. Mat
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