scholarly journals Auslander-Reiten triangles in subcategories

2008 ◽  
Vol 3 (3) ◽  
pp. 583-601 ◽  
Author(s):  
Peter Jørgensen
Keyword(s):  
Triangulated Category ◽  
Triangulated Categories ◽  
Approximation Properties ◽  
Finite Dimensional ◽  
Image Position ◽  
Finite Dimensional Algebras

AbstractThis paper studies Auslander-Reiten triangles in subcategories of triangulated categories. The main theorem shows that the Auslander-Reiten triangles in a subcategory are closely connected with the approximation properties of the subcategory. Namely, let C be an object in the subcategory C of the triangulated category T, and letbe an Auslander-Reiten triangle in T. Then under suitable assumptions, there is an Auslander-Reiten trianglein C if and only if there is a minimal right-C-approximation of the form.The theory is used to give a new proof of the existence of Auslander-Reiten sequences over finite dimensional algebras.

The Classification of Algebras by Dominant Dimension

1968 ◽  
Vol 20 ◽  
pp. 398-409 ◽  
Author(s):  
Bruno J. Mueller
Keyword(s):  
Exact Sequence ◽  
Infinite Sequence ◽  
Dominant Dimension ◽  
Finite Dimensional ◽  
Image Position ◽  
Finite Dimensional Algebras

Nakayama proposed to classify finite-dimensional algebras R over a field according to how long an exact sequenceof projective and injective R-R-bimodules Xi they allow. He conjectured that if there exists an infinite sequence of this type, then R must be quasi-Frobenius; and he proved this when R is generalized uniserial (17).

A Generalization of Commutative and Alternative Rings II

1972 ◽  
Vol 24 (4) ◽  
pp. 728-733 ◽  
Author(s):  
Erwin Kleinfeld
Keyword(s):  
Linear Function ◽  
Commutative Rings ◽  
Finite Dimensional ◽  
Image Position ◽  
Finite Dimensional Algebras

We shall call a linear function on the elements of a ring R skew-symmetric if it vanishes whenever at least two of the variables are equal. Here we shall study rings R of characteristic not 2 which satisfy the following two identities:1(2)is skew-symmetric.Both of these identities hold in alternative rings. The fact that F(w, x, y, z) is skew-symmetric in alternative rings is an important tool in the study of such rings. It is also obvious that both identities hold in commutative rings. But unlike other recent generalizations of commutative and alternative rings it turns out that there exist simple, finite dimensional algebras of degree two which are neither alternative nor commutative and satisfy (1) and (2).

Derived equivalences from cohomological approximations and mutations of Φ-Yoneda algebras

2013 ◽  
Vol 143 (3) ◽  
pp. 589-629 ◽  
Author(s):  
Wei Hu ◽  
Steffen Koenig ◽  
Changchang Xi
Keyword(s):  
Triangulated Categories ◽  
Endomorphism Rings ◽  
Infinite Dimensional ◽  
Derived Equivalences ◽  
Finite Dimensional ◽  
New Construction ◽  
Yoneda Algebras ◽  
Frobenius Categories ◽  
Finite Dimensional Algebras

A new construction of derived equivalences is given, which relates different endomorphism rings and, more generally, cohomological endomorphism rings, including higher extensions, of objects in triangulated categories. These objects need to be connected by certain universal maps that are cohomological approximations and that exist in very general circumstances. The construction turns out to be applicable to a wide variety of situations, covering finite-dimensional algebras as well as certain infinite-dimensional algebras, Frobenius categories and n-Calabi–Yau categories.

scholarly journals Dimensions of triangulated categories

2008 ◽  
Vol 1 (2) ◽  
pp. 193-256 ◽  
Author(s):  
Raphaël Rouquier
Keyword(s):  
Finite Dimension ◽  
Derived Category ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Coherent Sheaves ◽  
Finite Dimensional

AbstractWe define a dimension for a triangulated category. We prove a representability Theorem for a class of functors on finite dimensional triangulated categories. We study the dimension of the bounded derived category of an algebra or a scheme and we show in particular that the bounded derived category of coherent sheaves over a variety has a finite dimension.

scholarly journals Dimensions of triangulated categories

2007 ◽  
Vol 1 (2) ◽  
pp. 193-256 ◽  
Author(s):  
Raphaël Rouquier
Keyword(s):  
Finite Dimension ◽  
Derived Category ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Coherent Sheaves ◽  
Finite Dimensional

AbstractWe define a dimension for a triangulated category. We prove a representability Theorem for a class of functors on finite dimensional triangulated categories. We study the dimension of the bounded derived category of an algebra or a scheme and we show in particular that the bounded derived category of coherent sheaves over a variety has a finite dimension.

On the centre of a triangulated category

2011 ◽  
Vol 54 (2) ◽  
pp. 443-466 ◽  
Author(s):  
Henning Krause ◽  
Yu Ye
Keyword(s):  
Representation Theory ◽  
Triangulated Category ◽  
Basic Properties ◽  
Finite Dimensional ◽  
Finite Dimensional Algebras

AbstractWe discuss some basic properties of the graded centre of a triangulated category and compute examples arising in representation theory of finite-dimensional algebras.

DEGENERATION-LIKE ORDERS IN TRIANGULATED CATEGORIES

2005 ◽  
Vol 04 (05) ◽  
pp. 587-597 ◽  
Author(s):  
BERNT TORE JENSEN ◽  
XIUPING SU ◽  
ALEXANDER ZIMMERMANN
Keyword(s):  
Partial Order ◽  
Natural Generalization ◽  
Derived Categories ◽  
Derived Category ◽  
Closed Field ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Finite Dimensional

In an earlier paper we defined a relation ≤Δ between objects of the derived category of bounded complexes of modules over a finite dimensional algebra over an algebraically closed field. This relation was shown to be equivalent to the topologically defined degeneration order in a certain space [Formula: see text] for derived categories. This space was defined as a natural generalization of varieties for modules. We remark that this relation ≤Δ is defined for any triangulated category and show that under some finiteness assumptions on the triangulated category ≤Δ is always a partial order.

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