scholarly journals THE GRADED CENTER OF A TRIANGULATED CATEGORY

2016 ◽  
Vol 102 (1) ◽  
pp. 74-95
Author(s):  
JON F. CARLSON ◽  
PETER WEBB
Keyword(s):  
Group Algebra ◽  
Special Kind ◽  
Module Category ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Stable Module Category ◽  
Natural Transformations ◽  
Stable Module ◽  
Serre Functor

With applications in mind to the representations and cohomology of block algebras, we examine elements of the graded center of a triangulated category when the category has a Serre functor. These are natural transformations from the identity functor to powers of the shift functor that commute with the shift functor. We show that such natural transformations that have support in a single shift orbit of indecomposable objects are necessarily of a kind previously constructed by Linckelmann. Under further conditions, when the support is contained in only finitely many shift orbits, sums of transformations of this special kind account for all possibilities. Allowing infinitely many shift orbits in the support, we construct elements of the graded center of the stable module category of a tame group algebra of a kind that cannot occur with wild block algebras. We use functorial methods extensively in the proof, developing some of this theory in the context of triangulated categories.

scholarly journals Recollements, comma categories and morphic enhancements

2021 ◽  
pp. 1-25
Author(s):  
Xiao-Wu Chen ◽  
Jue Le
Keyword(s):  
Module Category ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Comma Category ◽  
The Right ◽  
Projective Lines ◽  
Triangle Functor

For each recollement of triangulated categories, there is an epivalence between the middle category and the comma category associated with a triangle functor from the category on the right to the category on the left. For a morphic enhancement of a triangulated category $\mathcal {T}$ , there are three explicit ideals of the enhancing category, whose corresponding factor categories are all equivalent to the module category over $\mathcal {T}$ . Examples related to inflation categories and weighted projective lines are discussed.

scholarly journals Recollement of homotopy categories and Cohen-Macaulay modules

2011 ◽  
Vol 8 (3) ◽  
pp. 507-542 ◽  
Author(s):  
Osamu Iyama ◽  
Kiriko Kato ◽  
Jun-ichi Miyachi
Keyword(s):  
Homotopy Category ◽  
Module Category ◽  
Gorenstein Ring ◽  
Projective Modules ◽  
Finitely Generated ◽  
Stable Module Category ◽  
Quotient Category ◽  
Stable Module

AbstractWe study the homotopy category of unbounded complexes with bounded homologies and its quotient category by the homotopy category of bounded complexes. In the case of the homotopy category of finitely generated projective modules over an Iwanaga-Gorenstein ring, we show the existence of a new structure in the above quotient category, which we call a triangle of recollements. Moreover, we show that this quotient category is triangle equivalent to the stable module category of Cohen-Macaulay T2(R)-modules.

scholarly journals The generating hypothesis for the stable module category of a p-group

2007 ◽  
Vol 310 (1) ◽  
pp. 428-433 ◽  
Author(s):  
David J. Benson ◽  
Sunil K. Chebolu ◽  
J. Daniel Christensen ◽  
Ján Mináč
Keyword(s):  
Module Category ◽  
Stable Module Category ◽  
Stable Module ◽  
Generating Hypothesis

scholarly journals Phantom Maps in the Stable Module Category

1998 ◽  
Vol 201 (2) ◽  
pp. 686-702 ◽  
Author(s):  
Gilles Ph Gnacadja
Keyword(s):  
Module Category ◽  
Stable Module Category ◽  
Phantom Maps ◽  
Stable Module

scholarly journals Almost split morphisms in subcategories of triangulated categories

2021 ◽  
pp. 2250239
Author(s):  
Francesca Fedele
Keyword(s):  
Full Description ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Cluster Category ◽  
Indecomposable Object ◽  
Type Formula ◽  
Dynkin Type ◽  
Serre Functor

For a suitable triangulated category [Formula: see text] with a Serre functor [Formula: see text] and a full precovering subcategory [Formula: see text] closed under summands and extensions, an indecomposable object [Formula: see text] in [Formula: see text] is called Ext-projective if Ext[Formula: see text]. Then there is no Auslander–Reiten triangle in [Formula: see text] with end term [Formula: see text]. In this paper, we show that if, for such an object [Formula: see text], there is a minimal right almost split morphism [Formula: see text] in [Formula: see text], then [Formula: see text] appears in something very similar to an Auslander–Reiten triangle in [Formula: see text]: an essentially unique triangle in [Formula: see text] of the form [Formula: see text] where [Formula: see text] is an indecomposable not in [Formula: see text] and [Formula: see text] is a [Formula: see text]-envelope of [Formula: see text]. Moreover, under some extra assumptions, we show that removing [Formula: see text] from [Formula: see text] and replacing it with [Formula: see text] produces a new subcategory of [Formula: see text] closed under extensions. We prove that this process coincides with the classic mutation of [Formula: see text] with respect to the rigid subcategory of [Formula: see text] generated by all the indecomposable Ext-projectives in [Formula: see text] apart from [Formula: see text]. When [Formula: see text] is the cluster category of Dynkin type [Formula: see text] and [Formula: see text] has the above properties, we give a full description of the triangles in [Formula: see text] of the form [Formula: see text] and show under which circumstances replacing [Formula: see text] by [Formula: see text] gives a new extension closed subcategory.

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