scholarly journals The dual of the homotopy category of projective modules satisfies Brown representability

2014 ◽  
Vol 46 (4) ◽  
pp. 765-770 ◽  
Author(s):  
George Ciprian Modoi
Keyword(s):  
Homotopy Category ◽  
Projective Modules ◽  
Brown Representability

scholarly journals Pure derived and pure singularity categories

2019 ◽  
Vol 19 (06) ◽  
pp. 2050117
Author(s):  
Tianya Cao ◽  
Wei Ren
Keyword(s):  
Global Dimension ◽  
Derived Categories ◽  
Derived Category ◽  
Homotopy Category ◽  
Projective Modules ◽  
Strong Assumption ◽  
Singularity Category

Firstly, we compare the bounded derived categories with respect to the pure-exact and the usual exact structures, and describe bounded derived category by pure-projective modules, under a fairly strong assumption on the ring. Then, we study Verdier quotient of bounded pure derived category modulo the bounded homotopy category of pure-projective modules, which is called a pure singularity category since we show that it reflects the finiteness of pure-global dimension of rings. Moreover, invariance of pure singularity in a recollement of bounded pure derived categories is studied.

scholarly journals AC-GORENSTEIN RINGS AND THEIR STABLE MODULE CATEGORIES

2018 ◽  
Vol 107 (02) ◽  
pp. 181-198
Author(s):  
JAMES GILLESPIE
Keyword(s):  
The Other ◽  
Homotopy Category ◽  
Gorenstein Ring ◽  
Model Structure ◽  
Projective Modules ◽  
Stable Module Category ◽  
Stable Module ◽  
Coherent Rings ◽  
Quillen Equivalent ◽  
Compactly Generated

We introduce what is meant by an AC-Gorenstein ring. It is a generalized notion of Gorenstein ring that is compatible with the Gorenstein AC-injective and Gorenstein AC-projective modules of Bravo–Gillespie–Hovey. It is also compatible with the notion of $n$ -coherent rings introduced by Bravo–Perez. So a $0$ -coherent AC-Gorenstein ring is precisely a usual Gorenstein ring in the sense of Iwanaga, while a $1$ -coherent AC-Gorenstein ring is precisely a Ding–Chen ring. We show that any AC-Gorenstein ring admits a stable module category that is compactly generated and is the homotopy category of two Quillen equivalent abelian model category structures. One is projective with cofibrant objects that are Gorenstein AC-projective modules while the other is an injective model structure with fibrant objects that are Gorenstein AC-injectives.

scholarly journals Brown representability often fails for homotopy categories of complexes

2011 ◽  
Vol 9 (1) ◽  
pp. 151-160 ◽  
Author(s):  
George Ciprian Modoi ◽  
Jan Šťovíček
Keyword(s):  
Abelian Groups ◽  
Homotopy Category ◽  
Brown Representability ◽  
Localizing Subcategory

AbstractWe show that for the homotopy categoryK(Ab) of complexes of abelian groups, both Brown representability and Brown representability for the dual fail. We also provide an example of a localizing subcategory ofK(Ab) for which the inclusion intoK(Ab) does not have a right adjoint.

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 Homotopy category of N -complexes of projective modules

2016 ◽  
Vol 220 (6) ◽  
pp. 2414-2433 ◽  
Author(s):  
P. Bahiraei ◽  
R. Hafezi ◽  
A. Nematbakhsh
Keyword(s):  
Homotopy Category ◽  
Projective Modules

scholarly journals Homotopy category of projective complexes and complexes of Gorenstein projective modules

2014 ◽  
Vol 399 ◽  
pp. 423-444 ◽  
Author(s):  
Javad Asadollahi ◽  
Rasool Hafezi ◽  
Shokrollah Salarian
Keyword(s):  
Homotopy Category ◽  
Projective Modules ◽  
Gorenstein Projective Modules

scholarly journals Categorical action of the extended braid group of affine type A

2017 ◽  
Vol 19 (03) ◽  
pp. 1650024 ◽  
Author(s):  
Agnès Gadbled ◽  
Anne-Laure Thiel ◽  
Emmanuel Wagner
Keyword(s):  
Braid Group ◽  
Type A ◽  
Homotopy Category ◽  
Projective Modules ◽  
Intersection Numbers ◽  
Finitely Generated ◽  
Type Formula ◽  
Affine Type ◽  
Cyclic Quiver

Using a quiver algebra of a double cyclic quiver, we construct a faithful categorical action of the extended braid group of affine type [Formula: see text] on its bounded homotopy category of finitely generated projective modules. The algebra is trigraded and we identify the trigraded dimensions of the space of morphisms of this category with intersection numbers coming from the topological origin of the group.

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