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Stable Infinity Categories [article]

Jacob Lurie
2009 arXiv   pre-print
We prove that the homotopy category of a stable infinity category is triangulated, and that the collection of stable infinity categories is closed under a variety of constructions.  ...  this stable infinity category by a universal mapping property.  ...  If C is a stable ∞-category, then the opposite ∞-category C op is also stable. Remark 2.14.  ... 
arXiv:math/0608228v5 fatcat:d76mnnq6wjhozcuc4yh6cjx3se

Stable homotopy categories

Alex Heller
1968 Bulletin of the American Mathematical Society  
Accordingly, the "stable categories" of Puppe are referred to below as triangulated categories (the word "stable" is itself used in a quite different way, cf. §1).  ...  If F: Gt' ->0t is a stable equivalence of stable additive categories then AHi?-l (A) gives a bijection ïa^ïCfc'.  ...  An additive category d is a torsion category if each object A is a torsion object, i.e. one such that for some integer h, h-lA = 0. LEMMA 17.1. JET 2 3 is a torsion category.  ... 
doi:10.1090/s0002-9904-1968-11871-3 fatcat:fifndjqtnzdodhburigfb2ibim

Realizing stable categories as derived categories

Kota Yamaura
2013 Advances in Mathematics  
Then the stable category mod Z A of the category of Z-graded A-modules is a triangulated category (cf. [1] ).  ...  Moreover the stable category mod Z/ Z A of the category of Z/ Z-graded A-modules is a triangulated category (cf. [1] ).  ... 
doi:10.1016/j.aim.2013.08.017 fatcat:ylbfkq4c3jafxi6qp2557477he

Realizing stable categories as derived categories [article]

Kota Yamaura
2012 arXiv   pre-print
First we show that there exists a triangle-equivalence between the stable category of Z-graded A-modules and the derived category of a certain algebra Γ of finite global dimension.  ...  Secondly we show that if A has Gorenstein parameter ℓ, then there exists a triangle-equivalence between the stable category of Z/ℓZ-graded A-modules and a derived-orbit category of Γ, which is a triangulated  ...  Realizing stable categories as derived-orbit categories In this section, we compare the stable categories of self-injective algebras and derivedorbit categories.  ... 
arXiv:1201.5487v1 fatcat:ffqd4cyz4baixba2v4sfespfwi

Noncommutative stable homotopy and stable infinity categories

Snigdhayan Mahanta
2015 Journal of Topology and Analysis (JTA)  
We show that the triangulated category NSH is topological as defined by Schwede using the formalism of (stable) infinity categories.  ...  The noncommutative stable homotopy category NSH is a triangulated category that is the universal receptacle for triangulated homology theories on separable C^*-algebras.  ...  stable homotopy category.  ... 
doi:10.1142/s1793525315500077 fatcat:zxlajtrcyrblrg4duzayvdn7ca

Stable model categories are categories of modules

Stefan Schwede, Brooke Shipley
2003 Topology  
A stable model category is a setting for homotopy theory where the suspension functor is invertible.  ...  The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring.  ...  the original stable model category C.  ... 
doi:10.1016/s0040-9383(02)00006-x fatcat:3zf5q5nic5bubdadti3t233cum

Stable categories of Cohen-Macaulay modules and cluster categories [article]

Claire Amiot
2015 arXiv   pre-print
As a byproduct, we give a triangle equivalence between the stable category of graded Cohen-Macaulay R-modules and the derived category of Λ.  ...  By Auslander's algebraic McKay correspondence, the stable category of Cohen-Macaulay modules over a simple singularity is equivalent to the 1-cluster category of the path algebra of a Dynkin quiver (i.e  ...  In [IO09] , it was shown that the stable categories of modules over d-preprojective algebras of (d − 1)-representationfinite algebras are triangle equivalent to generalized d-cluster categories of stable  ... 
arXiv:1104.3658v3 fatcat:cfoc26l4sbb4jlydl5ebxf7jna

The N-Stable Category [article]

Jeremy R. B. Brightbill, Vanessa Miemietz
2021 arXiv   pre-print
We identify this "N-stable category" via the monomorphism category and prove Buchweitz's theorem for N-complexes over a Frobenius exact abelian category.  ...  A well-known theorem of Buchweitz provides equivalences between three categories: the stable category of Gorenstein projective modules over a Gorenstein algebra, the homotopy category of acyclic complexes  ...  ); and c) stab(A), the stable category of A.  ... 
arXiv:2109.07728v2 fatcat:rxqourb5tvht7chtxdlbd25pxa

Balance in Stable Categories

Pedro Nicolás, Manuel Saorín
2009 Algebras and Representation Theory  
We study when the stable category of an abelian category modulo a full additive subcategory is balanced and, in case the subcategory is functorially finite, we study a weak version of balance.  ...  The results in this second case apply very neatly to (generalizations of) hereditary abelian categories.  ...  stable categories.  ... 
doi:10.1007/s10468-008-9113-6 fatcat:3u6lmvmq55emzppytifs362hnm

Recollements in stable ∞-categories [article]

Domenico Fiorenza, Fosco Loregian
2016 arXiv   pre-print
From this we deduce a generalized associative property for n-fold gluing t_0...t_n, valid in any stable ∞-category.  ...  Such a classical result, well-known in the setting of triangulated categories, is recasted in the setting of stable ∞-categories and the properties of the associated (∞-categorical) factorization systems  ...  Version 1 of the present paper is sensibly different from the present one; the unexpected (and actually undue) symmetric behavior of stable recollements (Lemma 4.3 of version 1, therein called the Rorschach  ... 
arXiv:1507.03913v2 fatcat:2nohutatlvgyljezbudd6p4zli

Stable categories and reconstruction [article]

Jeremy Rickard, Raphael Rouquier
2010 arXiv   pre-print
More precisely, given two self-injective algebras A and B and an equivalence between their stable categories, consider the set S of images of simple B-modules inside the stable category of A.  ...  That set satisfies some obvious properties of Hom-spaces and it generates the stable category of A. Keep now only S and A. Can B be reconstructed ?  ...  More precisely, given two self-injective algebras A and B and an equivalence between their stable categories, consider the set S of images of simple B-modules inside the stable category of A.  ... 
arXiv:1008.1976v1 fatcat:wwkjdyhmoveabf3tqnxcjxvqgu

Stable categories and reconstruction

Jeremy Rickard, Raphaël Rouquier
2017 Journal of Algebra  
More precisely, given two self-injective algebras A and B and an equivalence between their stable categories, consider the set S of images of simple B-modules inside the stable category of A.  ...  Let M be an A-module with a decomposition M ∼ M 1 ⊕ M 2 in the stable category.  ...  Assume now that Hom There is a surjective map g : M → M/M 1 with filtrable kernel such that the composition M can −→ M g −→ M/M 1 is equal to the canonical map M → M/M 1 in the stable category, by Proposition  ... 
doi:10.1016/j.jalgebra.2016.05.018 fatcat:2cj3q253dbbzrjhzxou2mokkuy

The Stable Monomorphism Category of a Frobenius category [article]

Xiao-Wu Chen
2009 arXiv   pre-print
For a Frobenius abelian category A, we show that the category Mon(A) of monomorphisms in A is a Frobenius exact category; the associated stable category Mon(A) modulo projective objects is called the stable  ...  As an application, we give two characterizations to the stable category of Ringel-Schmidmeier (RS3).  ...  We show that Mon(A) is a Frobenius exact category and then the stable category Mon(A) modulo projective objects is triangulated; it is called the stable monomorphism category of A.  ... 
arXiv:0911.1987v2 fatcat:4iql6vwv4bfivkxnx4nuibnjky

Differential Graded Categories are k-linear Stable Infinity Categories [article]

Lee Cohn
2016 arXiv   pre-print
We describe a comparison between pretriangulated differential graded categories and certain stable infinity categories.  ...  We show the underlying infinity category of this model category is equivalent to the infinity category of k-linear stable infinity categories.  ...  stable ∞-categories.  ... 
arXiv:1308.2587v2 fatcat:z6fgeufggjgiriaow6oruitwb4

The stable monomorphism category of a Frobenius category

Xiao-Wu Chen
2011 Mathematical Research Letters  
For a Frobenius abelian category A, we show that the category Mon(A) of monomorphisms in A is a Frobenius exact category; the associated stable category Mon(A) modulo projective objects is called the stable  ...  As an application, we give two characterizations to the stable category of Ringel-Schmidmeier.  ...  stable monomorphism category Mon(A).  ... 
doi:10.4310/mrl.2011.v18.n1.a9 fatcat:xqtvl36nczewlcyewygt57n7nm
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