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22
Cohomological quotients and smashing localizations
 Amer. J. Math
"... Abstract. The quotient of a triangulated category modulo a subcategory was defined by Verdier. Motivated by the failure of the telescope conjecture, we introduce a new type of quotients for any triangulated category which generalizes Verdier’s construction. Slightly simplifying this concept, the coh ..."
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Abstract. The quotient of a triangulated category modulo a subcategory was defined by Verdier. Motivated by the failure of the telescope conjecture, we introduce a new type of quotients for any triangulated category which generalizes Verdier’s construction. Slightly simplifying this concept, the cohomological quotients are flat epimorphisms, whereas the Verdier quotients are Ore localizations. For any compactly generated triangulated category S, a bijective correspondence between the smashing localizations of S and the cohomological quotients of the category of compact objects in S is established. We discuss some applications of this theory, for instance the problem of lifting chain complexes along a ring homomorphism. This is motivated by some consequences in algebraic Ktheory and demonstrates the relevance of the telescope
Failure Of Brown Representability In Derived Categories
"... Let T be a triangulated category with coproducts, T c T the full subcategory of compact objects in T. If T is the homotopy category of spectra, Adams proved the following in [1]: All homological functors fT c g op ! Ab are the restrictions of representable functors on T, and all natural tr ..."
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Let T be a triangulated category with coproducts, T c T the full subcategory of compact objects in T. If T is the homotopy category of spectra, Adams proved the following in [1]: All homological functors fT c g op ! Ab are the restrictions of representable functors on T, and all natural transformations are the restrictions of morphisms in T. It has been something of a mystery, to what extent this generalises to other triangulated categories. In [36], it was proved that Adams' theorem remains true as long as T c is countable, but can fail in general. The failure exhibited was that there can be natural transformations not arising from maps in T. A puzzling open problem remained: Is every homological functor the restriction of a representable functor on T? In a recent paper, Beligiannis [5] made some progress. But in this article, we settle the problem. The answer is no. There are examples of derived categories T = D(R) of rings, and homological functors fT c g op ! Ab which are not restrictions of representables. Contents
AuslanderReiten Theory Via Brown Representability
 KTHEORY
"... We develop an AuslanderReiten theory for triangulated categories which is based on Brown's representability theorem. ..."
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Cited by 9 (1 self)
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We develop an AuslanderReiten theory for triangulated categories which is based on Brown's representability theorem.
Decomposing Thick Subcategories Of The Stable Module Category
 Math. Ann
, 1999
"... . Let mod kG be the stable category of finitely generated modular representations of a finite group G over a field k. We prove a KrullSchmidt theorem for thick subcategories of modkG. It is shown that every thick tensorideal C of mod kG (i.e. a thick subcategory which is a tensor ideal) has a (usu ..."
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. Let mod kG be the stable category of finitely generated modular representations of a finite group G over a field k. We prove a KrullSchmidt theorem for thick subcategories of modkG. It is shown that every thick tensorideal C of mod kG (i.e. a thick subcategory which is a tensor ideal) has a (usually infinite) unique decomposition C = ` i2I C i into indecomposable thick tensorideals. This decomposition follows from a decomposition of the corresponding idempotent kGmodule EC into indecomposable modules. If C = CW is the thick tensorideal corresponding to a closed homogeneous subvariety W of the maximal ideal spectrum of the cohomology ring H (G; k), then the decomposition of C reflects the decomposition W = S n i=1 W i of W into connected components. Introduction In modular representation theory of finite groups, one frequently passes to the stable module category which is a triangulated category. Following ideas from stable homotopy theory, Benson, Carlson, and Rickard s...
Andrew Ranicki, Noncommutative localisation in algebraic K–theory
"... This article establishes, for an appropriate localisation of associative rings, a long exact sequence in algebraic K–theory. The main result goes as follows. Let A be an associative ring and let A − → B be the localisation with respect to a set σ of (B, B) maps between finitely generated projective ..."
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This article establishes, for an appropriate localisation of associative rings, a long exact sequence in algebraic K–theory. The main result goes as follows. Let A be an associative ring and let A − → B be the localisation with respect to a set σ of (B, B) maps between finitely generated projective A–modules. Suppose that Tor A n vanishes for all n> 0. View each map in σ as a complex (of length 1, meaning one nonzero map between two nonzero objects) in the category of perfect complexes Dperf (A). Denote by 〈σ 〉 the thick subcategory generated by these complexes. Then the canonical functor Dperf (A) − → Dperf (B) induces (up to direct factors) an equivalence D perf (A)/〈σ 〉 − → Dperf (B). As a consequence, one obtains a homotopy fibre sequence K(A, σ) −−−− → K(A) −−−− → K(B) (up to surjectivity of K0(A) − → K0(B)) of Waldhausen K–theory spectra. In subsequent articles [26, 27] we will present the K – and L–theoretic consequences of the main theorem in a form more suitable for the applications to surgery. For example if, in addition to the vanishing of Tor A n (B, B), we also assume that every map in σ is a monomorphism, then there is a description of the homotopy fiber of the
SUPPORT VARIETIES – AN IDEAL APPROACH
, 2005
"... Abstract. We define support varieties in an axiomatic setting using the prime spectrum of a lattice of ideals. A key observation is the functoriality of the spectrum and that this functor admits an adjoint. We assign to each ideal its support and can classify ideals in terms of their support. Applic ..."
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Abstract. We define support varieties in an axiomatic setting using the prime spectrum of a lattice of ideals. A key observation is the functoriality of the spectrum and that this functor admits an adjoint. We assign to each ideal its support and can classify ideals in terms of their support. Applications arise from studying abelian or triangulated tensor categories. Specific examples from algebraic geometry and modular representation theory are discussed, illustrating the power of this approach which is inspired by recent work of Balmer. Contents
Brown Representability And Flat Covers
, 1999
"... this paper is devoted to proving the main result. To this end we need to recall our assumptions on the triangulated category T : ..."
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this paper is devoted to proving the main result. To this end we need to recall our assumptions on the triangulated category T :
The AuslanderReiten formula for complexes of modules
"... Abstract. An AuslanderReiten formula for complexes of modules is presented. This formula contains as a special case the classical Auslander Reiten formula. The AuslanderReiten translate of a complex is described explicitly, and various applications are discussed. 1. ..."
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Abstract. An AuslanderReiten formula for complexes of modules is presented. This formula contains as a special case the classical Auslander Reiten formula. The AuslanderReiten translate of a complex is described explicitly, and various applications are discussed. 1.
Homotopy Theory Of Modules And Gorenstein Rings
, 1998
"... Homotopy Categories [12], and Brown Representability Theorem is applicable. We note that our results on injective homotopy generalize some recent results of Jørgensen [25]. In Section 6, inspired from the construction of the stable homotopy category of spectra [32], we study the existence of a stabl ..."
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Homotopy Categories [12], and Brown Representability Theorem is applicable. We note that our results on injective homotopy generalize some recent results of Jørgensen [25]. In Section 6, inspired from the construction of the stable homotopy category of spectra [32], we study the existence of a stable homotopy category associated to the projective or injective homotopy of a ring . Since the stable module categories are not in general triangulated, it is useful in many cases to replace them by their stabilizations [8], [19], which are triangulated categories, and this can be done in a universal way. We say that a ring has a projective, resp. injective, stable homotopy category if the stabilization of Mod(), resp. of Mod(), is compactly generated. We prove that in case is right Gorenstein in the sense of [8], and the ring is left coherent and right perfect or right Morita, then such a stable homotopy category exists and can be described as the triangulated stable category of CohenMacaula...
EXISTENCE OF GORENSTEIN PROJECTIVE RESOLUTIONS
, 2004
"... Abstract. Gorenstein rings are important to mathematical areas as diverse as algebraic geometry, where they encode information about singularities of spaces, and homotopy theory, through the concept of model categories. In consequence, the study of Gorenstein rings has led to the advent of a whole b ..."
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Abstract. Gorenstein rings are important to mathematical areas as diverse as algebraic geometry, where they encode information about singularities of spaces, and homotopy theory, through the concept of model categories. In consequence, the study of Gorenstein rings has led to the advent of a whole branch of homological algebra, known as Gorenstein homological algebra. This paper solves one of the open problems of Gorenstein homological algebra by showing that socalled Gorenstein projective resolutions exist over quite general rings, thereby enabling the definition of a Gorenstein version of derived functors. An application is given to the theory of Tate cohomology. Gorenstein rings are important mathematical objects originating in the work of Grothendieck and his pupils. The study of Gorenstein