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18
Weight filtrations via commuting automorphisms
 KTheory
, 1995
"... Abstract. We consider a filtration of the Ktheory space for a regular noetherian ring proposed by Goodwillie and Lichtenbaum and show that its successive quotients are geometric realizations of explicit simplicial abelian groups. The filtration in weight t involves ttuples of commuting automorphis ..."
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Cited by 25 (2 self)
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Abstract. We consider a filtration of the Ktheory space for a regular noetherian ring proposed by Goodwillie and Lichtenbaum and show that its successive quotients are geometric realizations of explicit simplicial abelian groups. The filtration in weight t involves ttuples of commuting automorphisms of projective Rmodules. It remains to show that the Adams operations act appropriately on the filtration. 1. Introduction. Much of the interest in algebraic Ktheory today arises because it points the way toward development of a motivic cohomology theory. According to the philosophy of Beilinson, if R is a regular noetherian ring and X = Spec(R), then motivic cohomology groups H m (X, Z(t)), once defined, ought to appear in a spectral sequence
Quillen Closed Model Structures for Sheaves
, 1995
"... In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2groupoids, the category of bisim ..."
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Cited by 14 (0 self)
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In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2groupoids, the category of bisimplicial sheaves and the category of simplicial sheaves of groupoids. Subsequently, the homotopy theories of these categories are related to the homotopy theory of simplicial sheaves. 1 Introduction There are two ways of trying to generalize the well known closed model structure on the category of simplicial sets to the category of simplicial objects in a Grothendieck topos. One way is to concentrate on the local aspect, and to use the Kanfibrations as a starting point. In [14] Heller showed that for simplicial presheaves there is a local (there called right) closed model structure. In [2] K. Brown showed that for a topological space X the category of "locally fibrant" sheaves of spectra on ...
Exterior Power Operations on Higher KTheory
"... We construct operations on higher algebraic Kgroups induced by operations such as exterior power on any suitable exact category, without appeal to the plusconstruction of Quillen. ..."
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Cited by 12 (3 self)
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We construct operations on higher algebraic Kgroups induced by operations such as exterior power on any suitable exact category, without appeal to the plusconstruction of Quillen.
Noncommutative Localization And Chain Complexes I. Algebraic K And LTheory
, 2001
"... The noncommutative (Cohn) localization # 1 R of a ring R is defined for any collection # of morphisms of f.g. projective left Rmodules. We exhibit # 1 R as the endomorphism ring of R in an appropriate triangulated category. We use this expression to prove that if Tor 1 then every bounded f. ..."
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Cited by 6 (1 self)
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The noncommutative (Cohn) localization # 1 R of a ring R is defined for any collection # of morphisms of f.g. projective left Rmodules. We exhibit # 1 R as the endomorphism ring of R in an appropriate triangulated category. We use this expression to prove that if Tor 1 then every bounded f.g.
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
Weight filtrations in algebraic Ktheory
 In Motives, volume 55 of Proceedings of Symposia in Pure Mathematics
, 1994
"... Abstract. We survey briefly some of the Ktheoretic background related to the theory of mixed motives and motivic cohomology. 1. Introduction. The recent search for a motivic cohomology theory for varieties, described elsewhere in this volume, has been largely guided by certain aspects of the higher ..."
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Cited by 5 (2 self)
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Abstract. We survey briefly some of the Ktheoretic background related to the theory of mixed motives and motivic cohomology. 1. Introduction. The recent search for a motivic cohomology theory for varieties, described elsewhere in this volume, has been largely guided by certain aspects of the higher algebraic Ktheory developed by Quillen in 1972. It is the purpose of this article to explain the sense in which the previous statement is true, and to explain
Adams Operations for Bivariant KTheory and a Filtration Using Projective Links
, 1997
"... We establish the existence of Adams operations on the members of a filtration of Ktheory which is defined using products of projective lines. We also show that this filtration induces the gamma filtration on the rational Kgroups of a smooth variety over a field of characteristic zero. ..."
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We establish the existence of Adams operations on the members of a filtration of Ktheory which is defined using products of projective lines. We also show that this filtration induces the gamma filtration on the rational Kgroups of a smooth variety over a field of characteristic zero.
HIGHER ARITHMETIC KTHEORY
"... The aim of this paper is to provide a new definition of higher Ktheory in Arakelov geometry and to show that it enjoys the same formal properties as the higher algebraic Ktheory of schemes. Let X be a proper arithmetic variety, that is, let X be a regular scheme which is flat, proper ..."
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Cited by 2 (1 self)
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The aim of this paper is to provide a new definition of higher Ktheory in Arakelov geometry and to show that it enjoys the same formal properties as the higher algebraic Ktheory of schemes. Let X be a proper arithmetic variety, that is, let X be a regular scheme which is flat, proper
Erratum To "The Loop Space Of The QConstruction"
, 1996
"... map jZj F \Gamma! jY j is a quasifibration, and the fiber over any point in the open simplex of jY j corresponding to ae is homeomorphic to jZ ae j. it reads Then the map jjA 7! jZ(A; \Delta)j jj F \Gamma! jY j is a quasifibration, and the fiber over any point in the open simplex of jjY jj co ..."
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map jZj F \Gamma! jY j is a quasifibration, and the fiber over any point in the open simplex of jY j corresponding to ae is homeomorphic to jZ ae j. it reads Then the map jjA 7! jZ(A; \Delta)j jj F \Gamma! jY j is a quasifibration, and the fiber over any point in the open simplex of jjY jj corresponding to ae is homeomorphic to jZ ae j. The proof then proceeds as sketched originally, with the analogue of the first diagram in Quillen's proof being the following map of cocartesian squares of spaces, which results easily from Segal's description of jjY jj. @ \Del
CATEGORICAL AND SIMPLICIAL CONSTRUCTIONS AND DUALIZATION IN ALGEBRAIC KTHEORY
"... Abstract. Given an exact category A, we introduce a bisimplicial set W A which is selfdual and contains both the Gconstruction of A and its dualization. We prove that the embeddings of G.A and G op.A into W A are homotopy equivalences. If A is an exact category with duality, we calculate the induc ..."
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Abstract. Given an exact category A, we introduce a bisimplicial set W A which is selfdual and contains both the Gconstruction of A and its dualization. We prove that the embeddings of G.A and G op.A into W A are homotopy equivalences. If A is an exact category with duality, we calculate the induced action of duality on K1(A). We also survey on the selfduality property of some known constructions. Exact categories with duality is an environment for the theory of Witt groups of schemes. On the other hand, one can study their Ktheory. In summer ’99 Marek Szyjewski pointed out to me that it might be useful for his work on Witt groups [Sz12] to calculate the induced action of duality on K1 by using my description of K1 in terms of double short exact sequences [Ne24]. Though the answer [A f1 ⇒ B f2 g1 ⇒ C] g2