Results 1  10
of
25
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 ..."
Abstract

Cited by 33 (2 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 26 (0 self)
 Add to MetaCart
(Show Context)
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. ..."
Abstract

Cited by 16 (2 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 15 (1 self)
 Add to MetaCart
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
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. ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
(Show Context)
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.
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 ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
(Show Context)
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
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 ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
(Show Context)
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