Results 1  10
of
17
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 ..."
Abstract

Cited by 14 (2 self)
 Add to MetaCart
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
Realizability of algebraic Galois extensions by strictly commutative ring spectra
"... Abstract. We discuss some of the basic ideas of Galois theory for commutative Salgebras originally formulated by John Rognes. We restrict attention to the case of finite Galois groups and to global Galois extensions. We describe parts of the general framework developed by Rognes. Central rôles are ..."
Abstract

Cited by 9 (6 self)
 Add to MetaCart
Abstract. We discuss some of the basic ideas of Galois theory for commutative Salgebras originally formulated by John Rognes. We restrict attention to the case of finite Galois groups and to global Galois extensions. We describe parts of the general framework developed by Rognes. Central rôles are played by the notion of strong duality and a trace mapping constructed by Greenlees and May in the context of generalized Tate cohomology. We give some examples where algebraic data on coefficient rings ensures strong topological consequences. We consider the issue of passage from algebraic Galois extensions to topological ones applying obstruction theories of Robinson and GoerssHopkins to produce topological models for algebraic Galois extensions and the necessary morphisms of commutative Salgebras. Examples such as the complex Ktheory spectrum as a KOalgebra indicate that more exotic phenomena occur in the topological setting. We show how in certain cases topological abelian Galois extensions are classified by the same Harrison groups as algebraic ones and this leads to computable Harrison groups for such spectra. We end by proving an analogue of Hilbert’s theorem 90 for the units associated with a Galois extension.
Universal Toda brackets of ring spectra
, 2006
"... Abstract. We construct and examine the universal Toda bracket of a highly structured ring spectrum R. This invariant of R is a cohomology class in the Mac Lane cohomology of the graded ring of homotopy groups of R which carries information about R and the category of Rmodule spectra. It determines ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
Abstract. We construct and examine the universal Toda bracket of a highly structured ring spectrum R. This invariant of R is a cohomology class in the Mac Lane cohomology of the graded ring of homotopy groups of R which carries information about R and the category of Rmodule spectra. It determines for example all triple Toda brackets of R and the first obstruction to realizing a module over the homotopy groups of R by an Rmodule spectrum. For periodic ring spectra, we study the corresponding theory of higher universal Toda brackets. The real and complex Ktheory spectra serve as our main examples. 1.
REALISIBILITY OF ALGEBRAIC GALOIS EXTENSIONS BY STRICTLY COMMUTATIVE RING SPECTRA
, 2004
"... Abstract. We describe some of the basic ideas of Galois theory for commutative Salgebras originally formulated by John Rognes. We restrict attention to the case of finite Galois groups. We describe the general framework developed by Rognes. Central rôles are played by the notion of strong duality a ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Abstract. We describe some of the basic ideas of Galois theory for commutative Salgebras originally formulated by John Rognes. We restrict attention to the case of finite Galois groups. We describe the general framework developed by Rognes. Central rôles are played by the notion of strong duality and a trace or norm mapping constructed by Greenlees and May in the context of generalized Tate cohomology. We give some examples where algebraic data on coefficient rings ensures strong topological consequences. We consider the issue of passage from algebraic Galois extensions to topological ones applying obstruction theories of Robinson and GoerssHopkins to produce topological models for algebraic Galois extensions and the necessary morphisms of commutative Salgebras. Examples such as the Real Ktheory spectrum as a KOalgebra indicate that more exotic phenomena occur in the topological setting. We show how in certain cases topological abelian Galois extensions are classified by the same Harrison groups as algebraic ones and this leads to computable Harrison groups for such spectra. We consider the Tate spectrum associated to a GGalois extension and show that it is trivial, thus generalising an analogous result for algebraic Galois extensions. We end by proving an analogue of Hilbert’s theorem 90 for the units associated with a Galois extension.
INVERTIBLE MODULES FOR COMMUTATIVE SALGEBRAS WITH RESIDUE FIELDS
, 2004
"... Abstract. The aim of this note is to understand invertible modules over a commutative Salgebra in the sense of Elmendorf, Kriz, Mandell & May in some very wellbehaved cases. Our main result shows that as long as the commutative Salgebra R has ‘reductions mod m’ for all maximal ideals m ⊳ R∗, and ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract. The aim of this note is to understand invertible modules over a commutative Salgebra in the sense of Elmendorf, Kriz, Mandell & May in some very wellbehaved cases. Our main result shows that as long as the commutative Salgebra R has ‘reductions mod m’ for all maximal ideals m ⊳ R∗, and Noetherian localisations (R∗)m, then for every invertible Rmodule U, U ∗ = π∗U is an invertible R∗module.
HOMOLOGICAL DIMENSIONS OF RING SPECTRA
"... Abstract. We define homological dimensions for Salgebras, the generalized rings that arise in algebraic topology. We compute the homological dimensions of a number of examples, and establish some basic properties. The most difficult computation is the global dimension of real Ktheory KO and its co ..."
Abstract
 Add to MetaCart
Abstract. We define homological dimensions for Salgebras, the generalized rings that arise in algebraic topology. We compute the homological dimensions of a number of examples, and establish some basic properties. The most difficult computation is the global dimension of real Ktheory KO and its connective version ko at the prime 2. We show that the global dimension of KO is 1, 2, or 3, and the global dimension of ko is 4 or 5.
STABLE COHOMOLOGY OVER LOCAL RINGS
, 2007
"... Abstract. For a commutative noetherian ring R with residue field k stable cohomology modules dExt n R (k, k) have been defined for each n ∈ Z, but their meaning has remained elusive. It is proved that the krank of any dExt n R (k, k) characterizes important properties of R, such as being regular, c ..."
Abstract
 Add to MetaCart
Abstract. For a commutative noetherian ring R with residue field k stable cohomology modules dExt n R (k, k) have been defined for each n ∈ Z, but their meaning has remained elusive. It is proved that the krank of any dExt n R (k, k) characterizes important properties of R, such as being regular, complete intersection, or Gorenstein. These numerical characterizations are based on results concerning the structure of Zgraded kalgebra carried by stable cohomology. It is shown that in many cases it is determined by absolute cohomology through a canonical homomorphism of algebras ExtR (k, k) → dExt R (k, k). Some techniques developed in the paper are applicable to the study of stable cohomology