Results 1  10
of
95
Stable model categories are categories of modules
 TOPOLOGY
, 2003
"... 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. In this paper we develop methods for ..."
Abstract

Cited by 78 (16 self)
 Add to MetaCart
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. In this paper we develop methods for deciding when two stable model categories represent ‘the same homotopy theory’. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a ‘ring spectrum with several objects’, i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard’s work on derived equivalent rings. We also include a proof of the model category equivalence of modules over the EilenbergMac Lane spectrum HR and (unbounded) chain complexes of Rmodules for a ring R.
Algebraic topology and modular forms
 Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), Higher Ed
, 2002
"... The problem of describing the homotopy groups of spheres has been fundamental to algebraic topology for around 80 years. There were periods when specific computations were important and periods when the emphasis favored theory. Many ..."
Abstract

Cited by 24 (2 self)
 Add to MetaCart
The problem of describing the homotopy groups of spheres has been fundamental to algebraic topology for around 80 years. There were periods when specific computations were important and periods when the emphasis favored theory. Many
Morita theory in abelian, derived and stable model categories, Structured ring spectra
 London Math. Soc. Lecture Note Ser
, 2004
"... These notes are based on lectures given at the Workshop on Structured ring spectra and ..."
Abstract

Cited by 19 (0 self)
 Add to MetaCart
These notes are based on lectures given at the Workshop on Structured ring spectra and
Calculus III: Taylor Series
, 2003
"... We study functors from spaces to spaces or spectra that preserve weak homotopy equivalences. For each such functor we construct a universal nexcisive approximation, which may be thought of as its nexcisive part. Homogeneous functors, meaning nexcisive functors with trivial (n − 1)excisive part, ..."
Abstract

Cited by 18 (0 self)
 Add to MetaCart
We study functors from spaces to spaces or spectra that preserve weak homotopy equivalences. For each such functor we construct a universal nexcisive approximation, which may be thought of as its nexcisive part. Homogeneous functors, meaning nexcisive functors with trivial (n − 1)excisive part, can be classified: they correspond to symmetric functors of n variables that are reduced and 1excisive in each variable. We discuss some important examples, including the identity functor and Waldhausen’s algebraic Ktheory.
Moduli Spaces of Commutative Ring Spectra
, 2003
"... Let E be a homotopy commutative ring spectrum, and suppose the ring of cooperations E#E is flat over E# . We wish to address the following question: given a commutative E# algebra A in E#Ecomodules, is there an E# ring spectrum X with E#X = A as comodule algebras? We will formulate this as ..."
Abstract

Cited by 17 (0 self)
 Add to MetaCart
Let E be a homotopy commutative ring spectrum, and suppose the ring of cooperations E#E is flat over E# . We wish to address the following question: given a commutative E# algebra A in E#Ecomodules, is there an E# ring spectrum X with E#X = A as comodule algebras? We will formulate this as a moduli problem, and give a way  suggested by work of Dwyer, Kan, and Stover  of dissecting the resulting moduli space as a tower with layers governed by appropriate AndreQuillen cohomology groups. A special case is A = E#E itself. The final section applies this to discuss the LubinTate or Morava spectra En .
PARTITION COMPLEXES, TITS BUILDINGS AND SYMMETRIC PRODUCTS
, 1999
"... A partition complex is a geometric object associated to the poset of equivalence relations on a ®nite set; a Tits building is a geometric object associated to the poset of subspaces of a vector space. The two objects are formally alike, but aside from that they do not seem to have much in common. Fo ..."
Abstract

Cited by 16 (3 self)
 Add to MetaCart
A partition complex is a geometric object associated to the poset of equivalence relations on a ®nite set; a Tits building is a geometric object associated to the poset of subspaces of a vector space. The two objects are formally alike, but aside from that they do not seem to have much in common. For instance, they have
Homology and cohomology of E∞ ring spectra
 MATHEMATISCHE ZEITSCHRIFT
, 2005
"... Every homology or cohomology theory on a category of E∞ ring spectra is Topological André–Quillen homology or cohomology with appropriate coefficients. Analogous results hold more generally for categories of algebras over operads. ..."
Abstract

Cited by 16 (0 self)
 Add to MetaCart
Every homology or cohomology theory on a category of E∞ ring spectra is Topological André–Quillen homology or cohomology with appropriate coefficients. Analogous results hold more generally for categories of algebras over operads.
Local cohomology and support for triangulated categories, arXiv: math.KT/0702610
"... Abstract. We propose a new method for defining a notion of support for objects in any compactly generated triangulated category admitting small coproducts. This approach is based on a construction of local cohomology functors on triangulated categories, with respect to a central ring of operators. S ..."
Abstract

Cited by 15 (7 self)
 Add to MetaCart
Abstract. We propose a new method for defining a notion of support for objects in any compactly generated triangulated category admitting small coproducts. This approach is based on a construction of local cohomology functors on triangulated categories, with respect to a central ring of operators. Suitably specialized one recovers, for example, the theory for commutative noetherian rings due to Foxby and Neeman, the theory of Avramov and Buchweitz for complete intersection local rings, and varieties for representations of finite groups according to Benson, Carlson, and Rickard. We give explicit examples of objects whose triangulated support and cohomological support differ. In the case of group representations, this leads to a counterexample to a conjecture of Benson.