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32
Algebraic Cobordism
, 2007
"... 1.1. Pushforwards, pullbacks and first Chern classes 20 ..."
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Cited by 35 (3 self)
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1.1. Pushforwards, pullbacks and first Chern classes 20
Floer's Infinite Dimensional Morse Theory And Homotopy Theory
, 1996
"... This paper is a progress report on our efforts to understand the homotopy theory underlying Floer homology; its objectives are as follows: ..."
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Cited by 28 (6 self)
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This paper is a progress report on our efforts to understand the homotopy theory underlying Floer homology; its objectives are as follows:
Homotopy theory of comodules over a Hopf algebroid
, 2003
"... Given a good homology theory E and a topological space X, E∗X is not just an E∗module but also a comodule over the Hopf algebroid (E∗, E∗E). We establish a framework for studying the homological algebra of comodules over a wellbehaved Hopf algebroid (A, Γ). That is, we construct ..."
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Cited by 13 (3 self)
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Given a good homology theory E and a topological space X, E∗X is not just an E∗module but also a comodule over the Hopf algebroid (E∗, E∗E). We establish a framework for studying the homological algebra of comodules over a wellbehaved Hopf algebroid (A, Γ). That is, we construct
Algebraic Ktheory of rings from a topological viewpoint
"... Because of its strong interaction with almost every part of pure mathematics, algebraic Ktheory has had a spectacular development since its origin in the late fifties. The objective of this paper is to provide the basic definitions of the algebraic Ktheory of rings and an overview of the main clas ..."
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Cited by 5 (1 self)
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Because of its strong interaction with almost every part of pure mathematics, algebraic Ktheory has had a spectacular development since its origin in the late fifties. The objective of this paper is to provide the basic definitions of the algebraic Ktheory of rings and an overview of the main classical theorems. Since the algebraic Kgroups of a ring R are the homotopy groups of a topological space associated with the general linear group over R, it is obvious that many general results follow from arguments from homotopy theory. This paper is essentially devoted to some of them: it explains in particular how methods from stable homotopy theory, group cohomology and Postnikov theory can be used in algebraic Ktheory.
Commuting homotopy limits and smash products
 KTheory
, 2003
"... Abstract. In general the processes of taking a homotopy inverse limit of a diagram of spectra and smashing spectra with a fixed space do not commute. In this paper we investigate under what additional assumptions these two processes do commute. In fact we deal with an equivariant generalization that ..."
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Cited by 4 (3 self)
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Abstract. In general the processes of taking a homotopy inverse limit of a diagram of spectra and smashing spectra with a fixed space do not commute. In this paper we investigate under what additional assumptions these two processes do commute. In fact we deal with an equivariant generalization that involves spectra and smash products over the orbit category of a discrete group. Such a situation naturally occurs if one studies the equivariant homology theory associated to topological cyclic homology. The main theorem of this paper will play a role in the generalization of the results obtained by Bökstedt, Hsiang and Madsen about the algebraic Ktheory Novikov Conjecture to the assembly map for the family of virtually cyclic subgroups. to Hyman Bass on his seventieth birthday 1.
Stable étale realization and étale cobordism
 Adv. Math
"... We show that there is a stable homotopy theory of profinite spaces and use it for two main applications. On the one hand we construct an étale topological realization of the stable A 1homotopy theory of smooth schemes over a base field of arbitrary characteristic in analogy to the complex realizati ..."
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Cited by 4 (4 self)
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We show that there is a stable homotopy theory of profinite spaces and use it for two main applications. On the one hand we construct an étale topological realization of the stable A 1homotopy theory of smooth schemes over a base field of arbitrary characteristic in analogy to the complex realization functor for fields of characteristic zero. On the other hand we get a natural setting for étale cohomology theories. In particular, we define and discuss an étale topological cobordism theory for schemes. It is equipped with an AtiyahHirzebruch spectral sequence starting from étale cohomology. Finally, we construct maps from algebraic to étale cobordism and discuss algebraic cobordism with finite coefficients over an algebraically closed field after inverting a Bott element. 1
KrullSchmidt decompositions for thick subcategories
, 2005
"... Abstract. Following Krause [Kra99], we prove KrullSchmidt theorems for thick subcategories of various triangulated categories: derived categories of rings, noetherian stable homotopy categories, stable module categories over Hopf algebras, and the stable homotopy category of spectra. In all these c ..."
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Cited by 3 (1 self)
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Abstract. Following Krause [Kra99], we prove KrullSchmidt theorems for thick subcategories of various triangulated categories: derived categories of rings, noetherian stable homotopy categories, stable module categories over Hopf algebras, and the stable homotopy category of spectra. In all these categories, it is shown that the thick ideals of small objects decompose uniquely into indecomposable thick ideals. Some consequences of these decomposition results are also discussed. In particular, it is shown that all these decompositions respect Ktheory. 1.
GEOMETRIC CRITERIA FOR LANDWEBER EXACTNESS
"... Abstract. The purpose of this paper is to give a new presentation of some of the main results concerning Landweber exactness in the context of the homotopy theory of stacks. We present two new criteria for Landweber exactness over a flat Hopf algebroid. The first criterion is used to classify stacks ..."
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Cited by 2 (2 self)
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Abstract. The purpose of this paper is to give a new presentation of some of the main results concerning Landweber exactness in the context of the homotopy theory of stacks. We present two new criteria for Landweber exactness over a flat Hopf algebroid. The first criterion is used to classify stacks arising from Landweber exact maps of rings. Using as extra input only Lazard’s theorem and Cartier’s classification of ptypical formal group laws, this result is then applied to deduce many of the main results concerning Landweber exactness in stable homotopy theory and to compute the Bousfield classes of certain BPalgebra spectra. The second criterion can be regarded as a generalization of the Landweber exact functor theorem and we use it to give a proof of the original theorem. 1.