scholarly journals The smash product for derived categories in stable homotopy theory

2012 ◽  
Vol 230 (4-6) ◽  
pp. 1531-1556 ◽  
Author(s):  
Michael A. Mandell
Keyword(s):  
Homotopy Theory ◽  
Derived Categories ◽  
Stable Homotopy ◽  
Smash Product ◽  
Stable Homotopy Theory

scholarly journals Stable homotopical algebra and Γ-spaces

1999 ◽  
Vol 126 (2) ◽  
pp. 329-356 ◽  
Author(s):  
STEFAN SCHWEDE
Keyword(s):  
Homotopy Theory ◽  
Model Category ◽  
Homotopy Category ◽  
Stable Homotopy ◽  
Develop Model ◽  
Smash Product ◽  
Spectral Sequences ◽  
Homotopical Algebra ◽  
Stable Homotopy Theory ◽  
Set Up

In this paper we advertise the category of Γ-spaces as a convenient framework for doing ‘algebra’ over ‘rings’ in stable homotopy theory. Γ-spaces were introduced by Segal [Se] who showed that they give rise to a homotopy category equivalent to the usual homotopy category of connective (i.e. (−1)-connected) spectra. Bousfield and Friedlander [BF] later provided model category structures for Γ-spaces. The study of ‘rings, modules and algebras’ based on Γ-spaces became possible when Lydakis [Ly] introduced a symmetric monoidal smash product with good homotopical properties. Here we develop model category structures for modules and algebras, set up (derived) smash products and associated spectral sequences and compare simplicial modules and algebras to their Eilenberg–MacLane spectra counterparts.

v 1 - and v 2 -Periodicity in Stable Homotopy Theory

1981 ◽  
Vol 103 (4) ◽  
pp. 615 ◽  
Author(s):  
Donald M. Davis ◽  
Mark Mahowald
Keyword(s):  
Homotopy Theory ◽  
Stable Homotopy ◽  
Stable Homotopy Theory

Spectra of derived module homomorphisms

1987 ◽  
Vol 101 (2) ◽  
pp. 249-257 ◽  
Author(s):  
Alan Robinson
Keyword(s):  
Spectral Sequence ◽  
Homotopy Theory ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Stable Homotopy Theory ◽  
Ring Spectrum ◽  
Homotopy Invariant ◽  
New Construction

We introduce a new construction in stable homotopy theory. If F and G are module spectra over a ring spectrum E, there is no well-known spectrum of E-module homomorphisms from F to G. Such a construction would not be homotopy invariant, and therefore would not serve much purpose. We show that, provided the rings and modules have A∞ structures, there is a spectrum RHomE(F, G) of derived module homomorphisms which has very pleasant properties. It is homotopy invariant, exact in each variable, and its homotopy groups form the abutment of a hypercohomology-type spectral sequence.

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