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4 results

  • Introduction

  • David Barnes, Queen's University Belfast, Constanze Roitzheim, University of Kent, Canterbury
  • Book:
    Foundations of Stable Homotopy Theory
    Published online:
    09 March 2020
    Print publication:
    26 March 2020, pp 1-6

  • Summary

    In the crudest sense, stable homotopy theory is the study of those homotopy invariant constructions of spaces which are preserved by suspension. In this chapter, we show how there are naturally occurring situations which exhibit stable behaviour. We will discuss several historic attempts at constructing a “stable homotopy category” where this stable behaviour can be studied, and we relate these to the more developed notions of spectra and the Bousfield–Friedlander model structure. Of course, if one only wants to perform calculations of stable homotopy groups, to have certain spectral sequences or similar, then one does not need much of the formalism of model categories of spectra. But as soon as one wishes to move away from those tasks and consider other stable homotopy theories (such as G–equivariant stable homotopy theory for some group G) or to make serious use of a symmetric monoidal smash product in the context of “Brave New Algebra”, then the advantages of the more formal setup become overwhelming.

  • Dimensions of triangulated categories

  • Raphaël Rouquier
  • Journal:
    Journal of K-Theory / Volume 1 / Issue 2 / April 2008
    Published online by Cambridge University Press:
    30 November 2007, pp. 193-256
    Print publication:
    April 2008

  • We define a dimension for a triangulated category. We prove a representability Theorem for a class of functors on finite dimensional triangulated categories. We study the dimension of the bounded derived category of an algebra or a scheme and we show in particular that the bounded derived category of coherent sheaves over a variety has a finite dimension.