Higher Toda brackets and the Adams spectral sequence in triangulated categories
@article{Christensen2015HigherTB, title={Higher Toda brackets and the Adams spectral sequence in triangulated categories}, author={J. Daniel Christensen and Martin Frankland}, journal={arXiv: Algebraic Topology}, year={2015} }
The Adams spectral sequence is available in any triangulated category equipped with a projective or injective class. Higher Toda brackets can also be defined in a triangulated category, as observed by B. Shipley based on J. Cohen's approach for spectra. We provide a family of definitions of higher Toda brackets, show that they are equivalent to Shipley's, and show that they are self-dual. Our main result is that the Adams differential $d_r$ in any Adams spectral sequence can be expressed as an…
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References
SHOWING 1-10 OF 48 REFERENCES
The decomposition of stable homotopy.
- MathematicsProceedings of the National Academy of Sciences of the United States of America
- 1967
It is shown that, under the operation of higher Toda bracket (which will be defined in ? 2), certain classes in the stable homotopy ring of spheres w*(S) generate all of w-(S), called the Hopf classes, which are those detected by primary cohomology operations.
On the differentials in the Adams spectral sequence
- Mathematics
- 1964
In this paper, we shall prove a result which identifies the differentials in the Adams spectral sequence (see (1,2)) with certain cohomology operations of higher kinds, in the sense of (4). This…
Ideals in triangulated categories: phantoms, ghosts and skeleta
- Mathematics
- 1998
We begin by showing that in a triangulated category, specifying a projective class is equivalent to specifying an ideal I of morphisms with certain properties and that if I has these properties, then…
Homology and fibrations I Coalgebras, cotensor product and its derived functors
- Mathematics
- 1965
The s tudy of the relations between the homology structure of the base space, the total space and the fiber of a fibration offers ample opportunity for application of homological algebra. This series…
On the algebraic classification of module spectra
- Mathematics
- 2013
Using methods developed by Franke in [7], we obtain algebraic classification results for modules over certain symmetric ring spectra (S ‐algebras). In particular, for any symmetric ring spectrum R…
Triangulated Categories: Algebraic versus topological triangulated categories
- Mathematics
- 2010
The most commonly known triangulated categories arise from chain complexes in an abelian category by passing to chain homotopy classes or inverting quasi-isomorphisms. Such examples are called…
Universal Toda brackets of ring spectra
- Mathematics
- 2007
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…
triangulated categories
- Mathematics
- 2007
For a self-orthogonal module T , the relation between the quotient triangulated category Db(A)/K b(addT ) and the stable category of the Frobenius category of T -Cohen-Macaulay modules is…
Uniqueness of Massey Products on the Stable Homotopy of Spheres
- MathematicsCanadian Journal of Mathematics
- 2012
The product on the stable homotopy ring of spheres π* s can be defined by composing, smashing or joining maps. Each of these three points of view is used in Section 2 to define Massey products on π*…
An Algebraic Model for Rational S1‐Equivariant Stable Homotopy Theory
- Mathematics
- 2001
Greenlees dened an abelian categoryA whose derived category is equivalent to the rational S 1 -equivariant stable homotopy category whose objects represent rational S 1 - equivariant cohomology…