4,915 research outputs found
Phantom maps and chromatic phantom maps
In the first part, we determine conditions on spectra X and Y under which
either every map from X to Y is phantom, or no nonzero maps are. We also
address the question of whether such all or nothing behaviour is preserved when
X is replaced with V smash X for V finite. In the second part, we introduce
chromatic phantom maps. A map is n-phantom if it is null when restricted to
finite spectra of type at least n. We define divisibility and finite type
conditions which are suitable for studying n-phantom maps. We show that the
duality functor W_{n-1} defined by Mahowald and Rezk is the analog of
Brown-Comenetz duality for chromatic phantom maps, and give conditions under
which the natural map Y --> W_{n-1}^2 Y is an isomorphism.Comment: 18 page
Higher Toda brackets and the Adams spectral sequence in triangulated categories
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 in any Adams
spectral sequence can be expressed as an -fold Toda bracket and as an
order cohomology operation. We also show how the result
simplifies under a sparseness assumption, discuss several examples, and give an
elementary proof of a result of Heller, which implies that the three-fold Toda
brackets in principle determine the higher Toda brackets.Comment: v2: Added Section 7, about an application to computing maps between
modules over certain ring spectra. Minor improvements elsewhere. v3: Minor
updates throughout; closely matches published versio
Ghost numbers of Group Algebras
Motivated by Freyd's famous unsolved problem in stable homotopy theory, the
generating hypothesis for the stable module category of a finite group is the
statement that if a map in the thick subcategory generated by the trivial
representation induces the zero map in Tate cohomology, then it is stably
trivial. It is known that the generating hypothesis fails for most groups.
Generalizing work done for -groups, we define the ghost number of a group
algebra, which is a natural number that measures the degree to which the
generating hypothesis fails. We describe a close relationship between ghost
numbers and Auslander-Reiten triangles, with many results stated for a general
projective class in a general triangulated category. We then compute ghost
numbers and bounds on ghost numbers for many families of -groups, including
abelian -groups, the quaternion group and dihedral -groups, and also give
a general lower bound in terms of the radical length, the first general lower
bound that we are aware of. We conclude with a classification of group algebras
of -groups with small ghost number and examples of gaps in the possible
ghost numbers of such group algebras.Comment: 28 pages; v2 improves introduction and has many other minor changes
throughout. appears in Algebras and Representation Theory, 201
Quillen model structures for relative homological algebra
An important example of a model category is the category of unbounded chain
complexes of R-modules, which has as its homotopy category the derived category
of the ring R. This example shows that traditional homological algebra is
encompassed by Quillen's homotopical algebra. The goal of this paper is to show
that more general forms of homological algebra also fit into Quillen's
framework. Specifically, a projective class on a complete and cocomplete
abelian category A is exactly the information needed to do homological algebra
in A. The main result is that, under weak hypotheses, the category of chain
complexes of objects of A has a model category structure that reflects the
homological algebra of the projective class in the sense that it encodes the
Ext groups and more general derived functors. Examples include the "pure
derived category" of a ring R, and derived categories capturing relative
situations, including the projective class for Hochschild homology and
cohomology. We characterize the model structures that are cofibrantly
generated, and show that this fails for many interesting examples. Finally, we
explain how the category of simplicial objects in a possibly non-abelian
category can be equipped with a model category structure reflecting a given
projective class, and give examples that include equivariant homotopy theory
and bounded below derived categories.Comment: 29 pages. v4: Published in Math. Proc. Cambridge Philos. Soc. v5:
Minor corrections to published version appear on last pag
- …