Ideals in Triangulated Categories: Phantoms, Ghosts and Skeleta

TL;DR

This paper explores the structure of ideals in triangulated categories, introduces spectral sequences related to these ideals, and applies these concepts to stable homotopy theory, algebraic categories, and classical problems like the Hopf invariant.

Contribution

It establishes the equivalence between projective classes and ideals in triangulated categories, and connects these to spectral sequences and classical homotopy problems.

Findings

Ghosts induce a stable Lusternik-Schnirelmann category

Calculation of stable category for low-dimensional real projective spaces

Phantom maps can be described as lim^1 groups in derived categories

Abstract

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 so does each of its powers. We show how a projective class leads to an Adams spectral sequence and give some results on the convergence and collapsing of this spectral sequence. We use this to study various ideals. In the stable homotopy category we examine phantom maps, skeletal phantom maps, superphantom maps, and ghosts. (A ghost is a map which induces the zero map of homotopy groups.) We show that ghosts lead to a stable analogue of the Lusternik-Schnirelmann category of a space, and we calculate this stable analogue for low-dimensional real projective spaces. We also give a relation between ghosts and the Hopf and Kervaire invariant problems. In the case of A-infinity modules over an A-infinity ring spectrum, the ghost spectral sequence is a universal coefficient spectral sequence. From the phantom projective class we derive a generalized Milnor sequence for filtered diagrams of finite spectra, and from this it follows that the group of phantom maps from X to Y can always be described as a lim^1 group. The last two sections focus on algebraic examples. In the derived category of an abelian category we study the ideal of maps inducing the zero map of homology groups and find a natural setting for a result of Kelly on the vanishing of composites of such maps. We also explain how pure exact sequences relate to phantom maps in the derived category of a ring and give an example showing that phantoms can compose non-trivially.