Finiteness Structures in Chromatic Derived Categories

The project lies in stable homotopy theory which is a major part of modern algebraic topology. Within stable homotopy theory the core problem is the structure of the stable homotopy category. Algebraic topology is a particularly interesting branch of mathematics since it strongly links geometry and algebra. Geometric objects are studied with algebraic methods and through stable homotopy theory, the geometric and algebraic aspects are brought into a common framework. It has a multidisciplinary role in mathematics, with applications to differential geometry, homological algebra, algebraic geometry and mathematical physics amongst others. The stable homotopy category is the category of spectra up to homotopy. A spectrum can be thought of as an infinite loop space or as a topological space up to suspension. Spectra are also the representing objects of cohomology theories. The stable homotopy category is triangulated making it richer in structure than the analogous homotopy category of topological spaces.A tool to study the stable homotopy category is Bousfield localisation. For a spectrum/cohomology theory E, one can consider the E-local stable homotopy category. It has the same objects as the stable homotopy category, but those morphisms of spectra that induce isomorphisms in the cohomology theory E are formally inverted. Hence they become isomorphisms in the E-local homotopy category. The result is a category that is especially sensitive to phenomena associated with the cohomology theory E. If two spectra have an E-homology isomorphism between them, they are also isomorphic objects in the E-local stable homotopy category. The main focus of study concerns two highly specific families of cohomology theories and involves comparing the triangulated structures of their localisations. This becomes particularly relevant when one considers some central examples of cohomology theories, namely so-called chromatic cohomology theories: the Johnson-Wilson theories E(n) and the Morava K-theories K(n) for each n. Localisation with respect to these theories carries important structural information about the stable homotopy category itself. For example, the Thick Subcategory Theorem says that the K(n)-localisations are the elementary building blocks of the stable homotopy category. Further, the theories E(n) and K(n) detect nilpotent or periodic maps of spectra. Crucially, the Chromatic Convergence Theorem states that a spectrum is the homotopy limit of all its E(n)-localisations and thus, as n increases, the E(n)-local stable homotopy category gives a better and better approximation to the stable homotopy category itself. The outstanding open problem in this area of homotopy theory is the Telescope Conjecture. It asserts a major finiteness property for the E(n)-local stable homotopy category, and implies that the category of E(n)-acyclic pectra is generated by finite spectra (i.e. spectra that can be built from the sphere in finitely many steps). For n=1, the conjecture is known to be true. For n>1 it remains open. We plan to relate the conjecture to innovative notions in the subject whose applications have not yet been fully exploited. This will give new strategies for studying what is generally regarded as a highly formidable conjecture.In very recent years, the Telescope Conjecture has become a focus of interest to researchers dealing with model categories from an algebraic point of view. However, these approaches do not exploit the full potential of classical homotopy theory. On the other hand, the homotopy-theoretic approaches came to a halt before model categories became the powerful tool in algebraic topology that they are now. Our goal here is to combine those two and hence add some new and valuable insight to this open problem.