Local Stable Homotopy Category Research Articles

Invertible objects in Franke's comodule categories

We study the Picard group of Franke's category of quasi-periodic $E_0E$-comodules for $E$ a 2-periodic Landweber exact cohomology theory of height $n$ such as Morava $E$-theory, showing that for $2p-2 > n^2+n$, this group is infinite cyclic, generated by the suspension of the unit. This is analogous to, but independent of, the corresponding calculations by Hovey and Sadofsky in the $E$-local stable homotopy category. We also give a computation of the Picard group of $I_n$-complete quasi-periodic $E_0E$-comodules when $E$ is Morava $E$-theory, as studied by Barthel-Schlank-Stapleton for $2p-2 \ge n^2$ and $p-1 \nmid n$, and compare this to the Picard group of the $K(n)$-local stable homotopy category, showing that they agree up to extension.

Auslander–Reiten Sequences, Brown–Comenetz Duality, and the K(n)-local Generating Hypothesis

In this paper, we construct a version of Auslander–Reiten sequences for the K(n)-local stable homotopy category. In particular, the role of the Auslander–Reiten translation is played by the local Brown–Comenetz duality functor. As an application, we produce counterexamples to the K(n)-local generating hypothesis for all heights n > 0 and all primes. Furthermore, our methods apply to other triangulated categories, as for example the derived category of quasi-coherent sheaves on a smooth projective scheme.

Monoidality of Frankeʼs exotic model

We discuss the monoidal structure on Frankeʼs algebraic model for the K ( p ) -local stable homotopy category at odd primes and show that its Picard group is isomorphic to the integers.

On the algebraic classification of $K$-local spectra

In 1996, Jens Franke proved the equivalence of certain triangulated categories possessing an Adams spectral sequence. One particular application of this theorem is that the K_(p)-local stable homotopy category at an odd prime can be described as the derived category of an abelian category. We explain this proof from a topologist’s point of view.

Rigidity and exotic models for theK–local stable homotopy category

Can the model structure of a stable model category be recovered from the triangulated structure of its homotopy category? This paper introduces a new positive example for this, namely the K-local stable homotopy at the prime 2. For odd primes, however, this is not true: we discuss a counterexample given by Jens Franke and show how such exotic models for the K-local stable homotopy category at odd primes can be detected.

Morava $E$-theory of filtered colimits

Morava E-theory E V n* (-) is a much-studied theory in algebraic topology, but it is not a homology theory in the usual sense, because it fails to preserve coproducts (resp. filtered homotopy colimits). The object of this paper is to construct a spectral sequence to compute the Morava E-theory of a coproduct (resp. filtered homotopy colimit). The E 2 -term of this spectral sequence involves the derived functors of direct sum (resp. filtered colimit) in an appropriate abelian category. We show that there are at most n - 1 (resp. n) of these derived functors. When n = 1, we recover the known result that homotopy commutes with an appropriate version of direct sum in the K(1)-local stable homotopy category.

The discrete module category for the ring of [formula omitted]-theory operations

We study the category of discrete modules over the ring of degree-zero stable operations in p -local complex K -theory, where p is an odd prime. We show that the K ( p ) -homology of any space or spectrum is such a module, and that this category is isomorphic to a category defined by Bousfield and used in his work on the K ( p ) -local stable homotopy category. We give a simple construction of cofree discrete modules and construct the analogue in the category of discrete modules of a four-term exact sequence due to Bousfield.

𝐸(2)-invertible spectra smashing with the Smith-Toda spectrum 𝑉(1) at the prime 3

Let L 2 L_2 denote the Bousfield localization functor with respect to the Johnson-Wilson spectrum E ( 2 ) E(2) . A spectrum L 2 X L_2X is called invertible if there is a spectrum Y Y such that L 2 X ∧ Y = L 2 S 0 L_2X\wedge Y=L_2S^0 . Hovey and Sadofsky, Invertible spectra in the E ( n ) E(n) -local stable homotopy category, showed that every invertible spectrum is homotopy equivalent to a suspension of the E ( 2 ) E(2) -local sphere L 2 S 0 L_2S^0 at a prime p > 3 p>3 . At the prime 3 3 , it is shown, A relation between the Picard group of the E ( n ) E(n) -local homotopy category and E ( n ) E(n) -based Adams spectral sequence, that there exists an invertible spectrum X X that is not homotopy equivalent to a suspension of L 2 S 0 L_2S^0 . In this paper, we show the homotopy equivalence v 2 3 : Σ 48 L 2 V ( 1 ) ≃ V ( 1 ) ∧ X v_2^3\colon \Sigma ^{48}L_2V(1)\simeq V(1)\wedge X for the Smith-Toda spectrum V ( 1 ) V(1) . In the same manner as this, we also show the existence of the self-map β : Σ 144 L 2 V ( 1 ) → L 2 V ( 1 ) \beta \colon \Sigma ^{144}L_2V(1)\to L_2V(1) that induces v 2 9 v_2^9 on the E ( 2 ) ∗ E(2)_* -homology.

On the chromatic tower

We fix a prime p and work in the p - local stable homotopy category. Then, Hopkins' chromatic splitting conjecture essentially predicts the information of a p - completed finite spectrum, obtained using the first ( n + 1) Morava K - theories K (0), K (1), . . . , K ( n ), may be obtained using a single higher Morava K - theory K ( m ). ( m ≥ n +1). However, in spite of its importance, this conjecture is very difficult and subtle. Actually, Devinatz noted the conjecture is false, as soon as we omit the finiteness assumption to include such a nice infinite spectrum as the p - completed BP spectrum. In this paper, we prove a result which reconciles Hopkins' chromatic splitting conjecture and Devinatz' observation about the p - completed BP spectrum. For the p - completion of "nice" spectra, including finite spectra and the BP - spectrum, our result essentially claims that the information obtained using the first ( n +1) Morava K - theories K (0), K (1), . . . , K ( n ), may be obtained using any m - k consecutive higher Morava K - theories K ( k + 1), K ( k + 2), . . . , K ( m - 1), K ( m ) with m - k ≥ n + s 0 + 1. Here, n 0 is the Hopkins-Ravenel (Hovey-Sadofsky) uniform horizontal vanishing line for the E ( n )-based standard Adams-Novikov spectral sequence.

Gross–Hopkins duality

We give a new and simpler proof of a result of Hopkins and Gross relating Brown-Comenetz duality to Spanier-Whitehead duality in the K(n)-local stable homotopy category.

Rigidity of the $K(1)$-local stable homotopy category

One goal of algebraic topology is to find algebraic invariants that classify topological spaces up to homotopy equivalence. The notion of homotopy is not only restricted to topology. It also appears in algebra, for example as a chain homotopy between two maps of chain complexes. The theory of model categories,introduced by D. Quillen [Qui06], provided us with a powerful common language to represent different notions of homotopy. Quillen’s work transformed algebraic topology from the study of topological spaces into a wider setting useful in many areas of mathematics, such as homological algebra and algebraic geometry, where homotopy theoretic approaches led to interesting results. In brief, a model structure on a category C is a choice of three distinguished classes of morphisms: weak equivalences, fibrations, and cofibrations, satisfying certain axioms. We can pass to the homotopy category Ho(C) associated to a model category C by inverting the weak equivalences, i.e. by making them into isomorphisms. While the axioms allow us to define the homotopy relations between classes of morphisms in C, the classes of fibrations and cofibrations provide us with a solution to the set-theoretic issues arising in general localisations of categories. Even though it is sometimes sufficient to work in the homotopy category, looking at the homotopy level alone does not provide us with enough higher order structure information. For example, homotopy (co)limits are not usually a homotopy invariant, and in order to define them we need the tools provided by the model category. This is where the question of rigidity may be asked: if we just had the structure of the homotopy category, how much of the underlying model structure can we recover? This question of rigidity has been investigated during the last decade, and an extremely small list of examples have been studied, which leaves us with a lot of open questions regarding this fascinating subject. Ou goal is to investigate one of the open questions which have not been answered before. In this thesis, we prove rigidity of the K(1)-local stable homotopy category Ho(LK(1)Sp) at p = 2. In other words, we show that recovering higher order structure information, which is meant to be lost on the homotopy level,is possible by just looking at the triangulated structure of Ho(LK(1)Sp). This new result does not only add one more example to the list of known examples of rigidity in stable homotopy theory, it is also the first studied case of rigidity in the world of Morava K-theory.