Neil P. Strickland

Products on

In [2, Chapter V], Elmendorf, Kriz, Mandell and May (hereafter referred to as EKMM) use their new technology of modules over highly structured ring spectra to give new constructions of MU -modules such as BP , K(n) and so on, which makes it much easier to analyse product structures on these spectra. Unfortunately, their construction only works in its simplest form for modules over MU [12 ]∗ that are concentrated in degrees divisible by 4; this guarantees that various obstruction groups are trivial. In the present paper we extend the EKMM results to the cases where p = 2 or the homotopy groups are allowed to be nonzero in all even degrees. In this context the obstruction groups are nontrivial. We shall show that there are never any obstructions to associativity. We prove in Section 7 that the obstructions to commutativity are given by a certain power operation; this was inspired by a parallel result of Mironov in Baas-Sullivan theory [6]. In Section 8 we shall use formal group theory to derive various formulae for this power operation. In Section 9 we deduce a number of results about realising 2-local MU∗-modules as MU -modules.

MU

We study phantom maps and homology theories in a stable homotopy category S via a certain Abelian category A. We express the group P(X, Y) of phantom maps X → Y as an Ext group in A, and give conditions on X or Y which guarantee that it vanishes. We also determine P(X, HB). We show that any composite of two phantom maps is zero, and use this to reduce Margoliss axiomatisation conjecture to an extension problem. We show that a certain functor J → A is the universal example of a homology theory with values in an AB 5 category and compare this with some results of Freyd.

-modules

Abstract We compute the completed E(n) cohomology of the classifying spaces of the symmetric groups, and relate the answer to the theory of finite subgroups of formal groups.

Phantom maps and homology theories

We discuss various moduli problems involving the classification of finite subgroups or related structures on formal groups of finite height n. We show that many moduli schemes are smooth or at least Cohen-Macaulay. Moreover, many maps between such schemes are finite and flat, and their degrees can be predicted by thinking of (QpZp)n as a “discrete model” for the formal group.

Morava E-theory of symmetric groups

Abstract 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.

Finite subgroups of formal groups

We exhibit examples of triangulated categories which are neither the stable category of a Frobenius category nor a full triangulated subcategory of the homotopy category of a stable model category. Even more drastically, our examples do not admit any non-trivial exact functors to or from these algebraic respectively topological triangulated categories.

Gross–Hopkins duality

Abstract We study a natural inner product on K ( n ) / BG and relate it to Frobenius algebras, TQFT’s and HopkinsKuhn-Ravenel character theory.

Triangulated categories without models

Abstract Following Quillen [26, 27], we use the methods of algebraic geometry to study the ring E ∗ (BG) where E is a suitable complete periodic complex oriented theory and G is a finite group: we describe its variety in terms of the formal group associated to E, and the category of abelian p-subgroups of G. Our results considerably extend those of Hopkins–Kuhn–Ravenel [16], and this enables us to obtain information about the associated homology of BG. For example if E is the complete 2-periodic version of the Johnson–Wilson theory E(n) the irreducible components of the variety of the quotient E ∗ (BG)/I k by the invariant prime ideal I k =(p, v 1 , …, v k-1 ) correspond to conjugacy classes of abelian p-subgroups of rank ⩽n−k. Furthermore, if we invert vk the decomposition of the variety into irreducible pieces corresponding to minimal primes becomes a decomposition into connected components, corresponding to the fact that the ring splits as a product.

K(N)-local duality for finite groups and groupoids

Abstract We give a new proof of a special case of a theorem Hopkins and the authors, relating the Morava K-theory of BU〈6〉 to the theory of cubical structures on formal groups. In the process we relate the Morava K-theory of the Eilenberg-MacLane space K( Z ,3) to the theory of Weil pairing, and we appeal to results of algebraic geometers about biextensions.

VARIETIES AND LOCAL COHOMOLOGY FOR CHROMATIC GROUP COHOMOLOGY RINGS

The extended-power spectrum DS has two coproducts and two products, which interact in an intricate way. Given an H∞ ring spectrum E, the resulting algebraic structure on E∗DS0 gives a framework in which to encode information about power operations. (However, we will not study power operations in this paper). Fix a prime p and an integer n > 0. We shall take E to be a suitable completed and extended version of E(n). To be more precise, we let W be the Witt ring of Fpn , and consider the following graded ring: E∗ = W [[u1, . . . , un−1]][u, u −1] The generators uk have degree 0, and u has degree −2. We take u0 = p and un = 1 and uk = 0 for k > n. There is a map BP ∗ −→ E∗ sending vk to u k−1uk. Using this, we define a functor from spectra to E∗-modules by