Paul G. Goerss

Localization theories for simplicial presheaves

This work was motivated in part by the following question of Soule: given a simplicial presheaf X on a site C, how does one produce a map of simplicial presheaves X → LHZX in such a way that each of the maps in sections X(U) → LHZX(U), U ∈ C, is an integral homology localization map in the sense of Bousfield? Secondly, if Y is a simplicial presheaf which is integrally homology local in a suitable sense, is it the case that the map X → LHZX induces an isomorphism

(PRE-)Sheaves of Ring Spectra Over the Moduli Stack of Formal Group Laws

In the first part of this article, I will state a realization problem for diagrams of structured ring spectra, and in the second, I will discuss the moduli space which parametrizes the problem. While some of what I say is quite general, the ring spectra I have in mind will arise from the chromatic point of view, which uses the geometry of formal groups to organize stable homotopy theory. Thus, a subsidiary aim here is to reemphasize this connection between algebraic geometry and homotopy theory.

On Hopkins’ Picard groups for the prime 3 and chromatic level 2

We give a calculation of Picard groups Pic2 of K(2)-local invertible spectra and Pic(L2) of E(2)-local invertible spectra, both at the prime 3. The main contribution of this paper is to calculation the subgroup κ2 of invertible spectra X with (E2)∗X ∼= (E2)∗S as twisted modules over the Morava stabilizer group G2.

Some generalized brown-gitler spectra

Brown-Gitler spectra for the homology theories associated with the spectra KZp, to, and bu are constructed. Complexes adapted to the new Brown- Gitler spectra are produced and a spectral sequence converging to stable maps into these spectra is constructed and examined.

Relations among homotopy operations for simplicial commutative algebras

The homotopy groups of a simplicial commutative algebra over the field with two elements support natural operations. Here we write the relations among these operations in an admissible form. Let A be a simplicial commutative algebra over the field F2. Then A is, among other things, a simplicial set and, as such, has homotopy groups. In fact, 7,rA -H0(A, a) where 0 = di: A, -n A, I is the sum (or alternating sum) of the face operators. By the Eilenberg-Zilber Theorem, 7r.A is a graded commutative F2 algebra. Cartan [2], Bousfield [1], and Dwyer [3] have pointed out the existence of natural operations on these homotopy groups. Indeed, Dwyer proved the following theorem. Theorem 1. Let A be a simplicial commutative F2 algebra. Then there are natural operations i 7rA 7rAn+iA, 2 I1, then x2 = O. Received by the editors October 12, 1993 and, in revised form, January 20, 1994; the second author presented the contents of this paper at the 92nd Annual Meeting of the AMS, January 7-1 1, 1986, New Orleans, LA. 1991 Mathematics Subject Classification. Primary 1 8G30, 55S99.

The homology of homotopy inverse limits

Abstract The homology of a homotopy inverse limit can be studied by a spectral sequence which has as the E 2 term the derived functor of limit in the category of coalgebras. These derived functors can be computed using the theory of Dieudonne modules if one has a diagram of connected abelian Hopf algebras.

The Homotopy of L 2 V(1) for the Prime 3

Let V(1) be the Toda-Smith complex for the prime 3. We give a complete calculation of the homotopy groups of the L2-localization of V(1) by making use of the higher real K-theory EO 2 of Hopkins and Miller and related homotopy fixed point spectra. In particular we resolve an ambiguity which was left in an earlier approach of Shimomura whose computation was almost complete but left an unspecified parameter still to be determined.

André-Quillen cohomology and the homotopy groups of mapping spaces: understanding the E2-term of the Bousfield-Kan spectral sequence

Abstract Let X and Y be two topological spaces. We note that a discussion of the homotopy groups of the space of continuous functions from X to Y leads to a discussion of the algebra derivations from the cohomology algebra of Y to the cohomology algebra of X . This observation, and the Bousfield-Kan spectral sequence, allow us to apply the commutative algebra cohomology of Andre and Quillen to the study of the homotopy type of spaces of maps.

Towers and injective cohomology algebras

Since Carlssons paper [5] and the work of Lannes and Zarati [9], topo1ogists have learned to recognize many spaces Y so H*Y is injective in U. For example, if G is a finite group with p-Sylow subgroup (Z/p)kn the H*BG is injective. Further infinite examples include EB(z/p)k and any of the many wedge summands of these spaces. A complete classification of U-injectives is given in [18] and, from this, it can be seen that the only such injectives which are not A-unbounded are the Brown-Gitler modules. These have been adequately discussed elsewhere (see [6], among many) An immediate consequence of Theorem A will be the following result. Suppose Y satisfies the same hypotheses as in Theorem A.

Homotopy and homology of simplicial abelian Hopf algebras

Abstract. Let A be a simplicial bicommutative Hopf algebra over the field