John Greenlees

Derived functors of I-adic completion and local homology

In recent topological work [2], we were forced to consider the left derived functors of the I-adic completion functor, where I is a finitely generated ideal in a commutative ring A. While our concern in [2] was with a particular class of rings, namely the Burnside rings A(G) of compact Lie groups G, much of the foundational work we needed was not restricted to this special case. The essential point is that the modules we consider in [2] need not be finitely generated and, unless G is finite, say, the ring ,4(G) is not Noetherian. There seems to be remarkably little information in the literature about the behavior of I-adic completion in this generality. We presume that interesting non-Noetherian commutative rings and interesting non-finitely generated modules arise in subjects other than topology. We have therefore chosen to present our algebraic work separately, in the hope that it may be of value to mathematicians working in other fields. One consequence of our study, explained in Section 1, is that I-adic completion is exact on a much larger class of modules than might be expected from the key role played by the Artin-Rees lemma and that the deviations from exactness can be computed in terms of torsion products. However, the most interesting consequence, discussed in Section 2, is that the left derived functors of I-adic completion usually can be computed in terms of certain local homology groups, which are defined in a fashion dual to the definition of the classical local cohomology groups of Grothendieck. These new local homology groups may well be relevant to algebraists and algebraic geometers. In particular, we obtain a universal coefficients theorem for calculating these groups from local cohomology in Section 3; the classical local duality spectral sequence is a very special case.

LOCALIZATION AND COMPLETION THEOREMS FOR MU-MODULE SPECTRA

Let G be a finite extension of a torus. Working with highly structured ring and module spectra, let M be any module over MU ; examples include all of the standard homotopical MU -modules, such as the Brown-Peterson and Morava K-theory spectra. We shall prove localization and completion theorems for the computation of M∗(BG) and M∗(BG). The G-spectrum MUG that represents stabilized equivariant complex cobordism is an algebra over the equivariant sphere spectrum SG, and there is an MUG-module MG whose underlying MU -module is M . This allows the use of topological analogues of constructions in commutative algebra. The computation ofM∗(BG) andM∗(BG) is expressed in terms of spectral sequences whose respective E2 terms are computable in terms of local cohomology and local homology groups that are constructed from the coefficient ring MU ∗ and its module M ∗ . The central feature of the proof is a new norm map in equivariant stable homotopy theory, the construction of which involves the new concept of a global I∗-functor with smash product.

K-HOMOLOGY OF UNIVERSAL SPACES AND LOCAL COHOMOLOGY OF THE REPRESENTATION RING

0. PROLOGUE IN THIS note we calculate the K-homology of the classifying space BG of a finite group G by expressing it as the Grothendieck local cohomology of the representation ring R(G) at the augmentation ideal. in symbols, we show Ki(BG+) z H:(R(G)) (0.0) for i = 0, 1 where J = ker(R(G) --* Z) is the augmentation ideal. This illuminates various calculations of Wilson [22] and Knapp [ 173. This result is a special case of the more general (5.2), which calculates the equivariant K-homology of other universal spaces in precisely analogous terms. In fact G. Wilson dcduccs a formula for K,(BG+) from the Atiyah-Scgal completion theorem [3] and Atiyah’s universal coelficicnt thcorcm for K-theory, and (0.0) is easily deduced from [22] (1.2). There is also a more recent approach to (0.0) via cohomology in [12]. By contrast, our approach is to prove the homological theorem (0.0) directly and to deduce the Atiyah-Segal theorem from it. Accordingly, we obtain a new proof of the Atiyah-Segal theorem which uses little more than equivariant Bott periodicity (see also [I]). In addition, our explicit recognition of local cohomology places many powerful algebraic techniques at our disposal. It also provides analogous statements for other theories although (for example) the statement for stable homotopy is false whilst its cohomological counterpart (i.e. the Segal conjecture) is true.

Finiteness in derived categories of local rings

New homotopy invariant finiteness conditions on modules over commutative rings are introduced, and their properties are studied systematically. A number of finiteness results for classical homological invariants like flat dimension, injective dimension, and Gorenstein dimension, are established. It is proved that these specialize to give results concerning modules over complete intersection local rings. A noteworthy feature is the use of techniques based on thick subcategories of derived categories.

The Tate spectrum ofv n -periodic complex oriented theories

In this paper we work primarily in the equivariant stable category of [16], but we try to provide numerous reference points and nonequivariant versions of results for readers unfamiliar with that world. We discuss the Tare spectrum tG(X). Here X is a G-spectrum and to(X) is a new G-spectrum, covariantly functorial in X. Throughout this paper we assume that G is a finite group (though the functor to(X) is defined whenever G is compact Lie). Our introduction is in three parts; first we review a bit of equivariant stable homotopy theory from [16], then we briefly discuss the Tate spectrum of [11], and then we state our results. The rest of the paper is organized as follows. In Section 2 we prove some lemmas required for our Theorem 1.1 (and the theorem itself when G is cyclic), and in Section 3 we prove our main theorem, Theorem 1.1, for arbitrary finite G. Section 4 applies Theorem 1.1 to prove the remainder of our results about the Tate theory of complex oriented vn-periodic spectra. Recall that a G-spectrum X is a set of spaces indexed over G-invariant finite dimensional subspaces of some complete G-universe U. (U is an infinite dimensional G invariant inner product space containing a countably infinite direct sum of regular representations of G as a subspace.) The usual structure maps x(v) ,s2Wx(v G w)

Equivariant Formal Group Laws

Motivated by complex oriented equivariant cohomology theories, we give a natural algebraic definition of an

Representing Tate cohomology of G -spaces

A

Commutative algebra in group cohomology

-equivariant formal group law for any abelian compact Lie group

Rational SO(2)-equivariant spectra

A

Some remarks on the structure of Mackey functors

. The complex oriented cohomology of the classifying space for line bundles gives an example. We also show how the choice of a complete flag gives rise to a basis and a means of calculation. This allows us to deduce that there is a universal ring