First steps in brave new commutative algebra

This is an expository account of completion and local cohomology in brave new commutative algebra, especially as it applies to completion theorems and their duals in equivariant topology. The first part is fairly direct, but the second considers Morita theory and Gorenstein ring spectra. This draws on work of the author with Benson, Dwyer, Iyengar and May, amongst others

Rational torus-equivariant homotopy I: calculating groups of stable maps

We construct an abelian category A(G) of sheaves over a category of closed subgroups of the r-torus G and show it is of finite injective dimension. It can be used as a model for rational $G$-spectra in the sense that there is a homology theory \piA_*: G-spectra/Q --> A(G) on rational G-spectra with values in A(G), and the associated Adams spectral sequence converges for all rational $G$-spectra and collapses at a finite stage

Duality in algebra and topology

In this paper we take some classical ideas from commutative algebra, mostly ideas involving duality, and apply them in algebraic topology. To accomplish this we interpret properties of ordinary commutative rings in such a way that they can be extended to the more general rings that come up in homotopy theory. Amongst the rings we work with are the differential graded ring of cochains on a space, the differential graded ring of chains on the loop space, and various ring spectra, e.g., the Spanier-Whitehead duals of finite spectra or chromatic localizations of the sphere spectrum. Maybe the most important contribution of this paper is the conceptual framework, which allows us to view all of the following dualities: Poincare duality for manifolds, Gorenstein duality for commutative rings, Benson-Carlson duality for cohomology rings of finite groups, Poincare duality for groups, Gross-Hopkins duality in chromatic stable homotopy theory, as examples of a single phenomenon. Beyond setting up this framework, though, we prove some new results, both in algebra and topology, and give new proofs of a number of old results

An algebraic model for free rational G-spectra

We show that for any compact Lie group G with identity component N and component group W = G/N, the category of free rational G-spectra is equivalent to the category of torsion modules over the twisted group ring H ∗ (BN)[W]. This gives an algebraic classification of rational G-equivariant cohomology theories on free G-spaces and a practical method for calculating the groups of natural transformations between them

Rational S^1-equivariant elliptic cohomology

For each elliptic curve A over the rational numbers we construct a 2-periodic S^1-equivariant cohomology theory E whose cohomology ring is the sheaf cohomology of A; the homology of the sphere of the representation z^n is the cohomology of the divisor A(n) of points with order dividing n. The construction proceeds by using the algebraic models of the author's AMS Memoir ``Rational S^1 equivariant homotopy theory.'' and is natural and explicit in terms of sheaves of functions on A. This is Version 5.2 of a paper of long genesis (this should be the final version). The following additional topics were first added in the Fourth Edition: (a) periodicity and differentials treated (b) dependence on coordinate (c) relationship with Grojnowksi's construction and, most importantly, (d) equivalence between a derived category of O_A-modules and a derived category of EA-modules. The Fifth Edition included (e) the Hasse square and (f) explanation of how to calculate maps of EA-module spectra

An algebraic model for rational torus-equivariant spectra

We show that the category of rational G-spectra for a torus G is Quillen equivalent to an explicit small and practical algebraic model, thereby providing a universal de Rham model for rational G-equivariant cohomology theories. The result builds on the first author's Adams spectral sequence, the second author's functors making rational spectra algebraic. There are several steps, some perhaps of wider interest (1) isotropy separation (replacing the category of G-spectra by modules over a diagram of isotropically simple ring G-spectra) (2) passage to fixed points on ring and module categories (replacing diagrams of ring G-spectra by diagrams of ring spectra) (3) replacing diagrams of ring spectra by diagrams of differential graded algebras (4) rigidity (replacing diagrams of DGAs by diagrams of graded rings). Systematic use of cellularization of model categories is central

Ausoni–Bökstedt duality for topological Hochschild homology

We consider the Gorenstein condition for topological Hochschild homology, and show that it holds remarkably often. More precisely, if R is a commutative ring spectrum and R −→ k is a map to a field of characteristic p then, provided k is small as an R-module, T HH(R; k) is Gorenstein in the sense of [11]. In particular, this holds if R is a (conventional) regular local ring with residue field k of characteristic p. Using only B¨okstedt’s calculation of T HH(k), this gives a non-calculational proof of dualities observed by B¨okstedt [9] and Ausoni [3], Lindenstrauss-Madsen [17], AngeltweitRognes [4] and others

Stratifying derived categories of cochains on certain spaces

In recent years, Benson, Iyengar and Krause have developed a theory of stratification for compactly generated triangulated categories with an action of a graded commutative Noetherian ring. Stratification implies a classification of localizing and thick subcategories in terms of subsets of the prime ideal spectrum of the given ring. In this paper two stratification results are presented: one for the derived category of a commutative ring-spectrum with polynomial homotopy and another for the derived category of cochains on certain spaces. We also give the stratification of cochains on a space a topological content.Comment: 27 page

Towers and fibered products of model categories

Given a left Quillen presheaf of localized model structures, we study the homotopy limit model structure on the associated category of sections. We focus specifically on towers and fibered products of model categories. As applications we consider Postnikov towers of model categories, chromatic towers of spectra and Bousfield arithmetic squares of spectra. For spectral model categories, we show that the homotopy fiber of a stable left Bousfield localization is a stable right Bousfield localization

Circle-equivariant classifying spaces and the rational equivariant sigma genus

We analyze the circle-equivariant spectrum MString_C which is the equivariant analogue of the cobordism spectrum MU of stably almost complex manifolds with c_1=c_2=0. The second author has shown how to construct the ring T-spectrum EJ representing the T-equivariant elliptic cohomology associated to a rational elliptic curve J. In the case that J is a complex elliptic curve, we construct a map of ring T-spectra MString_C --> EJ which is the rational equivariant analogue of the sigma orientation of Ando-Hopkins-Strickland. Our method gives a proof of a conjecture of the first author