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Comodules and Landweber exact homology theories
 Adv. Math
"... Abstract. We show that, if E is a commutative MUalgebra spectrum such ..."
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Abstract. We show that, if E is a commutative MUalgebra spectrum such
The stack of formal groups in stable homotopy theory
 Adv. Math
"... We construct the algebraic stack of formal groups and use it to provide a new perspective onto a recent result of M. Hovey and N. Strickland on comodule categories for Landweber exact algebras. This leads to a geometric understanding of their results as well as to a generalisation. 1. ..."
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We construct the algebraic stack of formal groups and use it to provide a new perspective onto a recent result of M. Hovey and N. Strickland on comodule categories for Landweber exact algebras. This leads to a geometric understanding of their results as well as to a generalisation. 1.
Moduli problems for structured ring spectra
 DANIEL DUGGER AND BROOKE
, 2005
"... In this document we make good on all the assertions we made in the previous paper “Moduli spaces of commutative ring spectra ” [20] wherein we laid out a theory a moduli spaces and problems for the existence and uniqueness of E∞ring spectra. In that paper, we discussed the the HopkinsMiller theore ..."
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In this document we make good on all the assertions we made in the previous paper “Moduli spaces of commutative ring spectra ” [20] wherein we laid out a theory a moduli spaces and problems for the existence and uniqueness of E∞ring spectra. In that paper, we discussed the the HopkinsMiller theorem on the LubinTate or Morava spectra En; in particular, we showed how to prove that the moduli space of all E ∞ ring spectra X so that (En)∗X ∼ = (En)∗En as commutative (En) ∗ algebras had the homotopy type of BG, where G was an appropriate variant of the Morava stabilizer group. This is but one point of view on these results, and the reader should also consult [3], [38], and [41], among others. A point worth reiterating is that the moduli problems here begin with algebra: we have a homology theory E ∗ and a commutative ring A in E∗E comodules and we wish to discuss the homotopy type of the space T M(A) of all E∞ring spectra so that E∗X ∼ = A. We do not, a priori, assume that T M(A) is nonempty, or even that there is a spectrum X so that E∗X ∼ = A as comodules.
Quasicoherent sheaves on the moduli stack of formal groups
"... For years I have been echoing my betters, especially Mike Hopkins, and telling anyone who would listen that the chromatic picture of stable homotopy theory is dictated and controlled by the geometry of the moduli stack Mfg of smooth, onedimensional formal groups. Specifically, I would say that the ..."
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For years I have been echoing my betters, especially Mike Hopkins, and telling anyone who would listen that the chromatic picture of stable homotopy theory is dictated and controlled by the geometry of the moduli stack Mfg of smooth, onedimensional formal groups. Specifically, I would say that the height filtration of Mfg dictates a canonical and natural decomposition of a quasicoherent sheaf on Mfg, and this decomposition predicts and controls the chromatic decomposition of a finite spectrum. This sounds well, and is even true, but there is no single place in the literature where I could send anyone in order for him or her to get a clear, detailed, unified, and linear rendition of this story. This document is an attempt to set that right. Before going on to state in detail what I actually hope to accomplish here, I should quickly acknowledge that the opening sentences of this introduction and, indeed, this whole point of view is not original with me. I have already mentioned Mike Hopkins, and just about everything I’m going to say here is encapsulated in the table in section 2 of [15] and can be gleaned from the notes
Chromatic phenomena in the algebra of BP∗BP comodules
, 2002
"... Abstract. We describe the author’s research with Neil Strickland on the global algebra and global homological algebra of the category of BP∗BP ..."
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Abstract. We describe the author’s research with Neil Strickland on the global algebra and global homological algebra of the category of BP∗BP
SMASH PRODUCTS OF E(1)LOCAL SPECTRA AT AN ODD PRIME
, 2004
"... The two categories are not Quillen equivalent, and his proof uses systems of triangulated diagram categories rather than model categories. Our main result is that in the case n = 1 Franke’s functor maps the derived tensor product to the smash product. It can however not be an associative equivalence ..."
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The two categories are not Quillen equivalent, and his proof uses systems of triangulated diagram categories rather than model categories. Our main result is that in the case n = 1 Franke’s functor maps the derived tensor product to the smash product. It can however not be an associative equivalence of monoidal categories. The first part of our paper sets up a monoidal version of Franke’s systems of triangulated diagram categories and explores its properties. The second part applies these results to the specific construction of Franke’s functor in order to prove the above result. 1.
INFINITESIMAL THICKENINGS OF MORAVA KTHEORIES
, 2007
"... Abstract. A. Baker has constructed certain sequences of cohomology theories which interpolate between the Johnson–Wilson and the Morava Ktheories. We realize the representing sequences of spectra as sequences of MUalgebras. Starting with the fact that the spectra representing the Johnson–Wilson an ..."
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Abstract. A. Baker has constructed certain sequences of cohomology theories which interpolate between the Johnson–Wilson and the Morava Ktheories. We realize the representing sequences of spectra as sequences of MUalgebras. Starting with the fact that the spectra representing the Johnson–Wilson and the Morava Ktheories admit such structures, we construct the sequences by inductively forming singular extensions. Our methods apply to other pairs of MUalgebras as well. 1.
ON LEFT AND RIGHT MODEL CATEGORIES AND LEFT AND RIGHT BOUSFIELD LOCALIZATIONS
"... We verify the existence of left Bousfield localizations and of enriched left Bousfield localizations, and we prove a collection of useful technical results characterizing certain fibrations of (enriched) left Bousfield localizations. We also use such Bousfield localizations to construct a number of ..."
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We verify the existence of left Bousfield localizations and of enriched left Bousfield localizations, and we prove a collection of useful technical results characterizing certain fibrations of (enriched) left Bousfield localizations. We also use such Bousfield localizations to construct a number of new model categories, including models for the homotopy limit of right Quillen presheaves, for Postnikov towers in model categories, and for presheaves valued in a symmetric monoidal model category satisfying a homotopycoherent descent condition. We then verify the existence of right Bousfield localizations of right model categories, and we apply this to construct a model of the homotopy limit of a left Quillen presheaf as a right model category.
LOCAL COHOMOLOGY OF BP∗BPCOMODULES
"... Abstract. Given a spectrum X, we construct a spectral sequence of BP∗BPcomodules that converges to BP∗(LnX), where LnX is the Bousfield localization of X with respect to the JohnsonWilson theory E(n)∗. The E2term of this spectral sequence consists of the derived functors of an algebraic version o ..."
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Abstract. Given a spectrum X, we construct a spectral sequence of BP∗BPcomodules that converges to BP∗(LnX), where LnX is the Bousfield localization of X with respect to the JohnsonWilson theory E(n)∗. The E2term of this spectral sequence consists of the derived functors of an algebraic version of Ln. We show how to calculate these derived functors, which are closely related to local cohomology of BP∗modules with respect to the ideal In+1.
Interpolation categories for homology theories
, 2004
"... For a homological functor from a triangulated category to an abelian category satisfying some technical assumptions we construct a tower of interpolation categories. These are categories over which the functor factorizes and which capture more and more information according to the injective dimensio ..."
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For a homological functor from a triangulated category to an abelian category satisfying some technical assumptions we construct a tower of interpolation categories. These are categories over which the functor factorizes and which capture more and more information according to the injective dimension of the images of the functor. The categories are obtained by proving the existence of truncated versions of resolution or E2model structures. Examples of functors fitting in our framework are given by every generalized homology theory represented by a ring spectrum satisfying the AdamsAtiyah condition. The constructions are closely related to the modified Adams spectral sequence and give a very conceptual approach to the associated moduli problem and obstruction theory. As application we establish an isomorphism between certain E(n)local Picard groups and some Extgroups.