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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.
On the moduli stack of commutative, 1parameter formal Lie groups
, 2007
"... Abstract. We attempt to develop a general algebrogeometric study of the moduli stack of commutative, 1parameter formal Lie groups, in full comportment with the modern foundations of algebraic geometry. We emphasize the proalgebraic structure of this stack: it is the inverse limit, over varying n, ..."
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Abstract. We attempt to develop a general algebrogeometric study of the moduli stack of commutative, 1parameter formal Lie groups, in full comportment with the modern foundations of algebraic geometry. We emphasize the proalgebraic structure of this stack: it is the inverse limit, over varying n, of moduli stacks of nbuds, and these latter stacks are algebraic. Our main theorems pertain to the height stratification relative to fixed prime p on the stacks of formal Lie groups and of nbuds. Notably, we show that the stack of nbuds of height ≥ h is smooth and universally closed over Fp of dimension −h; we characterize the stratum of nbuds of (exact) height h and the stratum of formal Lie groups of (exact) height h as classifying stacks of certain groups, smooth algebraic in the bud case; and we obtain some structure results on these groups. We also obtain a second characterization of the stratum of formal Lie groups of height h as an inverse limit of classifying stacks of certain finite étale algebraic groups.
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
GEOMETRIC CRITERIA FOR LANDWEBER EXACTNESS
"... Abstract. The purpose of this paper is to give a new presentation of some of the main results concerning Landweber exactness in the context of the homotopy theory of stacks. We present two new criteria for Landweber exactness over a flat Hopf algebroid. The first criterion is used to classify stacks ..."
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Abstract. The purpose of this paper is to give a new presentation of some of the main results concerning Landweber exactness in the context of the homotopy theory of stacks. We present two new criteria for Landweber exactness over a flat Hopf algebroid. The first criterion is used to classify stacks arising from Landweber exact maps of rings. Using as extra input only Lazard’s theorem and Cartier’s classification of ptypical formal group laws, this result is then applied to deduce many of the main results concerning Landweber exactness in stable homotopy theory and to compute the Bousfield classes of certain BPalgebra spectra. The second criterion can be regarded as a generalization of the Landweber exact functor theorem and we use it to give a proof of the original theorem. 1.
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
βFAMILY CONGRUENCES AND THE fINVARIANT
"... Abstract. In previous work, the authors have each introduced methods for studying the 2line of the plocal AdamsNovikov spectral sequence in terms of the arithmetic of modular forms. We give the precise relationship between the congruences of modular forms introduced by the first author with the Q ..."
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Abstract. In previous work, the authors have each introduced methods for studying the 2line of the plocal AdamsNovikov spectral sequence in terms of the arithmetic of modular forms. We give the precise relationship between the congruences of modular forms introduced by the first author with the Qspectrum and the finvariant of the second author. This relationship enables us to refine the target group of the finvariant in a way which makes it more manageable for computations. 1.
ADJOINT PAIRS FOR QUASICOHERENT SHEAVES ON STACKS.
"... Abstract. In this paper we construct a pushforwardpullback adjoint pair for categories of quasicoherent sheaves, along a morphism of algebraic stacks, which is represented in algebraic stacks over the site C = Aff flat. The construction uses the characterization of algebraic stacks of [H3] and is ..."
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Abstract. In this paper we construct a pushforwardpullback adjoint pair for categories of quasicoherent sheaves, along a morphism of algebraic stacks, which is represented in algebraic stacks over the site C = Aff flat. The construction uses the characterization of algebraic stacks of [H3] and is based on the descent description of the category of quasicoherent sheaves given in [H2]. We show that an essentially immediate consequence of the presentation we give for this adjoint pair is the MillerRavenelMorava change of rings theorem and the algebraic chromatic convergence theorem. 1.
The AdamsNovikov Spectral Sequence and the Homotopy Groups of Spheres
, 2007
"... These are notes for a five lecture series intended to uncover largescale phenomena in the homotopy groups of spheres using the AdamsNovikov Spectral Sequence. The lectures were given in Strasbourg, May 7–11, ..."
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These are notes for a five lecture series intended to uncover largescale phenomena in the homotopy groups of spheres using the AdamsNovikov Spectral Sequence. The lectures were given in Strasbourg, May 7–11,
Motivic twisted Ktheory
, 2010
"... This paper sets out basic properties of motivic twisted Ktheory with respect to degree three motivic cohomology classes of weight one. Motivic twisted Ktheory is defined in terms of such motivic cohomology classes by taking pullbacks along the universal principal BGmbundle for the classifying spa ..."
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This paper sets out basic properties of motivic twisted Ktheory with respect to degree three motivic cohomology classes of weight one. Motivic twisted Ktheory is defined in terms of such motivic cohomology classes by taking pullbacks along the universal principal BGmbundle for the classifying space of the multiplicative group scheme. We show a Künneth isomorphism for homological motivic twisted Kgroups computing the latter as a tensor product of Kgroups over the Ktheory of BGm. The proof employs an Adams Hopf algebroid and a trigraded Torspectral sequence for motivic twisted Ktheory. By adopting the notion of an E∞ring spectrum to the motivic homotopy theoretic setting, we construct spectral sequences relating motivic (co)homology groups to twisted Kgroups. It generalizes various spectral sequences computing the algebraic Kgroups of schemes over fields. Moreover, we construct a Chern character between motivic twisted Ktheory and twisted periodized rational motivic cohomology, and show that it is a rational isomorphism. The paper includes a discussion of some open problems.