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Motivic Landweber Exactness
 DOCUMENTA MATH.
, 2009
"... We prove a motivic Landweber exact functor theorem. The main result shows the assignment given by a Landwebertype formula involving the MGLhomology of a motivic spectrum defines a homology theory on the motivic stable homotopy category which is representable by a Tate spectrum. Using a universal ..."
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We prove a motivic Landweber exact functor theorem. The main result shows the assignment given by a Landwebertype formula involving the MGLhomology of a motivic spectrum defines a homology theory on the motivic stable homotopy category which is representable by a Tate spectrum. Using a universal coefficient spectral sequence we deduce formulas for operations of certain motivic Landweber exact spectra including homotopy algebraic Ktheory. Finally we employ a Chern character between motivic spectra in order to compute rational algebraic cobordism groups over fields in terms of rational motivic cohomology groups and the Lazard ring.
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.
Realizing families of Landweber exact homology theories, from: “New topological contexts for Galois theory and algebraic geometry (BIRS 2008
, 2009
"... I discuss the problem of realizing families of complex orientable homology theories as families of E ∞ring spectra, including a recent result of Jacob Lurie emphasizing the role of pdivisible groups. 1 55N22; 55N34, 14H10 A few years ago, I wrote a paper [10] discussing a realization problem for f ..."
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I discuss the problem of realizing families of complex orientable homology theories as families of E ∞ring spectra, including a recent result of Jacob Lurie emphasizing the role of pdivisible groups. 1 55N22; 55N34, 14H10 A few years ago, I wrote a paper [10] discussing a realization problem for families of Landweber exact spectra. Since Jacob Lurie [27] now has a major positive result in this direction, it seems worthwhile to revisit these ideas. In brief, the realization problem can be stated as follows. Suppose we are given a flat morphism g: Spec(R) − → Mfg from an affine scheme to the moduli stack of smooth 1dimensional formal groups. Then we get a twoperiodic homology theory E(R, G) with E(R, G)0 ∼ = R and associated formal group G = Spf(E 0 CP ∞) isomorphic to the formal group classified by g. The higher homotopy groups of E(R, G) are zero in odd degrees and E(R, G)2n ∼ = ω ⊗n G where ωG is the module of invariant differentials for G. The module ωG is locally free of rank 1 over R, and free of rank 1 if G has a coordinate. In this case E(R, G) ∗ = R[u ±1] where u ∈ E(R, G)2 is a generator. The fact that g was flat implies E(R, G) is Landweber exact, even if G doesn’t have a coordinate. Now suppose we are given a flat morphism of stacks g: X − → Mfg. 1