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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.
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.