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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 ..."
Abstract

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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.
CORES OF SPACES, SPECTRA, AND E ∞ RING SPECTRA
"... Abstract. In a paper that has attracted little notice, Priddy showed that the BrownPeterson spectrum at a prime p can be constructed from the plocal sphere spectrum S by successively killing its odd dimensional homotopy groups. This seems to be an isolated curiosity, but it is not. For any space o ..."
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Abstract. In a paper that has attracted little notice, Priddy showed that the BrownPeterson spectrum at a prime p can be constructed from the plocal sphere spectrum S by successively killing its odd dimensional homotopy groups. This seems to be an isolated curiosity, but it is not. For any space or spectrum Y that is plocal and (n0 − 1)connected and has πn0 (Y) cyclic, there is a plocal, (n0 − 1)connected “nuclear ” CW complex or CW spectrum X and a map f: X → Y that induces an isomorphism on πn0 and a monomorphism on all homotopy groups. Nuclear complexes are atomic: a selfmap that induces an isomorphism on πn0 must be an equivalence. The construction of X from Y is neither functorial nor even unique up to equivalence, but it is there. Applied to the localization of MU at p, the construction yields BP. In 1999, the third author gave an April Fool’s talk on how to prove that BP is an E ∞ ring spectrum or, in modern language, a commutative Salgebra. As explained in [20], he gave a quite different April Fool’s talk on the same subject two years earlier. His new idea was to exploit the remarkable paper of Stewart Priddy [23], in which Priddy constructed BP by killing the odd degree homotopy groups of the sphere spectrum. The hope was that by mimicking Priddy’s construction in the category of commutative Salgebras, one might arrive at a construction of BP as a commutative Salgebra. As the first two authors discovered, that argument fails. However, the ideas are still interesting. As we shall explain, Priddy’s construction of BP is not an accidental fluke but rather a special case of a very general construction. The elementary space and spectrum level construction is given in Section 1. The more sophisticated E ∞ ring spectrum analogue and its specialization to MU are discussed in Section 2. It is a pleasure to thank Nick Kuhn and Fred Cohen for very illuminating emails. In particular, Example 1.10 is due to Cohen. 1. Cores of spaces and spectra