MOTIVIC LANDWEBER EXACT THEORIES AND THEIR EFFECTIVE COVERS

Let k be a field of characteristic 0, and let (F, R) be a Landweber exact formal group law. We consider a Landweber exact T-spectrum epsilon := R circle times(L) MGL and its effective cover f(0)epsilon -> epsilon with respect to Voevodsky's slice tower. The coefficient ring R-0 of f(0)epsilon is the subring of R consisting of elements of R of non-positive degree; the power series F is an element of R[[u, v]] has coefficients in R-0, although (F, R-0) is not necessarily Landweber exact. We show that the geometric part X bar right arrow f(0)epsilon*(X) := (f(0)epsilon)(2)*,*(X) of f(0)epsilon is canonically isomorphic to the oriented cohomology theory X bar right arrow R-0 circle times(L) Omega*(X), where Omega* is the theory of algebraic cobordism as defined in [12]. This recovers results of Dai-Levine [2] as the special case of algebraic K-theory and its effective cover, connective algebraic K-theory.