Motivic Landweber exact theories and their effective covers

Abstract

Let $k$ be a field of characteristic zero and let $(F,R)$ be a Landweber exact formal group law. We consider a Landweber exact $T$-spectrum $\mathcal{E}:=R\otimes_{\mathbb{L}}\text{MGL}$ and its effective cover $f_0\mathcal{E}\to \mathcal{E}$ with respect to Voevodsky's slice tower. The coefficient ring $R_0$ of $f_0\mathcal{E}$ is the subring of $R$ consisting of elements of $R$ of non-positive degree; the power series $F\in R[[u,v]]$ has coefficients in $R_0$ although $(F,R_0)$ is not necessarily Landweber exact. We show that the geometric part $X\mapsto f_0\mathcal{E}^*(X):=(f_0\mathcal{E})^{2*,*}(X)$ of $f_0\mathcal{E}$ is canonically isomorphic to the oriented cohomology theory $X\mapsto R_0 \otimes_{\mathbb{L}} \Omega^*(X)$, where $\Omega^*$ is the theory of algebraic cobordism, as defined by Levine-Morel. This recovers results of Dai-Levine as the special case of algebraic $K$-theory and its effective cover, connective algebraic $K$-theory.