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 $\mathcal{E}:=R\otimes_\mathbb{L}\mathrm{MGL}$ 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 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.

Keywords

algebraic cobordism, oriented theory, slice tower

2010 Mathematics Subject Classification

Primary 14C25, 19E15. Secondary 14F42, 19E08, 55P42.

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Published 18 May 2015