Motivic Landweber exact theories and their effective covers (Q2354987)

This article explores the relationship between two constructions in motivic homotopy theory, Landweber exact spectra and effective covers. For the first of these, recall that if \(k\) is a base field of characteristic zero and \((F, R)\) is a Landweber exact formal group law, then there exists an oriented motivic \(T\)-spectrum \(\mathcal{E} = R \otimes_{\mathbb{L}} MGL\) such that the associated formal group law is precisely \((F, R)\). For the second ingredient, recall that the motivic homotopy category can be filtered by effectivity: \(SH(k) \supset \dots \supset SH(k)^{\mathrm{eff}}(n) \supset SH(k)^{\mathrm{eff}}(n-1) \supset \dots\). The inclusion \(i_0: SH(k)^{\mathrm{eff}}(0) \hookrightarrow SH(k)\) has a right adjoint \(r_0\) and the connective cover functor is \(f_0 := i_0 r_0\). The author proves that the coefficients of \(f_0 \mathcal{E}\) are given precisely by the subring \(R_0 \subset R\) of elements of non-positive degree. Furthermore he shows how to recover the ``geometric part'' \(X \mapsto f_0\mathcal{E}^{2*,*}(X)\) in terms of just \(R_0\) and the algebraic cobordism cohomology theory \(\Omega^*\).