Given a spectrum $X$, we construct a spectral sequence of $BP_{*}BP$-comodules that converges to $BP_{*}(L_{n}X)$, where $L_{n}X$ is the Bousfield localization of $X$ with respect to the Johnson–Wilson theory $E(n)_{*}$. The $E_{2}$-term of this spectral sequence consists of the derived functors of an algebraic version of $L_{n}$. We show how to calculate these derived functors, which are closely related to local cohomology of $BP_{*}$-modules with respect to the ideal $I_{n + 1}$.

In this work we present a generalization of an exact sequence of normal bordism groups given in a paper by H. A. Salomonsen (Math. Scand.32 (1973), 87–111). This is applied to prove that if $h:M^n\to X^{n+k}$, $5\leq n\lt2k$, is a continuous map between two manifolds and $g:M^n\to BO$ is the classifying map of the stable normal bundle of $h$ such that $(h,g)_*:H_i(M,\mathbb{Z}_2)\to H_i(X\times BO,\mathbb{Z}_2)$ is an isomorphism for $i\lt n-k$ and an epimorphism for $i=n-k$, then $h$ bordant to an immersion implies that $h$ is homotopic to an immersion. The second remark complements the result of C. Biasi, D. L. Gonçalves and A. K. M. Libardi (Topology Applic.116 (2001), 293–303) and it concerns conditions for which there exist immersions in the metastable dimension range. Some applications and examples for the main results are also given.

AMS 2000 Mathematics subject classification: Primary 57R42. Secondary 55Q10; 55P60