Generalized homology and cohomolgy theories with coefficients

Abstract

For any Moore spectrum M and any homology theory \({{\mathcal {H}}}_*\), we associate a homology theory \({{\mathcal {H}}}_*^M\) which is related to \({{\mathcal {H}}}_*\) by a universal coefficient exact sequence of classical type. On the other hand the category of Moore spectra is not the category of \({\mathbb {Z}}\)-modules, but it can be identified to a full subcategory of an abelian category \({{\mathscr {D}}}\). We prove that \({{\mathcal {H}}}_*\) can be lifted to a homology theory \(\widehat{{\mathcal {H}}}_*\) with values in \({{\mathscr {D}}}\) and we give a new universal coefficient exact sequence relating \({{\mathcal {H}}}_*^M\) and \(\widehat{{\mathcal {H}}}_*\) which is in general more precise than the classical one. We prove also a similar result for cohomology theories and we illustrate its convenience by computing the real K-theory of Moore spaces.