\(E_ *\)-injective spectra and injective \(E_ *E\)-comodules
A spectrum \(W\) was defined by Ohkawa to be \(E_ *\)-injective if whenever \(f: X\to Y\) has \(E_ *f\) monic, then \(f^*: [Y,W]\to [X,W]\) is epic. In this paper, other characterizations of this property are presented. If \(E\) satisfies a technical condition which is satisfied by \(S\), \(HZ/p\), \(MO\), \(MU\), \(MSp\), \(KU\), and \(KO\), then each of the following four conditions is equivalent to \(W\) being \(E_ *\)-injective. (i.) \(W\) is a quasi \(E\)-module spectrum such that \(E_ *W\) is injective as an \(E_ *E\)-module. (ii.) \(W\) is an \(E_ *\)-local spectrum such that \(E_ *W\) is injective as an \(E_ *E\)-comodule. (iii.) \([X,W]\to \text{Hom}_{E_ *E}(E_ *X,E_ *W)\) is bijective for all \(X\). (iv.) \([X,W]\to\text{Hom}_{E_ *E}(E_ *X,E_ *W)\) is injective for any \(X\). Another result says that \(W\) is \(HG_ *\)-injective if and only if \(W\) is a generalized Eilenberg-MacLane spectrum \(\bigvee_ n\Sigma^ n HA_ n\) with \(A_ n\) a direct summand of \(\text{Hom}(G,D_ n)\) with \(D_ n\) divisible.
scientific article; zbMATH DE number 93859