Injective spectra with respect to the \(K\)-homologies
The author makes a detailed study of \(E_ *\)-injective spectra, in case \(E\) is the wedge \(KU\vee KO\vee KT\) of the spectra representing complex, real and self-conjugate \(K\)-theory, respectively. By an \(E_ *\)-injective spectrum it meant a \(CW\)-spectrum \(W\) for which a map \(f:X\to Y\) induces an epimorphism \(f^*[Y,W]\to [X,W]\) whenever \(f_ *:E_ * X\to E_ * Y\) is a monomorphism. In his paper [Osaka J. Math. 29, 41-62 (1992; Zbl 0758.55006)] the author has examined \(E_ *\)-injective spectra for a wide class of ring spectra. Here he makes use of \textit{A. K. Bousfield's} categories \(CRT\) and \(ACRT\) [J. Pure Appl. Algebra 66, 121-163 (1990; Zbl 0713.55007)] in which the ``united \(K\)-homology theory'' \(K_ *^{CRT}X=\{KU_ * X,KO_ * X,KT_ * X\}\) takes values for any \(CW\)- spectrum \(X\). The main result asserts the equivalence of the following four conditions on a \(CW\)-spectrum \(W\): (a) \(W\) is an \(E_ *\)-injective spectrum for \(E=KU\vee KO\vee KT\); (b) \(W\) is a not-necessarily- associative \(KO\)-module spectrum such that \(K_ *^{CRT}W\) is injective in the category \(CRT\); (c) \(W\) is a \(KO_ *\)-local spectrum such that \(K_ *^{CRT}W\) is injective in the category \(ACRT\); and (d) the canonical morphism \([X,W]\to \Hom_{ACRT}(K_ *^{CRT}X,K_ *^{CRT}W)\) is an isomorphism for any \(CW\)-spectrum \(X\). The results obtained are actually more general, since they are also proved for six simpler categories \(CR\), \(CT\), \(RT\), \(C\), \(R\) and \(T\) as well as for \(CRT\). A major point is that one can do homological algebra conveniently in these categories.
Bousfield's categories \(CRT\) and \(ACRT\)
scientific article; zbMATH DE number 108662