Rational \(\mathbb Z_{p}\)-equivariant spectra (Q633888)

In studying the homotopy theory of spaces with actions of a topological group \(G\), it is often useful to consider equivariant cohomology theories, and since these are represented by \(G\)-spectra, this leads naturally to the study of \(G\)-equivariant stable homotopy theory. The most highly studied cases are those of compact Lie groups and discrete groups, however profinite groups are increasingly important (not least in chromatic stable homotopy theory). The present paper focuses on the important case of the group of \(p\)-adic integers \(\mathbb Z_p=\lim_{n}\mathbb Z/p^n\) for a prime number~\(p\). The categories of \(\mathbb Z_p\) spaces and spectra have Quillen model structures introduced by Fausk, and the stable category can then be Bousfield localised to form the category of rational \(\mathbb Z_p\)-spectra (this turns out to be a monoidal model category with good properties, as shown in earlier work of the author). The main results involve an algebraic model for this model category, based on differential graded discrete rational \(\mathbb Z_p\)-modules, where a \textit{discrete module} is one for which the action of \(\mathbb Z_p\) has open (i.e., finite index) stabilizers. Then Theorem~7.1 asserts that there is a Quillen equivalence between these two categories. Unfortunately, the proof does not show that this is a monoidal equivalence. The methods used should generalise to other profinite groups, and the results are very similar in spirit to those of the author for rational \(O(2)\)-spectra.