There is a perfectly functional Eilenberg-Maclane spectrum HZ_p, although it returns the discrete ring. You can also take It looks like you want a limit of Eilenberg-Maclane way to attach spectra HZ/p^nto cohomology theories with topologized coefficient groups, but and I don't know how well-behaved it isif that's possible (in particular the wedge axiom in Brown representability smells funny).Alternatively, you can take
Random ideas that may or may not work:
- Take a limit of Eilenberg-Maclane spectra HZ/p^n
- Take a p-completion of HZ, and that might be well-behaved when fed p-complete spaces.