First of the questions about derived algebraic geometry from a noobie.


The way I understand it, every discrete ring $R$ corresponds to some ring spectrum whose $\pi\_0$ is $R$. Now consider $p$-adic numbers. They are a limit of discrete rings — what should correspond to them? How to generalize?


(+ what could be good tags for derived algebraic geometry? I was considering: e-infinity, infty-structures, math-0703204, derived-alggeom, derived-spaces, infty-topology, a-infinity-algebras (2 currently tagged))



$E_\infty$ spectrum corresponding to $\Bbb Z_p$
There is indeed an Eilenberg-Maclane spectrum $H\Bbb Z\_p$. It is equivalent to the $p$-completion of $H\Bbb Z$ and the (homotopy) limit of the $H\Bbb Z/p^k$. However, it doesn't remember the topology.


If you want to remember the topology on the p-adics you need to do something more complicated, such as view the inverse system $\{H\Bbb Z/p^k\}$ as a pro-object in spectra rather than actually taking the limit. This is something like what you would do if you wanted to talk about formal schemes.


