If you've got a complete DVR with fraction field K and residue field k, then the absolute Galois group of k is a quotient of the absolute Galois group of K.
So there's some kind of "specialization" map from the galois group of the larger field to that of the smaller.
This kind of happens for topological spaces too -- let's see if I can trigger that.
Anyway. His comment seems suggestive of the idea of "specialization" maps from one Morava stabilizer to the next (or, at least, from a subgroup of it).
There isn't really anything special about the multiplicative group here, though, right? Isn't the same construction going to always give my a map from the n'th Morava stabiliser algebra to truncated polynomials in F_{p^n}?
So somehow, if we want to end up over HF_p, which is this sort of infinite codimension closed subset of M_{fg}, we're going to have to mod out by p at some point.
But it also seems to be taking something close to classical (the Morava K-theories of a space only depend on their cohomologies beyong a certain stage) and asserting it as some kind of compatibility among actions of the Morava stabilizers for coherent sheaves on M_fgl.
\mu_{q^n-1} is a way to write roots of unity for those of us who have had so much category theory that we are uncomfortable identifying them with Z/(q^n-1)
@Drew: is it a familiar fact that G_2 (say over F_3, since that's your favourite prime), admits a map to F_9^\times, the cyclic group of order 8? I s'pose that if we're keeping with what Jack's written here, we should be looking at G_2 over F_9, and finding a map to F_{81}^\times = Z/80...
we have K(m)_* X = K(m)_* tensored with P(n)_*X over P(n)_*
and so P(n)_*X determines K(m)_* X, and moreover some of the "Morava stabilizer" action from K(m)_* K(m) is determined by the action of whatever lifts to P(n)_* P(n).
Craig, that doesn't appear obvious. But then I've only ever read about maximal finite subgroups. Then at n=2,p=3 there are only 2 conjugacy classes of max finite subgroups; C_2 and the (non-trivial) semi-direct product of C_3 and C_4
hrm. that's somehow unsatisfying, but it seems in the vein of what Jack is saying ... that there's some "master" information behind the scenes that governs both the Morava stabilizer action mod-p and the Steenrod coaction.
Tyler, possibly related: have you ever thought about a spectral sequence coming from a sequence of generalized Moore spectra: S-->M(p^i)-->M(p^{i},v_1^{j})-->...
Huh, maybe this C_8 is the (not so semi-)direct product of the Galois C_2 with the C_4 in the maximal finite. But that only works if that C_4 splits off of S_2...
Wait.... am i being stupid. Is the composition $M(p^{i_0})\to M(p^{i_1},v_1^{i_1})\to M(p^{i_0},v_1^{i_1},v_2^{i_2})$, where the maps are the cofibers, a trivial composition?
i had some other questions in the brick from last night that weren't literally on page 9, like: On / p is like End (Ga taken to order p^n), which sounds like the story involving A(n)
Two things: first O_n/ p is an algebra, and so acts on itself, giving a map O(n) / p ---> End(O(n)/p). But second: I think the fact that O_n/p is rank n over F_q, and not rank n^2 (the rank of End(O(n)/p) means that this map only realises O(n)/p as a subalgebra of End(O(n)/p) = Mat_n(F_q), and not the whole thing (unless n=1)
