@TomBachmann It's not necessarily the R/G-cohomology of BG. For example, let's say G is a finite subgroup of a monoid Δ and R = HQ[Δ], with the action of G (on the right). Then R/G = HQ[Δ/G], and [R/G, R/G]_{R} = (HQ[Δ/G])^hG is the Hecke algebra of double coset functions for G in Δ. On the other hand, the R/G-cohomology of BG is just R/G again because R/G is rational. E.g. the two are additively different for G an order two subgroup of Δ = the symmetric group on three letters.
You do always get an identification of it as (R/G)^{hG}.
But you have to figure out the action, and this always confuses me a little. The R-linear (right) action is a group map ρ:G -> GL_1(R) which acts by r * g = r ρ(g), and then you get a compatible left action by g * r = ρ(g^{-1}) r.
The action that you're taking homotopy fixed-points for is the latter one, I think.
Nope, there's no inverse and I can't edit it anymore. Should be ρ(g) r. (Forgot it was a right action.)
@TylerLawson So this problem goes away if I assume R to be commutative?
@RuneHaugseng I feel I must be overlooking something obvious, but doesn't "take homotopy groups" commute with filtered colimits, and n-truncated = no homotopy groups above n, so ... yes?
@RuneHaugseng doesn't this follow from the fact that the boundary of a simplex is a compact object?
X is a filtered colimit colim F where F(i) is n-truncated. Given a map ∂Δ^k -> X, it factors through some F(i) since ∂Δ^k is compact, and if k>n, then a lift to Δ^k exists to F(i) and ergo to X, ergo all k-boundaries for k>n admit fillers, so we're done. I think this works
@RuneHaugseng Yes. One cheap way of seeing directly (beyond what Harry and Tom already mentioned) is that a space $X$ is $n$-truncated iff the diagonal $X→X^{S^{n+1}}$ is an equivalence and $S^{n+1}$ is a compact object (HTT.5.5.6.17)
This also shows that the result is true whenever you are in an ∞-category where finite limits commute with filtered colimits (e.g. an ∞-topos)
Let R -> S be a morphism of E_oo-ring spectra. Then A = S smash S and B = S smash_R S form "Hopf algebroids over R" in an appropriate sense, right? Moreover I should be able to form the "derived cotensor" of A and S over B. Does this give me S smash R? Moreover, I feel like this whole "hopf algebroid in spectra" business, including the answer to the above question, should just be some kind of stupid thing about cosimplicial objects. Can I read about this somewhere?
A long time ago in this very chat I learned from @EricPeterson that Segal proved that the map BU(1) --> BU of infinite loop spaces is a rational equivalence. Paraphrasing a remark of Eric, since the splitting lemma for vector bundles tells you that any such bundle is decomposed as a sum of line bundles, it would be interesting to understand why BU(1) --> BU is not an equivalence mod p. i.e. to have a geometric explanation for the difference that appears mod p.
I haven't learned anything in that direction since that moment but I'm still intrigued, wondering if any of you here have anything to add
Could it be true that this map S[CP^infty]--->BU is n connective after inverting n! or something of the sort that its connectuvity is improved when you invert hiegjt and higher primes?