 12:51 AM
let's more explicitly nose around the v1-divisibility in the E := E(2)/p-homology of K(Z, 3), and pick any element x. Wuergler says inverting v1 on E gives something bousfield equivalent to K(1), for which K(Z, 3) is acyclic—so x has to be v1-power-torsion, lest it survive into K(1)-homology
v1-multiplication on E fits into a lexseq with K(2)_* K(Z, 3); if x is finitely v1-divisible, its last nonzero product v1^j x and its most divisible v1^-k x will both contribute to K(2)_—but one in even degree and one in odd degree. so the only option is for x to be v1-torsion and infinitely v1-divisible, and then its last nonzero v1^j x will contribute to K(2)_ K(Z, 3) by the kernel of the v1-mult map, so that x is in odd degree
so the hopf algebra coming out of topology lives all in odd degrees, so has no visible product or coproduct. not the kind of thing that will be easily in the image of an alternating algebra construction
this leg of the discussion has been mod p, but my ~ suspicion ~ is that trying to re-inject p into the picture will cause similar mess. you could try to play around with that part by knowing the mod-p homology of K(Z, 3), using an ASS to recover ku_* K(Z, 3), and quotienting/inverting p/v1 in whatever combination pleases you
if you do reinject p, then E(2) is BP-flat, so BP is guaranteed to be at least as gross. at the moment, we can only guess that it's at least as gross—but it feels like a good guess
a slightly different lesson you might distill from this is that outright taking the homology of something (or homotopy!—the same garbage litters the homotopy of the E(1)- and E(2)-local spheres) is asking for trouble, and it is "better" (as in: less nasty) to understand local data + gluing data than to be taking 'global sections' of whatever delocalized object
i suspect that there are clever ways around these kinds of complaints if you start to travel along those lines, but i never heard an example i found to be both complicated and compelling
odd-concentration aside, these kinds of divisibility towers where what made me skeptical about the construction you want living in the hopf algebra literature. well-behaved and over a field, fine; losing one of those two, hmm, maybe; infinitely divisible over a nonnoetherian base, godspeed

11 hours later… 11:45 AM
this is less algebraic topology than algebraic geometry, but i figure someone here might know an answer. fix r>0. what is an example of a smooth proper variety (necessarily of dimension geq r) over a field k of char p>0 whose hodge-de rham spectral sequence has nontrivial d_r-differential? (i guess this question has a homotopical analogue, too: one can ask for a smooth proper dg-k-linear category C whose tate spectral sequence HH(C)((u)) => HP(C) does not degenerate at E_r.)

3 hours later… 2:29 PM
@EricPeterson Thanks for the details!

6 hours later… 8:13 PM
@DenisNardin Sorry for pinging you. So once again regarding Rmk 7.2: How is exactly the hypothesis that the functor takes finite disjoint unions to products used to define the map? Obviously, we have functoriality in A[T(S)], but the problem is functoriality in the sheafification. So given s \in A[T](S), we want a map A[S] \to A[T]. By passing to a cover of S, I get element s_i \in A[T(S_i)]. I would guess that we want to descend the map \sum_i A[S_i] \to A[T] to a map A[S] \to A[T]
And that for this we need the hypothesis. But I am having trouble seeing where this is used.