In fact it seems that, if the characteristic of F_p were zero, knowing the cohomology of X(p) as a dg-algebra would tell you that it is a torus by rational homotopy theory, and the base would have to really be a classifying space.

That makes me think it should be possible to calculate the cohomology explicitly by thinking about the spectral sequence, maybe using the algebra structure and the p large assumption at some point. (Maybe we get stuck at a certain point if p is too small…)

The next thing to compute is E_{2}^{3,0} (where E_2 is the first page of your spectral sequence). This has a differential going to E_2^{1,1}, which you know vanishes, and then E_{3}^{3,0} has a differential going to E_{3}^{0,2}, which is the cokernel of the differential E_{2}^{2,1} => E_2^{0,2}. With your prexisting calculations, we see that this is H^0 (SL(F_p), L tensor M tensor M) => H^0( SL(F_p), L tensor wedge^2 M). The map is obtained from cup product with the map you wrote down, so it comes from the natural map M tensor M => wedge^2 M that arises from cup product in the formula you get from Lazard, which I guess is the obvious such map. If so then this splits in characteristic > 2 at least, and so H^3 vanishes in characteristic greater than two.

By some big inductive argument maybe you get H^i has a simple formula for i at most p, and the first really interesting cohomology (i.e. not stable in p and n simultaneously) shows up in degree p+1.

]]>I guess I’m not going. ]]>

– wasted effort from multiple experts independently repeating the same checking (since none post their results),

– less- or non-expert readers giving up,

– a reliance on folklore whose accessibility and existence attenuates with distance from the centers of learning,

– even in the centers, a critical dependence on a tiny number of uber-experts with encyclopedic understanding of foundations and literature who serve as de facto oracles on what’s what.

The last in particular is a ridiculous situation and, as is implicit in your comment on the difference between arithmetic geometry and machine learning (and Kevin Buzzard’s comment on not having to ask a human oracle) scales very badly with the complexity of dependencies in the literature.

When I was in grad school, there was no MathOverflow, and one of the papers that I wasted lots of time puzzling over was a long and dense article on arithmetic geometry. As it turned out, the central construct in the paper (a very promising thing for many applications) puzzled many other people, because it was critically wrong, but the exact reasons why were subtle, the repairs difficult and un-obvious, and public statements of the problems did not appear for 15 years after the publication. At the time I assumed it was me not understanding something. For political reasons I could not, or thought I should not, ask the author. Recently, I saw that someone asked about the same problem on MO, and got an answer and a link to a paper resolving this longstanding issue. I’d call that an instance of MO improving the literature, and having such things be systematized per-publication and not as patchwork of random MO questions would be even better.

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