Closed brankoju closed 3 years ago
You're absolutely right. Fabian also sent me an e-mail about this error and he confirmed that Lurie's lecture notes have a weird sign convention. Which is never mentioned :angry:. Misunderstanding Lurie's sign convention is also what made me think ku
should not be complex oriented (because as you said, we wouldn't obtain ku^0(ku)=KU^0(ku)
).
I've fixed this in the latest version, which went online a few minutes ago.
Just a small issue on page 175… you use KU instead of ku because you were concerned whether $ku$ is complex oriented. It should be complex oriented because the homotopy groups vanish in odd degrees, see e.g. lurie's lecture notes on chromatic homotopoy theory, lecture 4, remark 4 (one could also use remark 3 and that ku is the connective cover of KU )
I was a bit confused why we actually obtain
ku^0(ku)=KU^0(ku)
in the end. The reason is a sign error (which also appears in Lurie's lecture notes... or he has different conventions). I think thatE^*(BU(n)) \simeq \pi_{-*}(E)[[c_1, \dots , c_n]]
. In particularE^*(BU(n))
should be a module over\pi_{-*}(E)
(and not\pi_*(E)
) because a class inE^k(BU(n))
is represented by a mapS[-k] \to E
in spectra (and notS[k] \to E
)