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FlorianAdler / AlgebraBonn

Lecture Notes for Algebra Lectures at the University of Bonn
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complex oriented #7

Closed brankoju closed 3 years ago

brankoju commented 3 years 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 that E^*(BU(n)) \simeq \pi_{-*}(E)[[c_1, \dots , c_n]] . In particular E^*(BU(n)) should be a module over \pi_{-*}(E) (and not \pi_*(E)) because a class in E^k(BU(n)) is represented by a map S[-k] \to E in spectra (and not S[k] \to E )

FlorianAdler commented 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.