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I edited lambda-ring, added a definition from the thesis of John R. Hopkins. Later on I will add the definitions of Hazewinkel, too. This entry has a long (and very instructive) idea-section. Maybe I find time to fill in some more details to these ideas.
Thank you, this is very important topic we should have more in $n$Lab. John Baez was at some time interested, I think some of his ideas deserve more thought, and of course, now it is very actual with the developments in absolute algebraic geometry.
Maybe I find time to fill in some more details to these ideas.
I have done this now: lambda-ring. What was explained in prose I wrote in a more formal (looking) way. I also split the section containing the reading guide to Hazewinkel’s article from the main article. What is still on the to-do list is to merge lambda ring and special lambda-ring.
Couple of queries on this page. First, a minor one: I found a thesis due to a John Hopkinson, not John Hopkins. Do I have the right person (perhaps adding to the confusion is that the supervisor was Mike Hopkins)?^{1}
More importantly, Wilkerson’s theorem is stated as:
Let $A$ be an additively torsion-free commutative ring. Let $\{\psi_p\}$ be a commuting family of Frobenius lifts.
Then there is a unique $\lambda$-ring structure on $A$ whose Adams operations are the given Frobenius lifts $\{\psi_p\}$.
This is certainly true rationally, but I’m not sure that it is true integrally. The relationship between the Adams’ operations and the lambda operations is that the $n$th lambda operation is determined by the Adams’ operations (and lower lambda operations) upto a multiplier of $n!$. The correct statement (I believe) is that the Adams operations determine the lambda operations in the torsion-free setting, so long as they are already there. Hopkinson’s thesis states it this way, I’m unable to get a (free, electronic) copy of Wilkerson’s original paper to see what this contained.
Evidence for this is supplied by the fact that the link to the PDF on the page lambda ring has changed to its “visited” form after I downloaded John Hopkinson’s thesis via another route. ↩
Hmm, maybe there’s something about the $\psi_p$ being Frobenius lifts that makes all the difference. Haven’t yet tracked down a detailed proof, though.
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