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Witt vectors Part 1,
on Lambda-rings of Witt vectors.
Hazewinkel talks about both the ‘orthodox’ and ‘heterodox’ approaches to $\lambda$-rings. Indeed he starts out with a lot of material on Witt vectors and their relation to $p$-adics. The Lambda-rings only make their debut on page 87, where the operation of ‘taking the Witt vectors’ of a commutative ring is revealed to be the right adjoint to the forgetful functor from $\lambda$-rings to commutative rings.
He then goes ahead and defines $\lambda$-rings on page 88. At first his definition looks a bit frustrating, because Hazewinkel defines ‘$\lambda$-ring’ using the concept of ‘morphism of $\lambda$-rings’! But it’s not actually circular; it’s really just a trick to spare us certain ugly equations that appear in the usual definition.
Just for fun, note the unusual remark in footnote 62 on page 88: warning the reader to ‘steer clear’ of a certain book by two famous authors.
On page 92, Hazewinkel proves the Wilkerson theorem getting $\lambda$-rings from rings equipped with Adams operations $\psi_p$. And then, at the bottom of page 94, he goes heterodox and defines ‘$\psi$-rings’ to be commutative rings equipped with Adams operations — and notes that over a field of characteristic zero, $\lambda$-rings are the same as $\psi$-rings.
On page 97, he describes ‘taking the Witt vectors’ as a comonad on the category of commutative rings. This is just another way of talking about the right adjointness property he mentioned on page 87; now he’s proving it.
On page 98 he goes orthodox again, and shows that Symm, the ring of symmetric functions in countably many variables, is the free Lambda ring on one generator.
On page 102 starts explaining ‘plethysm’.
Last revised on June 9, 2022 at 05:18:13. See the history of this page for a list of all contributions to it.