nLab Hazewinkel, Witt vectors

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on Lambda-rings of Witt vectors.

  • Hazewinkel talks about both the ‘orthodox’ and ‘heterodox’ approaches to λ\lambda-rings. Indeed he starts out with a lot of material on Witt vectors and their relation to pp-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 λ\lambda-rings to commutative rings.

  • He then goes ahead and defines λ\lambda-rings on page 88. At first his definition looks a bit frustrating, because Hazewinkel defines ‘λ\lambda-ring’ using the concept of ‘morphism of λ\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 λ\lambda-rings from rings equipped with Adams operations ψ p\psi_p. And then, at the bottom of page 94, he goes heterodox and defines ‘ψ\psi-rings’ to be commutative rings equipped with Adams operations — and notes that over a field of characteristic zero, λ\lambda-rings are the same as ψ\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’.

category: reference

Last revised on June 9, 2022 at 05:18:13. See the history of this page for a list of all contributions to it.