nLab
Hazewinkel, Witt vectors

category: reference

This entry provides a reading guide to the text

• Hazewinkel, Witt vectors, pdf.

• 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’.