Witt vectors. Part 1

Publisher Summary This chapter discusses the aspects of Symm, which relate directly to the Witt vector constructions and their properties. The Witt vector construction is a very beautiful one. But it takes one out of the more traditional realms of algebra very quickly. The big and p -adic and truncated Witt vectors carry ring and algebra structures, and hence, naturally, are sometimes referred to as a “ring or algebra of Witt vectors” and then as “Witt ring” and “Witt algebra.” This is a bit unfortunate and potentially confusing because these phrases mostly carry other totally unrelated meanings. Mostly, Witt ring refers to a ring of equivalence classes of quadratic forms, with addition and multiplication induced by direct product and tensor product respectively. The functor represented by Symm—that is, the big Witt vector functor has a comonad structure and the associated coalgebras are precisely the λ-rings. All this by no means exhausts the manifestations of and structures carried by Symm.