Construction of the Ring of Witt Vectors

I will describe a functor A 7→ W (A) from the category of commutative rings to itself. The ring W (A) of ‘Witt vectors’ over A has many applications (to algebraic geometry, local rings, etc.), but I won’t discuss those. Convention: rings have 1’s that are respected by ring homomorphisms. By A I will always denote a commutative ring. The literature on the functor W is in a somewhat unsatisfactory state: nobody seems to have any interest in Witt vectors beyond applying them for a purpose, and they are often treated in appendices to papers devoting to something else; also, the construction usually depends on a set of implicit or unintelligible formulae. Apparently, anybody who wishes to understand Witt vectors needs to construct them personally. That is what is now happening to myself. One may compare the construction of W (A) to the construction of the polynomial ring A[X]: the ring operations in the latter are also defined by formulae, but those are both explicit and intelligible. In addition, A[X] can be thought of in a conceptual way: it is an A-algebra that represents the forgetful functor from the category of A-algebras to the category of sets. It is quite possible thatW (A) also represents some functor, and that this helps in constructing W ; but I never saw a satisfactory treatment along these lines. For W (A), the arrows run in the opposite direction: A is a W (A)-algebra rather than the other way around, and if W (A) represents a functor then most likely it is a contravariant one. If the only available way to construct W is by implicit formulae, then one is doomed to using those formulae whenever one wishes to prove any result about Witt vectors. The theory as found in the literature is indeed formula-ridden. My treatment depends also on a formula (see (ii) below), but it is both explicit and intelligible. One may be hopeful that my approach will pass the test of allowing a smooth development of the entire theory of Witt vectors. For example, one can use it to construct an important morphism W →W ◦W that turns each W (A) into a ‘lambda-ring’. I start by defining a ring Λ(A) that is isomorphic to W (A), the only difference being notational. Let A[[T ]] be the ring of power series in one