Construction of the ring of Witt vectors

The paper contains an intelligible construction of the ring W(A) of Witt vectors over an arbitrary commutative ring A.

I will describe a functor A → 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 devoted 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 that W (A) also represents some functor, Editor's note. This paper is published here exactly as it had appeared (as a preprint) in 2002 and therefore may be viewed also as a historical document. The abstract, MSC codes and keywords were provided by the editor.  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.
This is a multiplicative group, and is a functor from the category of commutative rings to the category of abelian groups. The multiplication on (A) will serve as the "addition" in a new ring structure to be defined on (A).
Theorem There is a unique system of maps * = * A : (A) × (A) → (A), one for each commutative ring A, such that: (i) * is left and right distributive with respect to ×; (ii) for all A and all a, b ∈ A, one has commutes.

For each A, the map * A is T -adically continuous and makes (A) into a commutative ring with addition ×, multiplication * and unit element
Finally, is a functor from the category of commutative rings to itself.
The elements occurring in (ii) are sums of geometric progressions: Thus, on elements of this form, the operation * is given by coefficientwise multiplication, the "Hadamard product". The unit element (1 − T ) −1 has all coefficients equal to 1. One finds also other normalizations in the literature, leading to unit element 1 − T (invert all elements of (A)) or 1 + T (substitute −T for T ). My convention keeps the formulae simple, and leads for zeta functions of varieties X , Y over a finite field k to the pleasing formula I now first prove existence of the operations * A . For each n 0, put to be the subgroup of n (A) generated by {1 − aT : a ∈ A}. The strategy is to first make each M n (A) into a ring, next extend the ring structure to n (A) (this will require varying A), and finally pass to (A) by taking the projective limit.
Lemma 1 For each commutative ring A and non-negative integer n, the abelian group M n (A) has a unique composition * A satisfying property (ii) and making M n (A) into a commutative ring; also, M n is a functor from the category of commutative rings to itself, and the natural maps M n+1 → M n are morphisms of functors.
Example The map A → M 1 (A) sending a to 1 + aT (mod T 2 ) is bijective, and the ring structure on M 1 (A) makes it into an isomorphism of rings.
Proof For a ∈ A, the A-algebra endomorphism induces an element ϕ a of the endomorphism ring End n (A) of n (A). Clearly one has ϕ a ϕ b = ϕ ab for a, b ∈ A. Hence, if E ⊂ End n (A) denotes the additive subgroup generated by {ϕ a : a ∈ A}, then E is a commutative subring of End n (A). The natural action of E on n (A) makes n (A) into an E-module, and I write the action exponentially.
The map , since it is generated by the images of generators. The kernel is a left ideal I of E, and one obtains a group isomorphism Since E is commutative, I is a two-sided ideal of E, so E/I has a ring structure. One can now transport the ring structure from E/I to M n (A). All assertions in the lemma are then straightforward to verify.
Next I pass from M n (A) to n (A). It would be convenient if every monic polynomial over A were a product of linear factors, since then one had identities like 1 + a 1 T + · · · + a n T n = showing that n (A) = M n (A). This is true, for example, if A is an algebraically closed field. Also for A = R one can show that n (A) = M n (A). In general one must vary the ring. From (ii) one sees that A is free as an A-module, and that the map from A to A is injective.

Lemma 2 For each A, there is an A-algebra A such that
The lemma is much stronger than what I need. It would be enough to show that for each n and for each finite subset F ⊂ n (A) there exists a faithfully flat A-algebra A F,n with F ⊂ M n (A F,n ).

Repeating the construction, write A = (A ) , and inductively A (n) = (A (n−1) ) (where A (0) = A). It is now routine to verify that the A-algebra
has the properties stated in the lemma.
There are many ways of making other rings that do the job just as well, but the following lemma shows that there is no reason to care about this at all.
Lemma 3 Let A ⊂ B be commutative rings, n 0, and let u, v ∈ n (A) be such that u, v ∈ M n (B). Then u * B v and u * A v lie in n (A) and are equal.
Since one can write A = i∈I Ae i with e 0 = 1, one has C = i∈I Be i . From this one sees that there are inclusions B, A ⊂ C, and that inside C one has B ∩ A = A (elements of B can only at e 0 have a non-zero coefficient).
Therefore one has u * Each n is a functor from the category of commutative rings to itself, and the natural maps n+1 → n are morphisms of functors. Thus (A) = lim ← − n n (A) now gets a ring structure. This proves the existence part of the theorem, and also shows the additional properties of * A . The only thing left to prove is uniqueness.

Lemma 4 Let I and J be sets, and let
be a map, one for each commutative ring A, functorial in A. Then each ϑ A is continuous (where A has the discrete topology and A I and A J the product topologies); more precisely, for each j ∈ J there is a finite subset I j ⊂ I such that for all A there exists a commutative diagram the vertical maps being the obvious projections.
Proof The functor − I (taking A → A I ) from the category of commutative rings to the category of sets is isomorphic to the functor Rhom(Z[ X i : i ∈ I ], −) (taking A to the set of ring homomorphisms Z[ X i : i ∈ I ] → A). By Yoneda's lemma, the system of maps ϑ A corresponds to a ring homomorphism Z[ Lemma 4 now comes down to the statement that for every j ∈ J there is a finite subset I j ⊂ I such that the image of X j is in the subring Z[ X i : i ∈ I j ] of Z[ X i : i ∈ I ], and this is clear.
To (1) For all a ∈ A and u ∈ ( (the Hadamard product!). From this one can deduce that the ideal I occurring in the proof of Lemma 1 is 0.
(2) For a 1 , a 2 , b 1 , b 2 ∈ A one has (3) Let m, n be positive integers, and put l = lcm(m, n), g = gcd(m, n). Then for a, b ∈ A, one has Equivalently: if two collections of α's and β's satisfy then one has α,β (X − αβ) = X l − a l/m b l/n g . This is particularly easy to see if A is a field of characteristic 0. (4) For relatively prime positive integers m, n one has This is best understood through an interpretation of (Z) as a Burnside ring. Taking m = 14, n = 15 one concludes that (A) is not a domain for any A.
To conclude, I exhibit the relationship between the given construction of Witt vectors and the standard one.
Define the maps γ n : where u is the formal derivative of u with respect to T .

Proposition Each γ n is a ring homomorphism, functorial in A. The ring structure on the set (A) is characterized by being functorial in A and all γ n being ring homomorphisms.
Proof It is well-known that the logarithmic derivative u → u /u transforms multiplication into addition. For u = (1 − aT ) −1 one has This is multiplicative in a, so on elements of the form (1 − aT ) −1 each γ n transforms * into multiplication. Using functoriality and continuity one concludes that it gives a ring homomorphism. As for the last statement, with Yoneda's lemma one reduces the proof to the case of polynomial rings over Z, and one uses that for those rings the map u → T u /u is injective; the details are left to the reader. (1 − a m T m ) −1 for n = 0, 1, 2, . . . as well as the map The proof is routine. I can now relate the standard definition of W (A) to the construction given.
Here is a diagram in the category of commutative rings that is important in the theory of Witt vectors: The top horizontal map is ϕ. The right vertical map sends u to T u /u; by the proposition, it is a ring homomorphism if TA [[T ]] has the usual addition and Hadamard multiplication. The bottom horizontal map sends (a n ) n 1 to ∞ n=1 a n T n ; it is a ring isomorphism if ∞ n=1 A has componentwise ring operations. The left vertical map is defined by the commutativity of the diagram. By a straightforward computation, it sends (a n ) ∞ n=1 to (a (n) ) ∞ n=1 , where the "ghost components" a (n) are given by a (n) = d|n da n/d d .
By the proposition, the ring structure on W (A) is characterized by functoriality and by the ghost components being ring homomorphisms W (A) → A. This is often taken as the definition of W (A).

March 4, 2002
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