# , , and the ring of Witt vectors

No.08:

In the following, we use infinite sums and infinite products of elements of . They are defined as limits of sums and products with respect to the filtration topology defined in the usual way.

LEMMA 08.1   Let be any commutative ring. Then every element of is written uniquely as

PROOF.. We may use an expansion

to inductively determine . More precisely, for each , let us define a polynomial in the following way:

Then for any element , we define

Then it is easy to verify that an equation

holds.

COROLLARY 08.2   is topologically generated by

LEMMA 08.3   Let be positive integers. Let be the least common multiple of . Then we have

(Note that .)

PROOF.. let be positive integers. Let be the least common multiple of . We have,

DEFINITION 08.4   Let be any commutative ring. Then we define an addition and a multiplication on who satisfy the following requirements:
1. .
2. For any positive integer , Let be the least common multiple of . Then we have

3. for general , the multiplication is defined by first expressing as a formal -sum as in Lemma 8.3 and then applying the rule 2 formally to each  -summand''.

(Note that Lemma 8.1 guarantees the existence and the uniqueness of such multiplication .)

THEOREM 08.5   Let be any commutative ring. Then:
1. Any element of is written uniquely as

2. forms a commutative ring under the binary operations and . More precisely,
1. is an additive group with the zero element .
2. The multiplication is an associative commutative product on with the unit element .
3. The distributive law holds.
3. When , the ring is isomorphic to via the map .

PROOF.. When , the statements trivially hold. This implies in particular that rules such as distributivity and associativity hold for universal cases (that means, for formal power series with indeterminate coefficients). Thus we conclude by specialzation arguments that the rule also hold for any ring .

DEFINITION 08.6   For any commutative ring , elements of are called universal Witt vectors over . The ring is called the ring of universal Witt vectors over .

PROPOSITION 08.7   is uniquely determined by the following properties.
1. .
2. The multiplication is -biadditive. (That means, obeys the distributive law.)
3. .
4. is continuous.
5. is functorial.

PROOF.. We only need to prove the requirement (2) of Definition 8.4. With the help of distributive law, the requirement is satisfied if an equation

 (#)

holds for each .

To that aim, we first deal with a special case where , , algebraically independent over . In that case we may easily decompose the polynomials and and then we use the distributive law to see that the requirement actually holds. Indeed, let us put

and compute as follows.

We second deal with a case where , algebraically independent over . In that case we take a look at an inclusion

and consider . It is easy to see that is injection so that the equation (#) is also true in this case. The general case now follows from specialization argument.