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,
, 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
PROOF..
let
be positive integers. Let
be the least common multiple of
. We have,
(Note that Lemma 8.1 guarantees the
existence and the uniqueness of such multiplication
.)
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
.
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.
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