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,
, and the ring of Witt vectors
No.07:
From here on, we make use of several notions of category theory.
Readers who are unfamiliar with the subject is advised to see
a book such as [1] for basic definitions and first properties.
Let
be a prime number.
For any commutative ring
of characteristic
, we want to
construct a ring
of characteristic 0
in such a way that:
-
.
-
is a functor. That means,
- For any ring homomorphism
between rings of characterisic
,
there is given a unique ring homomorphism
.
-
should furthermore
commutes with compositions of homomorphisms.
To construct
, we construct a new addition and multiplication on a
-module
. The ring
will then be called
the ring of Witt vectors.
The treatment here
essentially follows the treatment which appears in [2, VI,Ex.46-49],
with a slight modification (which may or may not be good-it may even be wrong)
by the author.
We first introduce a nice idea of Witt.
DEFINITION 07.1
Let

be a ring (of any characteristic).
Let

be an indeterminate.
We define the following copies of

.
LEMMA 07.2
and
are functors from the category of
rings to the category of sets. They are represented by
``polynomial rings in infinite indeterminates''
and
That means, there are functorial bijections
and
DEFINITION 07.3
We define the following ``universal elements''.
LEMMA 07.4
There is an well-defined map
If
contains an copy of
, then the map is a bijection. The inverse is
given by
PROOF..
To see that

is well defined (that is, ``defined over

''), we compute
as follows.
The rest should be obvious.
Note: the condition
is required to guarantee exictence of
exponential
and existence of the integration
.
DEFINITION 07.5
We equip
![$ \mathcal W_0(A)=T A[[T]]$](img36.png)
with the usual addition and the following
(unusual) multiplication:
It is easy to see that

forms a (unital associative)
commutative ring with these binary operations.
DEFINITION 07.6
Let

be a ring which contains a copy of

. Then we define ring structure on

by putting
PROOF..
easy
We may thus extend the definition
on
to cases
where the condition
is no longer satisfied.
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2008-07-08