In this entry we define a commutative ring, the Witt vectors, which is particularly useful in number theory, algebraic geometry and other areas of commutative algebra. The Witt vectors are named after Ernst Witt.
Let be a perfect ring of characteristic . The unique strict -ring (http://planetmath.org/PRing) with residue ring is called the ring of Witt vectors with coefficients in .
Next, we give an explicit construction of the Witt vectors.
Let be two sequences of indeterminates. For every polynomial in two variables there exist polynomials in the variables and
for all .
See , p. 40. ∎
Let (resp. ) be the polynomials associated with (resp. ) given by the previous proposition. We will use the polynomials , to define the addition and multiplication in a new ring. In the following proposition, the notation stands for the set of all sequences of elements in .
Let be a perfect ring of characteristic . We define a ring where the addition and multiplication, for , are defined by:
Then the ring concides with , the ring of Witt vectors with coefficients in .
Let be a perfect ring of characteristic . We define the ring of Witt vectors of length (over ) to be the ring , where, for :
It is clear from the definitions that:
In words, is the projective limit of the Witt vectors of finite length.
Let . Then . Thus:
the ring of -adic integers (http://planetmath.org/PAdicIntegers).
- 1 J. P. Serre, Local Fields, Springer-Verlag, New York.
|Date of creation||2013-03-22 15:14:31|
|Last modified on||2013-03-22 15:14:31|
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