Witt vectors

Witt vectors

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.

Theorem 1

.

Let

p

be a prime and let

K

be a

perfect ring

of

characteristic

p

. There exists a unique strict

p

-ring (

http://planetmath.org/PRing

)

W

⁢

(

K

)

with

residue

ring

K

.

Definition 1

.

Let

K

be a perfect ring of characteristic

p

. The unique strict

p

-ring (

http://planetmath.org/PRing

)

W

⁢

(

K

)

with residue ring

K

is called the ring of Witt vectors with

coefficients

in

K

.

Next, we give an explicit construction of the Witt vectors.

Definition 2

.

Let

p

be a

prime number

and let

{

X

i

}

i

=

0

∞

be a

sequence

of

indeterminates

. The

polynomials

W

n

∈

Z

⁢

[

X

1

,

…

,

X

n

]

given by:

W

0

=

X

0

,

W

1

=

X

0

p

+

p

⁢

X

1

,

W

n

=

X

0

p

n

+

p

⁢

X

1

p

n

-

1

+

…

+

p

n

⁢

X

n

=

∑

i

=

0

n

p

i

⁢

X

i

p

n

-

i

.

are called the Witt polynomials.

Proposition 1

.

Let

{

X

i

}

,

{

Y

i

}

be two sequences of indeterminates. For every polynomial in two

variables

Q

⁢

(

U

,

V

)

∈

Z

⁢

[

U

,

V

]

there exist polynomials

{

t

i

}

i

=

0

∞

in the variables

{

X

i

}

and

{

Y

i

}

t

i

∈

ℤ

⁢

[

{

X

i

}

,

{

Y

i

}

]

such that

W

n

⁢

(

t

0

,

t

1

,

t

2

,

…

,

t

n

)

=

Q

⁢

(

W

n

⁢

(

X

0

,

X

1

,

…

)

,

W

n

⁢

(

Y

0

,

Y

1

,

…

)

)

for all

n

≥

0

.

Proof.

See

[

1

]

, p. 40.
∎

Let

S

0

,

S

1

,

S

2

,

…

(resp.

P

0

,

P

1

,

P

2

,

…

) be the polynomials

t

0

,

t

1

,

t

2

,

…

associated with

Q

⁢

(

U

,

V

)

=

U

+

V

(resp.

Q

⁢

(

U

,

V

)

=

U

⋅

V

) given by the previous

proposition

. We will use the polynomials

S

i

,

P

i

to define the

addition

and

multiplication

in a new ring. In the following proposition, the notation

R

∞

stands for the set of all sequences

(

r

1

,

r

2

,

…

)

of elements in

R

.

Theorem 2

.

Let

K

be a perfect ring of characteristic

p

. We define a ring

W

=

(

K

∞

,

+

,

⋅

)

where the addition and multiplication, for

k

,

h

∈

K

∞

, are defined by:

k

+

h

=

(

S

0

⁢

(

k

,

h

)

,

S

1

⁢

(

k

,

h

)

,

…

)

,

k

⋅

h

=

(

P

0

⁢

(

k

,

h

)

,

P

1

⁢

(

k

,

h

)

,

…

)

.

Then the ring

W

concides with

W

⁢

(

K

)

, the ring of Witt vectors with coefficients in

K

.

Definition 3

.

Let

K

be a perfect ring of characteristic

p

. We define the ring of Witt vectors of length

n

(over

K

) to be the ring

W

n

⁢

(

K

)

=

(

K

n

-

1

,

+

,

⋅

)

, where, for

k

,

h

∈

K

n

-

1

:

k

+

h

=

(

S

0

⁢

(

k

,

h

)

,

…

,

S

n

-

1

⁢

(

k

,

h

)

)

,

k

⋅

h

=

(

P

0

⁢

(

k

,

h

)

,

…

,

P

n

-

1

⁢

(

k

,

h

)

)

.

It is clear from the

definitions

that:

W

⁢

(

𝕂

)

=

lim

←

⁡

W

n

⁢

(

𝕂

)

In words,

W

⁢

(

𝕂

)

is the projective limit of the Witt vectors of

finite length

.

Example 1

.

Let

𝕂

=

𝔽

p

. Then

W

n

⁢

(

𝔽

p

)

=

ℤ

/

p

n

⁢

ℤ

. Thus:

W

⁢

(

𝔽

p

)

=

ℤ

p

,

the ring of

p

-adic

integers

(

http://planetmath.org/PAdicIntegers

).

References

1

J. P. Serre,

Local Fields

,
Springer-Verlag, New York.

Title

Witt vectors

Canonical

name

WittVectors

Date of creation

2013-03-22 15:14:31

Last modified on

2013-03-22 15:14:31

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Definition

Classification

msc 13K05

Classification

msc 13J10

Defines

Witt polynomials

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