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 {Xi}∞i=0 be a sequence of indeterminates. The polynomials Wn∈Z[X1,…,Xn] given by:
W0 | = | X0, | ||
W1 | = | Xp0+pX1, | ||
Wn | = | Xpn0+pXpn-11+…+pnXn=n∑i=0piXpn-ii. |
are called the Witt polynomials.
Proposition 1.
Let {Xi},{Yi} be two sequences of indeterminates. For every polynomial in two variables Q(U,V)∈Z[U,V] there exist polynomials {ti}∞i=0 in the variables {Xi} and {Yi}
ti∈ℤ[{Xi},{Yi}] |
such that
Wn(t0,t1,t2,…,tn)=Q(Wn(X0,X1,…),Wn(Y0,Y1,…)) |
for all n≥0.
Proof.
See [1], p. 40. ∎
Let S0,S1,S2,… (resp. P0,P1,P2,…) be the polynomials t0,t1,t2,… associated with Q(U,V)=U+V (resp. Q(U,V)=U⋅V) given by the previous proposition. We will use the polynomials Si, Pi to define the addition and multiplication in a new ring. In the following proposition, the notation R∞ stands for the set of all sequences (r1,r2,…) 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=(S0(k,h),S1(k,h),…),k⋅h=(P0(k,h),P1(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 Wn(K)=(Kn-1,+,⋅), where, for k,h∈Kn-1:
k+h=(S0(k,h),…,Sn-1(k,h)),k⋅h=(P0(k,h),…,Pn-1(k,h)). |
It is clear from the definitions that:
W(𝕂)=lim←Wn(𝕂) |
In words, W(𝕂) is the projective limit of the Witt vectors of finite length.
Example 1.
Let 𝕂=𝔽p. Then Wn(𝔽p)=ℤ/pnℤ. Thus:
W(𝔽p)=ℤp, |
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 |