Witt vector

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In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors over the finite field of order p is the ring of p-adic integers.

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[edit] Motivation

Any p-adic integer can be written as a power series a0 + a1p1 + a2p² + ... where the a's are usually taken from the set {0, 1, 2, ..., p − 1}. This set of representatives is rather artificial, and Teichmüller suggested the more canonical set consisting of 0 together with the p − 1-th roots of 1: in other words, the p roots of

xp − x = 0.

These Teichmüller representatives can be identified with the elements of the finite field Fp of order p (by taking residues mod p), so this identifies the set of p-adic numbers with infinite sequences of elements of Fp.

We now have the following problem: given two infinite sequences of elements of Fp, identified with p-adic numbers using Teichmüller's representatives, describe their sum and product as p-adic numbers explicitly. This problem was solved by Witt using Witt vectors.

[edit] Construction of Witt rings

Fix a prime number p. A Witt vector over a commutative ring R is a sequence (X0, X1,X2,...) of elements of R. Define the Witt polynomials Wi by

W_0=X_0\,
W_1=X_0^p+pX_1
W_2=X_0^{p^2}+pX_1^p+p^2X_2

and in general

W_n=\sum_ip^iX_i^{p^{n-i}}.

Then Witt showed that there is a unique way to make the set of Witt vectors over any commutative ring R into a ring, called the ring of Witt vectors, such that

  • the sum and product are given by polynomials with integral coefficients that do not depend on R, and
  • Every Witt polynomial is a homomorphism from the ring of Witt vectors over R to R.

The first few polynomials giving the sum and product of Witt vectors can be written down explicitly. For example,

(X0, X1,...) + (Y0, Y1,...) = (X0+Y0, X1 + Y1 + (X0p + Y0p − (X0 + Y0)p)/p, ...)
(X0, X1,...) × (Y0, Y1,...) = (X0Y0, X0pY1 + Y0pX1 + p X1Y1, ...)

[edit] Examples

  • The Witt ring of any commutative ring R in which p is invertible is just isomorphic to RN (the product of a countable number of copies of R). In fact the Witt polynomials always give a homomorphism from the ring of Witt vectors to RN, and if p is invertible this homomorphism is an isomorphism.
  • The Witt ring of the finite field of order p is the ring of p-adic integers.
  • The Witt ring of a finite field of order pn is the unramified extension of degree n of the ring of p-adic integers.

[edit] Universal Witt vectors

The Witt polynomials for different primes p are special cases of universal Witt polynomials, which can be used to form a universal Witt ring (not depending on a choice of prime p). Define the universal Witt polynomials Wn for n≥1 by

W_1=X_1\,
W_2=X_1^2+2X_2
W_3=X_1^3+3X_3
W_4=X_1^{4}+2X_2^2+4X_4

and in general

W_n=\sum_{d|n}dX_d^{n/d}.

We can use these polynomials to define the ring of universal Witt vectors over any commutative ring R in much the same way as above (so the universal Witt polynomials are all homomorphisms to the ring R).

[edit] Ring schemes

The map taking a commutative ring R to the ring of Witt vectors over R (for a fixed prime p) is a functor from commutative rings to commutative rings, and is also representable, so it can be thought of as a ring scheme, called the Witt scheme, over Spec(Z).

Similarly the rings of truncated Witt vectors, and the rings of universal Witt vectors, correspond to ring schemes, called the universal Witt scheme and the truncated Witt schemes.

[edit] Commutative unipotent algebraic groups

Over an algebraically closed field of characteristic 0, any unipotent abelian connected algebraic group is isomorphic to a product of copies of the additive group Ga. The analogue of this for fields of characteristic p is false: the truncated Witt schemes are counterexamples. (We make them into algebraic groups by forgetting the multiplication and just using the additive structure.) However these are essentially the only counterexamples: over an algebraically closed field of characteristic p, any unipotent abelian connected algebraic group is isogenous to a product of truncated Witt group schemes.

[edit] See also

[edit] References