Witt vector

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.

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 not the only possible choice, and Teichmüller suggested an alternative set consisting of 0 together with the p − 1st roots of 1: in other words, the p roots of

x^p − x = 0.

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

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

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 W_i 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 + (X₀^p + Y₀^p − (X₀ + Y₀)^p)/p, ...)

(X₀, X₁,...) × (Y₀, Y₁,...) = (X₀Y₀, X₀^pY₁ + Y₀^pX₁ + p X₁Y₁, ...)

Examples

• The Witt ring of any commutative ring R in which p is invertible is just isomorphic to R^N (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 R^N, 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.

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 W_n 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).

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). The Witt scheme can be canonically identified with the spectrum of the ring of symmetric functions.

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

Moreover, the functor taking the commutative ring R to the set Rⁿ is represented by the affine space $\mathbb{A}_{\mathbb{Z}}^n$, and the ring structure on Rⁿ makes $\mathbb{A}_{\mathbb{Z}}^n$ into a ring scheme denoted $\underline{\mathcal{O}}^n$. From the construction of truncated Witt vectors it follows that their associated ring scheme $\mathbb{W}_n$ is the scheme $\mathbb{A}_{\mathbb{Z}}^n$ with the unique ring structure such that the morphism $\mathbb{W}_n\rightarrow \underline{\mathcal{O}}^n$ given by the Witt polynomials is a morphism of ring schemes.

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 G_a. 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.

See also

• Formal group
• Artin–Hasse exponential

References

• Dolgachev, I.V. (2001), "Witt vector", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=Witt_vector 
• Hazewinkel, Michiel (2009), "Witt vectors. I.", Handbook of algebra. Vol. 6, Amsterdam: Elsevier/North-Holland, pp. 319–472,, arXiv:0804.3888, ISBN 978-0444532572, MR2553661 
• Mumford, David, Lectures on Curves on an Algebraic Surface, Annals of Mathematics Studies, 59, Princeton, NJ: Princeton University Press, ISBN 978-0-691-07993-6 
• Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics, 67, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90424-5, MR554237 , section II.6
• Serre, Jean-Pierre (1988), Algebraic groups and class fields, Graduate Texts in Mathematics, 117, Berlin, New York: Springer-Verlag, ISBN 978-0-387-96648-9, MR918564 
• Witt, Ernst (1936), "Zyklische Körper und Algebren der Characteristik p vom Grad pⁿ. Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassenkörper der Charakteristik pⁿ" (in German), Journal für Reine und Angewandte Mathematik 176: 126–140, http://www.digizeitschriften.de/main/dms/img/?IDDOC=504725 
• Greenberg, M. J. (1969), Lectures on Forms in Many Variables, New York and Amsterdam, Benjamin, MR 241358, ASIN: B0006BX17M