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 is isomorphic to , the ring of adic integers. They have a highly non intuitive structure upon first glance because their additive and multiplicative structure depends on an infinite set of recursive formulas which do not behave like addition and multiplication formulas for standard p adic integers. The main idea This inspired them to ask what other fields had finite absolute Galois groups. In the midst of proving that no other such fields exist, they proved that degree extensions of a field of characteristic were the same as splitting fields of Artin–Schreier polynomials. These are by definition of the form By repeating their construction, they described degree extensions. Abraham Adrian Albert used this idea to describe degree extensions. Each repetition entailed complicated algebraic conditions to ensure that the field extension was normal. Schmid generalized further to non commutative cyclic algebras of degree In the process of doing so, certain polynomials related to the addition of adic integers appeared. Witt seized on these polynomials. By using them systematically, he was able to give simple and unified constructions of degree field extensions and cyclic algebras. Specifically, he introduced a ring now called , the ring of truncated typical Witt vectors. This ring has as a quotient, and it comes with an operator which is called the Frobenius operator because it reduces to the Frobenius operator on Witt observes that the degree analog of Artin–Schreier polynomials is

where To complete the analogy with Kummer theory, define to be the operator Then the degree extensions of are in bijective correspondence with cyclic subgroups of order , where corresponds to the field .

Any adic integer (an element of , not to be confused with ) can be written as a power series , where the are usually taken from the integer interval It is hard to provide an algebraic expression for addition and multiplication using this representation, as one faces the problem of carrying between digits. However, taking representative coefficients is only one of many choices, and Hensel himself (the creator of adic numbers) suggested the roots of unity in the field as representatives. These representatives are therefore the number together with the roots of unity; that is, the solutions of in , so that This choice extends naturally to ring extensions of in which the residue field is enlarged to with , some power of Indeed, it is these fields (the fields of fractions of the rings) that motivated Hensel's choice. Now the representatives are the solutions in the field to Call the field , with an appropriate primitive root of unity (over ). The representatives are then and for Since these representatives form a multiplicative set they can be thought of as characters. Some thirty years after Hensel's works Teichmüller studied these characters, which now bear his name, and this led him to a characterisation of the structure of the whole field in terms of the residue field. These Teichmüller representatives can be identified with the elements of the finite field of order by taking residues modulo in , and elements of are taken to their representatives by the Teichmüller character This operation identifies the set of integers in with infinite sequences of elements of
Taking those representatives the expressions for addition and multiplication can be written in closed form. We now have the following problem (stated for the simplest case: ): given two infinite sequences of elements of describe their sum and product as adic integers explicitly. This problem was solved by Witt using Witt vectors.

There is a better coefficient subset of which does yield closed formulas, the Teichmüller representatives: zero together with the roots of unity. They can be explicitly calculated (in terms of the original coefficient representatives ) as roots of through Hensel lifting, the adic version of Newton's method. For example, in to calculate the representative of one starts by finding the unique solution of in with ; one gets Repeating this in with the conditions and , gives and so on; the resulting Teichmüller representative of , denoted , is the sequenceThe existence of a lift in each step is guaranteed by the greatest common divisor in every

This algorithm shows that for every , there is exactly one Teichmüller representative with , which we denote Indeed, this defines the Teichmüller character as a (multiplicative) group homomorphism, which moreover satisfies if we let denote the canonical projection. Note however that is not additive, as the sum need not be a representative. Despite this, if in then in

As would be expected, the unit in the ring of Witt vectors is the elementAdding this element to itself gives a non trivial sequence, for example in ,sincewhich is not the expected behavior, since it doesn't equal But, when we reduce with the map , we get
Note if we have an element and an element thenshowing multiplication also behaves in a highly non trivial manner.

The Witt ring of any commutative ring in which is invertible is just isomorphic to (the product of a countable number of copies of ). In fact the Witt polynomials always give a homomorphism from the ring of Witt vectors to , and if is invertible this homomorphism is an isomorphism. The Witt ring of the finite field of order is the ring of adic integers written in terms of the Teichmüller representatives, as demonstrated above. The Witt ring of a finite field of order is the ring of integers of the unique unramified extension of degree of the ring of adic numbers Note for the th root of unity, hence .

Witt also provided another approach using generating functions.

The map taking a commutative ring to the ring of Witt vectors over (for a fixed prime ) 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 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 to the set is represented by the affine space , and the ring structure on makes into a ring scheme denoted From the construction of truncated Witt vectors, it follows that their associated ring scheme is the scheme with the unique ring structure such that the morphism given by the Witt polynomials is a morphism of ring schemes.

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 The analogue of this for fields of characteristic 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 , any unipotent abelian connected algebraic group is isogenous to a product of truncated Witt group schemes.

Notes on Witt vectors: a motivated approach Basic notes giving the main ideas and intuition. Best to start here! The Theory of Witt Vectors Elementary introduction to the theory. Complexe de de Rham Witt et cohomologie cristalline Note he uses a different but equivalent convention as in this article. Also, the main points in the introduction are still valid.

, section II.6