In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring.
They have a highly non-intuitive structure[1] 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[1] behind Witt vectors is that instead of using the standard p-adic expansion
.Then, the non-trivial additive and multiplicative structure in Witt vectors comes from using this map to give
In the 19th century, Ernst Eduard Kummer studied cyclic extensions of fields as part of his work on Fermat's Last Theorem.
Consequently, Kummer theory is never applicable to extensions whose degree is divisible by the characteristic.
The case where the characteristic divides the degree is today called Artin–Schreier theory because the first progress was made by Artin and Schreier.
Their initial motivation was the Artin–Schreier theorem, which characterizes the real closed fields as those whose absolute Galois group has order two.
Each repetition entailed complicated algebraic conditions to ensure that the field extension was normal.
[3] Schmid[4] generalized further to non-commutative cyclic algebras of degree
It can be difficult to provide an algebraic expression for addition and multiplication using this representation, as one faces the problem of carrying between digits.
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.
Taking those representatives, the expressions for addition and multiplication can be written in closed form.
, describe their sum and product as p-adic integers explicitly.
of p-adic integers can be understood as the inverse limit of the rings
through Hensel lifting, the p-adic version of Newton's method.
, to calculate the representative of 2, one starts by finding the unique solution of
.The existence of a lift in each step is guaranteed by the greatest common divisor
, one can expand every p-adic integer as a power series in p with coefficients taken from the Teichmüller representatives.
satisfying the basic aim of defining a simple additive structure.
At this step, one works with addition of the form This motivates the definition of Witt vectors.
Fix a prime number p. A Witt vector[5] over a commutative ring
That is, that there is a unique way to make the set of Witt vectors over any commutative ring
into a ring such that: In other words, The first few polynomials giving the sum and product of Witt vectors can be written down explicitly.
The Witt scheme can be canonically identified with the spectrum of the ring of symmetric functions.
However, these are essentially the only counterexamples: over an algebraically closed field of characteristic
André Joyal explicated the universal property of the (p-typical) Witt vectors.
creates limits and colimits and admits an explicitly describable left adjoint as a type of free functor; from this, it can be shown that
restricts to a fully faithful functor on the full subcategory of perfect rings of characteristic p. Its image then consists of those