Wikipedia / Witt vector
Wikipedia / 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 over the finite field of prime order p is isomorphic, the ring of p-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 behind Witt vectors is that instead of using the standard p-adic expansion, an expansion using the Teichmüller character can be considered instead;,which is a group morphism sending each element in the solution set can be expanded out in terms of roots of unity. A p-adic integer can then be expressed as an infinite sum,which gives a Witt vector. induces a commutative ring homomorphism.
History
In the 19th century, Ernst Eduard Kummer studied cyclic extensions of fields as part of his work on Fermat's Last Theorem. This led to the subject known as Kummer theory-th root of unity. Kummer theory classifies degree. Such fields are in bijection cyclic groups.
has characteristic. By expanding using binomial coefficients-th power, known here as the Frobenius homomorphism. 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. This inspired them to ask what other fields had finite absolute Galois groups. In the midst of proving were the same as splitting fields of Artin–Schreier polynomials extensions. Abraham Adrian Albert extensions. Each repetition entailed complicated algebraic conditions to ensure that the field extension was normal.
Schmid. In the process of doing so, certain polynomials, the -typical Witt vectors as a quotient analog of Artin–Schreier polynomials is
,
are in bijective correspondence with cyclic subgroups.
Motivation
) can be written as a power series are usually taken from the integer. It can be difficult to provide an algebraic expression is only one of many choices, and Hensel-adic numbers) suggested the roots of unity in the field as representatives in which the residue field. Indeed, it is these fields (the fields of fractions. 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 are taken to their representatives by the Teichmüller character.
, describe their sum and product as p-adic integers explicitly. This problem was solved by Witt using Witt vectors.
Detailed motivational sketch
using a construction which naturally generalizes to the Witt vector construction.
of p-adic integers can be understood as the inverse limit. That is, each successive element of the sequence is equal to the previous elements modulo a lower power of p.
can be expanded as (formal) power series
,
using the standard metric on the reals with the p-adic metric.
, the following definition can be considered for addition:
to simplify the given sum into a form with coefficients in the allowed set.
Representing elements in Fp as elements in the ring of Witt vectors W(Fp)
which does yield closed formulas, the Teichmüller representatives through Hensel lifting, the p-adic version of Newton's method, is the sequence.The existence of a lift in each step is guaranteed by the greatest common divisor.
. This defines the Teichmüller character as a (multiplicative) group homomorphism is not
Representing elements in Zp as elements in the ring of Witt vectors W(Fp)
, one can expand every p-adic integer as a power series in p. Then, if one has some arbitrary p and proceed likewise. This yields a sequence of congruences
so that
implies
for
.
This obtains a power series for each residue of modulo powers of .
,
since
, so the difference tends to 0 with respect to the p except the first one.
Additional properties of elements in the ring of Witt vectors motivating general definition
, as the first congruence implies. In particular,
and thus
.
, this gives
.
satisfying the basic aim of defining a simple additive structure.
this step can be cumbersome. Write
.
th power is not enough: one must take
.
At this step, one works with addition of the form
This motivates the definition of Witt vectors.
Construction of Witt rings
Fix a prime number p. A Witt vector. The Witt polynomials can be defined by
and in general
.
are called the ghost components define the ghost map is p-torsionfree, then the ghost map is injective-module of sequences (though note that the ghost map is not surjective is p-divisible).
The ring of (p-typical) Witt vectors into a ring such that:
, and
,.
In other words,
are given by polynomials with integer coefficients that do not depend on R, and
The first few polynomials giving the sum and product of Witt vectors can be written down explicitly. For example,
remains and the formula makes sense. The same consideration applies to the ensuing components.
Examples of addition and multiplication
As would be expected, the identity element is the element,since, thenshowing that multiplication also behaves in a highly non-trivial manner.
Examples
is invertible this homomorphism is an isomorphism.
-adic integers written in terms of the Teichmüller representatives, as demonstrated above.
is the ring of integers of the unique unramified extension.
can be described as
The Witt vectors are the inverse limit along the canonical projections .
Universal Witt vectors
by
and in general
.
is called the vector of ghost components.
These polynomials can be used to define the ring of universal Witt vectors or big Witt ring).
Generating functions
Witt also provided another approach using generating functions.
Definition
be a Witt vector and define
. Then
One can get the ghost components by taking the logarithmic derivative:
Sum
. So that
. Then
, one can show by induction
Product
is set, then
.
but
.
is the least common multiple), the series becomes
so that
,
So by similar induction,
.
Ring schemes
) 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 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.
is represented by the affine space 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 false: the truncated Witt schemes are counterexamples, any unipotent abelian connected algebraic group is isogenous to a product of truncated Witt group schemes.
Universal property
André Joyal explicated the universal property of the (p-typical) Witt vectors. The basic intuition is that the formation of Witt vectors is the universal way to deform a characteristic p ring to characteristic 0 together with a lift of its Frobenius endomorphism. To make this precise, define a -ring that is a p-derivation satisfies the relations
;
;
.
is p-ring as a suitable replacement for a Frobenius lift in the non-p-torsionfree case.
-structure assembles to a category. One then has a forgetful functorwhose right adjoint creates limits and colimits and admits an explicitly describable left adjoint as a type of free functor inherits local presentability by appealing to the adjoint functor theorem.
restricts to a fully faithful functor on the full subcategory of perfect rings of characteristic p is an isomorphism) and whose underlying ring is p-adically complete.
See also
References
Introductory
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.
Applications
, section II.6
References