This article is about of groups of Witt vectors and rings of Witt vectors; which are often called just Witt groups and Witt rings. However, there is also a different notion of Witt group? and Witt ring.
In components, a Witt vector is an infinite sequence of elements of a given commutative ring . There is a ring structure on the set of Witt vectors of and is therefore called the Witt ring of . The multiplication is defined by means of Witt polynomials for every natural number . If the characteristic of is the Witt ring of is sometimes called universal Witt ring to distinguish it from the case where is of prime characteristic and a similar but different construction is of interest.
A p-adic Witt vector is an infinite sequence of elements af a commutative ring of prime characteristic . There exists a ring structure whose construction parallels that in characteristic except that only Witt polynomials whose index is a power of are taken.
More abstractly, the ring of Witt vectors carries the structure of a Lambda-ring and the construction of the Witt ring on a commutative ring is right adjoint to the forgetful functor from Lambda-rings to commutative rings. Hence rings of Witt vectors are the co-free Lambda-rings.
The construction of Witt vectors gives a functorial way to lift a commutative ring of prime characteristic to a commutative ring of characteristic 0. Since this construction is functorial, it can be applied to the structure sheaf of an algebraic variety. In interesting special cases the resulting ring has even more desirable properties: If is a perfect field then is a discrete valuation?. This is partly due to the fact that the construction of involves a ring of power series and a ring of power series over a field is always a discrete valuation ring.
There is a generalization to non-commutative Witt vectors, however these only carry a group- but no ring structure.
In an expansion of a -adic number the are called digits. Usually these digits are defined to be taken elements of the set .
Equivalently the digits can be defined to be taken from the set . Elements from this set are called Teichmüller digits or Teichmüller representatives.
The set is in bijection with the finite field . The set of (countably) infinite sequences of elements in hence is in bijection to the set of -adic integers. There is a ring structure on called Witt ring structure such that all ”truncated expansion polynomials” called Witt polynomials are morphisms
We first give the
and then discuss the
Let be a commutative ring.
If the characteristic of is then the Witt ring of is defined defined by the addition
and the multiplication
If is of prime characteristic we index the defining formulas by instead of .
Here the are called addition polynomials and the are called multiplication polynomials, these are described below.
and in general . The value of the -th Witt polynomial in some element of the Witt ring of is sometimes called the -th phantom component of or the -th ghost component of .
Now let . This just an arbitrary two variable polynomial with coefficients in .
We can define new polynomials such that the following condition is met .
In short we’ll notate this . The first thing we need to do is make sure that such polynomials exist. Now it isn’t hard to check that the can be written as a -linear combination of the just by some linear algebra.
, and , etc. so we can plug these in to get the existence of such polynomials with coefficients in . It is a fairly tedious lemma to prove that the coefficients are actually in , so we won’t detract from the construction right now to prove it.
Define yet another set of polynomials , and by the following properties:
, and .
We now can construct , the ring of generalized Witt vectors over . Define to be the set of all infinite sequences with entries in . Then we define addition and multiplication by and .
is a functor
Composed with the forgetful functor
There is a nice trick to prove that is a ring when is a -algebra. Just define by . This is a bijection and the addition and multiplication is taken to component-wise addition and multiplication, so since this is the standard ring structure we know is a ring. Also, , so is the additive identity, which shows is the multiplicative identity, and , so we see is the additive inverse.
We can actually get this idea to work for any characteristic ring by considering the embedding . We have an induced injective map . The addition and multiplication is defined by polynomials over , so these operations are preserved upon tensoring with . We just proved above that is a ring, so since and and the map preserves inverses we get that the image of the embedding is a subring and hence is a ring.
Lastly, we need to prove this for positive characteristic rings. Choose a characteristic ring that surjects onto , say . Then since the induced map again preserves everything and is surjective, the image is a ring and hence is a ring.
This statement appears in (Hazewinkel 08, p. 87, p. 97).
This statement appears in (Hazewinkel 08, p. 98).
On untruncated -adic Witt vectors there are two operations, the Frobenius morphism and the Verschiebung morphism? satisfying relations (Lemma 1) being constitutive for the definition of the Dieudonné ring: In fact the Dieudonné ring is generated by two objects satisfying these relations.
Also the -truncations of a Witt ring are rings since by definition the ring operations (addition and multiplication) of the first components only involve the first components. We have and the projection map is a ring homomorphism. We also have operations on the truncated Witt rings.
For the truncated Witt rings and the shift operation the Verschiebung morphism equals the where is the restriction map.
The restriction map is given by .
The Frobenius endomorphism is given by . This is also a ring map, but only because of our necessary assumption that is of characteristic .
Just by brute force checking on elements we see a few relations between these operations, namely that and the multiplication by map.
For a -ring let denote the ideal in consisting of sequences of nilpotent elements in such that for large .
Let denote the Artin-Hasse exponential?. Then we have is a polynomial for and
a) is an ideal in .
b) is an morphism of group schemes.
c) The morphism
is bilinear and gives an isomorphism of group schemes
where denotes the Cartier dual of (maybe it is equivalently the Pontryagin dual of the underlying group of the (plain) ring ). That this map is a morphism of group schemes follows from the definition of the Cartier dual.
d) For and we have and
be the section of the restriction . sends in . Note that is not a morphism of groups.
We define the bilinear map
then is bilinear and gives an isomorphism
where the morphisms are
The group of universal (i.e. not -adic) Witt vectors equals i.e. the multiplicative group of power series in one variable with constant term .
Let be a perfect field of prime characteristic .
Then is a discrete valuation ring with maximal ideal generated by . From the above we see that . This clearly gives .
Also, . Thus the completion of with respect to the maximal ideal is just which shows that is a complete discrete valuation ring.
the -adic integers.
is the unique unramified extension? of of degree .
As an Abelian group is isomorphic to the group of curves in the one-dimensional multiplicative formal group. In this way there is a Witt-vector-like Abelian-group-valued functor associated to every one-dimensional formal group. For special cases, such as the Lubin–Tate formal groups, this gives rise to ring-valued functors called ramified Witt vectors. (eom)
Hochschild-Witt complex and Kaledin’s non-commutative Witt vectors
witt vectors were introduced in
In the context of formal group laws they were used in
Joseph Rabinoff, The theory of Witt vectors (pdf)