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[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 instead of using the standard
-adic expansion
![{\displaystyle a=a_{0}+a_{1}p+a_{2}p^{2}+\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2c90a4f43580c74471f42bb3cd61d68d410287c)
to represent an element in
, we can instead consider an expansion using the Teichmüller character
![{\displaystyle \omega :\mathbb {F} _{p}^{*}\to \mathbb {Z} _{p}^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2199e8152fcd7d182e7911217edd85ae975c1acf)
which sends each element in the solution set of
in
to an element in the solution set of
in
. That is, we expand out elements in
in terms of roots of unity instead of as profinite elements in
. We can then express a
-adic integer as an infinite sum
![{\displaystyle \omega (a)=\omega (a_{0})+\omega (a_{1})p+\omega (a_{2})p^{2}+\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c8a722e7211414e7ad8f49d7438b390fd4ed321)
which gives a Witt vector
![{\displaystyle (\omega (a_{0}),\omega (a_{1}),\omega (a_{2}),\ldots )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f2c2312d36f5ed3b817674597770ccf05744915)
Then, the non-trivial additive and multiplicative structure in Witt vectors comes from using this map to give
an additive and multiplicative structure such that
induces a commutative ring morphism.
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 now known as Kummer theory. Let
be a field containing a primitive
-th root of unity. Kummer theory classifies degree
cyclic field extensions
of
. Such fields are in bijection with order
cyclic groups
, where
corresponds to
.
But suppose that
has characteristic
. The problem of studying degree
extensions of
, or more generally degree
extensions, may appear superficially similar to Kummer theory. However, in this situation,
cannot contain a primitive
-th root of unity. Indeed, if
is a
-th root of unity in
, then it satisfies
. But consider the expression
. By expanding using binomial coefficients we see that the operation of raising to the
-th power, known here as the Frobenius homomorphism, introduces the factor
to every coefficient except the first and the last, and so modulo
these equations are the same. Therefore
. 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.[2] 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.[3]
Schmid[4] 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
![{\displaystyle F(x)-x-a,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c9db937808d3cb5f52c51a1c5c740365dbd2d54)
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
.
Motivation
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.
Detailed motivational sketch
We derive the ring of
-adic integers
from the finite field
using a construction which naturally generalizes to the Witt vector construction.
The ring
of
-adic integers can be understood as the inverse limit of the rings
taken along the obvious projections. Specifically, it consists of the sequences
with
such that
for
That is, each successive element of the sequence is equal to the previous elements modulo a lower power of p; this is the inverse limit of the projections
The elements of
can be expanded as (formal) power series in
![{\displaystyle a_{0}+a_{1}p^{1}+a_{2}p^{2}+\cdots ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0633ffa0d9ef7c12ce3765a32c3d996cd0e72d64)
where the coefficients
are taken from the integer interval
Of course, this power series usually will not converge in
using the standard metric on the reals, but it will converge in
with the
-adic metric. We will sketch a method of defining ring operations for such power series.
Letting
be denoted by
, one might consider the following definition for addition:
![{\displaystyle {\begin{aligned}c_{0}&\equiv a_{0}+b_{0}&&{\bmod {p}}\\c_{0}+c_{1}p&\equiv (a_{0}+b_{0})+(a_{1}+b_{1})p&&{\bmod {p}}^{2}\\c_{0}+c_{1}p+c_{2}p^{2}&\equiv (a_{0}+b_{0})+(a_{1}+b_{1})p+(a_{2}+b_{2})p^{2}&&{\bmod {p}}^{3}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6838654779e17f4f17fb8a0ded2123e6086154d)
and one could make a similar definition for multiplication. However, this is not a closed formula, since the new coefficients are not in the allowed set
Representing elements in Fp as elements in the ring of Witt vectors W(Fp)
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 sequence
![{\displaystyle \omega (2)=(2,7,57,\ldots )\in W(\mathbb {F} _{5}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb3d37e1d0549a20321362462c70ae99bccfe93e)
The 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
Representing elements in Zp as elements in the ring of Witt vectors W(Fp)
Because of this one-to-one correspondence given by
, one can expand every
-adic integer as a power series in
with coefficients taken from the Teichmüller representatives. An explicit algorithm can be given, as follows. Write the Teichmüller representative as
Then, if one has some arbitrary
-adic integer of the form
one takes the difference
leaving a value divisible by
. Hence,
. The process is then repeated, subtracting
and proceed likewise. This yields a sequence of congruences
![{\displaystyle {\begin{aligned}x&\equiv \omega (x_{0})&&{\bmod {p}}\\x&\equiv \omega (x_{0})+\omega (x'_{1})p&&{\bmod {p}}^{2}\\&\cdots \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22afa98340165d0a48c61c96a144cc724df45a7e)
So that
![{\displaystyle x\equiv \sum _{j=0}^{i}\omega ({\bar {x}}_{j})p^{j}{\bmod {p}}^{i+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72e587ca67106e61755d827dfb227a13223a162b)
and
implies:
![{\displaystyle \sum _{j=0}^{i'}\omega ({\bar {x}}_{j})p^{j}\equiv \sum _{j=0}^{i}\omega ({\bar {x}}_{j})p^{j}{\bmod {p}}^{i+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72ca3f017c50ec43c304cf4f8177d2545de60bcf)
for
![{\displaystyle {\bar {x}}_{i}:=m\left({\frac {x-\sum _{j=0}^{i-1}\omega ({\bar {x}}_{j})p^{j}}{p^{i}}}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5299feccd7bfff6d5101de3e1863d3d5636badab)
Hence we have a power series for each residue of x modulo powers of p, but with coefficients in the Teichmüller representatives rather than
. It is clear that
![{\displaystyle \sum _{j=0}^{\infty }\omega ({\bar {x}}_{j})p^{j}=x,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6790d8c7c79b1423817422aef86098061651cf8)
since
![{\displaystyle p^{i+1}\mid x-\sum _{j=0}^{i}\omega ({\bar {x}}_{j})p^{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c6d56848f53957aefff7332e76813606e3f61a9)
for all
as
so the difference tends to 0 with respect to the
-adic metric. The resulting coefficients will typically differ from the
modulo
except the first one.
Additional properties of elements in the ring of Witt vectors motivating general definition
The Teichmüller coefficients have the key additional property that
which is missing for the numbers in
. This can be used to describe addition, as follows. Consider the equation
in
and let the coefficients
now be as in the Teichmüller expansion. Since the Teichmüller character is not additive,
is not true in
. But it holds in
as the first congruence implies. In particular,
![{\displaystyle c_{0}^{p}\equiv (a_{0}+b_{0})^{p}{\bmod {p}}^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db94004d6593d20e4454346587d94754cceba6d5)
and thus
![{\displaystyle c_{0}-a_{0}-b_{0}\equiv (a_{0}+b_{0})^{p}-a_{0}-b_{0}\equiv {\binom {p}{1}}a_{0}^{p-1}b_{0}+\cdots +{\binom {p}{p-1}}a_{0}b_{0}^{p-1}{\bmod {p}}^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02e8a77da089bc10f054da51da4c005670ed8f0c)
Since the binomial coefficient
is divisible by
, this gives
![{\displaystyle c_{1}\equiv a_{1}+b_{1}-a_{0}^{p-1}b_{0}-{\frac {p-1}{2}}a_{0}^{p-2}b_{0}^{2}-\cdots -a_{0}b_{0}^{p-1}{\bmod {p}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4900d2e17d35e3d587d2ecf47b241f3a8d1e2d08)
This completely determines
by the lift. Moreover, the congruence modulo
indicates that the calculation can actually be done in
satisfying the basic aim of defining a simple additive structure.
For
this step is already very cumbersome. Write
![{\displaystyle c_{1}=c_{1}^{p}\equiv \left(a_{1}+b_{1}-a_{0}^{p-1}b_{0}-{\frac {p-1}{2}}a_{0}^{p-2}b_{0}^{2}-\cdots -a_{0}b_{0}^{p-1}\right)^{p}{\bmod {p}}^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f72f898573460e274658d539772216a35ef4679)
Just as for
a single
th power is not enough: one must take
![{\displaystyle c_{0}=c_{0}^{p^{2}}\equiv (a_{0}+b_{0})^{p^{2}}{\bmod {p}}^{3}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f14970ce1e83410cffa2d1a5b6955ddae8d66e8a)
However,
is not in general divisible by
but it is divisible when
in which case
combined with similar monomials in
will make a multiple of
.
At this step, it becomes clear that one is actually working with addition of the form
![{\displaystyle {\begin{aligned}c_{0}&\equiv a_{0}+b_{0}&&{\bmod {p}}\\c_{0}^{p}+c_{1}p&\equiv a_{0}^{p}+a_{1}p+b_{0}^{p}+b_{1}p&&{\bmod {p}}^{2}\\c_{0}^{p^{2}}+c_{1}^{p}p+c_{2}p^{2}&\equiv a_{0}^{p^{2}}+a_{1}^{p}p+a_{2}p^{2}+b_{0}^{p^{2}}+b_{1}^{p}p+b_{2}p^{2}&&{\bmod {p}}^{3}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6901ce68abfb9c4ad431f7fa9b9b16642f876f00)
This motivates the definition of Witt vectors.
Construction of Witt rings
Fix a prime number p. A Witt vector[5] over a commutative ring
(relative to the prime
) is a sequence
of elements of
. Define the Witt polynomials
by
![{\displaystyle W_{0}=X_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b49d6256693ca934e19c19add9e15fba9c2b595f)
![{\displaystyle W_{1}=X_{0}^{p}+pX_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c73145b5f0ef8feed6d6d58d53a2f8cc8d928b8)
![{\displaystyle W_{2}=X_{0}^{p^{2}}+pX_{1}^{p}+p^{2}X_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fda580cf30be55773e706e9c92de7db0a7c4290)
and in general
![{\displaystyle W_{n}=\sum _{i=0}^{n}p^{i}X_{i}^{p^{n-i}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c4517bd5ac45287b712b0e3428acc75dcac11c9)
The
are called the ghost components of the Witt vector
, and are usually denoted by
; taken together, the
define the ghost map to
. If
is p-torsionfree, then the ghost map is injective and the ghost components can be thought of as an alternative coordinate system for the
-module of sequences (though note that the ghost map is not surjective unless
is p-divisible).
The ring of (p-typical) Witt vectors
is defined by componentwise addition and multiplication of the ghost components. That is, that there is a unique way to make the set of Witt vectors over any commutative ring
into a ring such that:
- the sum and product are given by polynomials with integral coefficients that do not depend on
, and - projection to each ghost component is a ring homomorphism from the Witt vectors over
, to
.
In other words,
and
are given by polynomials with integral coefficients that do not depend on R, and
and ![{\displaystyle X^{(i)}Y^{(i)}=(XY)^{(i)}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff65e8f34635e00c2824622bde39731bf1a8f5ce)
The first few polynomials giving the sum and product of Witt vectors can be written down explicitly. For example,
![{\displaystyle (X_{0},X_{1},\ldots )+(Y_{0},Y_{1},\ldots )=(X_{0}+Y_{0},X_{1}+Y_{1}-((X_{0}+Y_{0})^{p}-X_{0}^{p}-Y_{0}^{p})/p,\ldots )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26a7dac83ae1f289c7d7e4d979436663c68b38bf)
![{\displaystyle (X_{0},X_{1},\ldots )\times (Y_{0},Y_{1},\ldots )=(X_{0}Y_{0},X_{0}^{p}Y_{1}+X_{1}Y_{0}^{p}+pX_{1}Y_{1},\ldots )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb9db60c1577c34c0b06297e999a26b18028a9df)
These are to be understood as shortcuts for the actual formulas. If for example the ring
has characteristic
, the division by
in the first formula above, the one by
that would appear in the next component and so forth, do not make sense. However, if the
-power of the sum is developed, the terms
are cancelled with the previous ones and the remaining ones are simplified by
, no division by
remains and the formula makes sense. The same consideration applies to the ensuing components.
Examples of addition and multiplication
As would be expected, the unit in the ring of Witt vectors
is the element
![{\displaystyle {\underline {1}}=(1,0,0,\ldots )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a6ee36067c9edcf010a9bfaabf28e3f40971713)
Adding this element to itself gives a non-trivial sequence, for example in
,
![{\displaystyle {\underline {1}}+{\underline {1}}=(2,4,\ldots )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31dec3ccb744890d789579da25c3327078e90d85)
since
![{\displaystyle {\begin{aligned}2&=1+1\\4&=-{\frac {32-1-1}{5}}\mod 5\\&\cdots \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2f5d692a6f2299b0119a09a25753162155e87b9)
which 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
then
![{\displaystyle {\underline {x}}a=(xa_{0},x^{p}a_{1},\ldots ,x^{p^{n}}a_{n},\ldots )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/407bd1c05fa5a5c6a17bb5d341f3b5e1566eb1d3)
showing multiplication also behaves in a highly non-trivial manner.
Examples
- 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
.
Universal Witt vectors
The Witt polynomials for different primes
are special cases of universal Witt polynomials, which can be used to form a universal Witt ring (not depending on a choice of prime
). Define the universal Witt polynomials
for
by
![{\displaystyle W_{1}=X_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2730085eb8deb8fe7d64638d97f69f0c61ac8900)
![{\displaystyle W_{2}=X_{1}^{2}+2X_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4ddcef3b5fb7f12f14320a706794e04275682fc)
![{\displaystyle W_{3}=X_{1}^{3}+3X_{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef0412404c31ece73ba5bdd442f35687f366f32b)
![{\displaystyle W_{4}=X_{1}^{4}+2X_{2}^{2}+4X_{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adf11e0de9cc6d5bc2472644c21f4b5003ae8240)
and in general
![{\displaystyle W_{n}=\sum _{d|n}dX_{d}^{n/d}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79645f6862a8dda714c862ff99e4d589a83c93ed)
Again,
is called the vector of ghost components of the Witt vector
, and is usually denoted by
.
We can use these polynomials to define the ring of universal Witt vectors or big Witt ring of any commutative ring
in much the same way as above (so the universal Witt polynomials are all homomorphisms to the ring
).
Generating Functions
Witt also provided another approach using generating functions.[6]
Definition
Let
be a Witt vector and define
![{\displaystyle f_{X}(t)=\prod _{n\geq 1}(1-X_{n}t^{n})=\sum _{n\geq 0}A_{n}t^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/598bc0e3271bdf71dfad118a6d7483da71157247)
For
let
denote the collection of subsets of
whose elements add up to
. Then
![{\displaystyle A_{n}=\sum _{I\in {\mathcal {I}}_{n}}(-1)^{|I|}\prod _{i\in I}{X_{i}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa0233a5782cf40e3be96edb0c9c82f603f0a945)
We can get the ghost components by taking the logarithmic derivative:
![{\displaystyle {\begin{aligned}-t{\frac {d}{dt}}\log f_{X}(t)&=-t{\frac {d}{dt}}\sum _{n\geq 1}\log(1-X_{n}t^{n})\\&=t{\frac {d}{dt}}\sum _{n\geq 1}\sum _{d\geq 1}{\frac {X_{n}^{d}t^{nd}}{d}}\\&=\sum _{n\geq 1}\sum _{d\geq 1}nX_{n}^{d}t^{nd}\\&=\sum _{m\geq 1}\sum _{d|m}dX_{d}^{m/d}t^{m}\\&=\sum _{m\geq 1}X^{(m)}t^{m}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5864a3e0a6fe28e1d26665fd0e88faa713191b14)
Sum
Now we can see
if
. So that
![{\displaystyle C_{n}=\sum _{0\leq i\leq n}A_{n}B_{n-i},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b113494f0783cd326a39ee6680c42df2666c5a13)
if
are the respective coefficients in the power series
. Then
![{\displaystyle Z_{n}=\sum _{0\leq i\leq n}A_{n}B_{n-i}-\sum _{I\in {\mathcal {I}}_{n},I\neq \{n\}}(-1)^{|I|}\prod _{i\in I}{Z_{i}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a52051d311454bf9b11fa644cc2d49c05971072b)
Since
is a polynomial in
and likewise for
, we can show by induction that
is a polynomial in
Product
If we set
then
![{\displaystyle -t{\frac {d}{dt}}\log f_{W}(t)=-\sum _{m\geq 1}X^{(m)}Y^{(m)}t^{m}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb1160d4fc33bc4df31da937210c323295b3fae)
But
.
Now 3-tuples
with
are in bijection with 3-tuples
with
, via
(
is the least common multiple), our series becomes
![{\displaystyle \sum _{d,e\geq 1}de\sum _{n\geq 1}\left(X_{d}^{\frac {[d,e]}{d}}Y_{e}^{\frac {[d,e]}{e}}t^{[d,e]}\right)^{n}=-t{\frac {d}{dt}}\log \prod _{d,e\geq 1}\left(1-X_{d}^{\frac {[d,e]}{d}}Y_{e}^{\frac {[d,e]}{e}}t^{[d,e]}\right)^{\frac {de}{[d,e]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/768c5828582ae9d466d18fb2986406f56f1e4fea)
So that
![{\displaystyle f_{W}(t)=\prod _{d,e\geq 1}\left(1-X_{d}^{\frac {[d,e]}{d}}Y_{e}^{\frac {[d,e]}{e}}t^{[d,e]}\right)^{\frac {de}{[d,e]}}=\sum _{n\geq 0}D_{n}t^{n},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/003da5264bd6ac84beb88dc37e9a1d2844a2c2bd)
where
are polynomials of
So by similar induction, suppose
![{\displaystyle f_{W}(t)=\prod _{n\geq 1}(1-W_{n}t^{n}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90dc2aca41efcc3c9d7789b9283addec882416b7)
then
can be solved as polynomials of
Ring schemes
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.
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
. 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.
Universal property
André Joyal explicated the universal property of the (p-typical) Witt vectors.[7] The basic intuition is that the formation of Witt vectors is the universal way to deform a characteristic
ring to characteristic 0 together with a lift of its Frobenius endomorphism.[8] To make this precise, define a
-ring
to consist of a commutative ring
together with a map of sets
that is a
-derivation, so that
satisfies the relations
;
;
.
The definition is such that given a
-ring
, if one defines the map
by the formula
, then
is a ring homomorphism lifting Frobenius on
. Conversely, if
is
-torsionfree, then this formula uniquely defines the structure of a
-ring on
from that of a Frobenius lift. One may thus regard the notion of
-ring as a suitable replacement for a Frobenius lift in the non
-torsionfree case.
The collection of
-rings and ring homomorphisms thereof respecting the
-structure assembles to a category
. One then has a forgetful functor
![{\displaystyle U:\mathrm {CRing} _{\delta }\to \mathrm {CRing} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfca509b4335f89b89d311fbac91f44e55bae4c5)
whose
right adjoint identifies with the functor
![{\textstyle W}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e95737ee2530885a10b104e9cd5331077e1c88d5)
of Witt vectors. In fact, the functor
![{\textstyle U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/087a7d8a39fe35012cbf2f561879f9e975cb4555)
creates limits and colimits and admits an explicitly describable left adjoint as a type of
free functor; from this, it is not hard to show that
![{\textstyle \mathrm {CRing} _{\delta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a0007e32eca4aaede654aad3c4bb573532cb779)
inherits
local presentability from
![{\displaystyle \mathrm {CRing} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aea2b3750f835e1403071232def2657d45b9a84)
so that one can construct the functor
![{\textstyle W}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e95737ee2530885a10b104e9cd5331077e1c88d5)
by appealing to the
adjoint functor theorem.
One further has that
restricts to a fully faithful functor on the full subcategory of perfect rings of characteristic p. Its essential image then consists of those
-rings that are perfect (in the sense that the associated map
is an isomorphism) and whose underlying ring is
-adically complete.[9]
See also
References
- ^ a b Fisher, Benji (1999). "Notes on Witt Vectors: a motivated approach" (PDF). Archived (PDF) from the original on 12 January 2019.
- ^ Artin, Emil and Schreier, Otto, Über eine Kennzeichnung der reell abgeschlossenen Körper, Abh. Math. Sem. Hamburg 3 (1924).
- ^ A. A. Albert, Cyclic fields of degree
over
of characteristic
, Bull. Amer. Math. Soc. 40 (1934). - ^ Schmid, H. L., Zyklische algebraische Funktionenkörper vom Grad pn über endlichen Konstantenkörper der Charakteristik p, Crelle 175 (1936).
- ^ Illusie, Luc (1979). "Complexe de de Rham-Witt et cohomologie cristalline". Annales scientifiques de l'École Normale Supérieure (in French). 12 (4): 501–661. doi:10.24033/asens.1374.
- ^ Lang, Serge (September 19, 2005). "Chapter VI: Galois Theory". Algebra (3rd ed.). Springer. pp. 330. ISBN 978-0-387-95385-4.
- ^ Joyal, André (1985). "δ-anneaux et vecteurs de Witt". C.R. Math. Rep. Acad. Sci. Canada. 7 (3): 177–182.
- ^ "Is there a universal property for Witt vectors?". MathOverflow. Retrieved 2022-09-06.
- ^ Bhatt, Bhargav (October 8, 2018). "Lecture II: Delta rings" (PDF). Archived (PDF) from the original on September 6, 2022.
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
- Mumford, David (1966-08-21), Lectures on Curves on an Algebraic Surface, Annals of Mathematics Studies, vol. 59, Princeton, NJ: Princeton University Press, ISBN 978-0-691-07993-6
- Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics, vol. 67, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90424-5, MR 0554237, section II.6
- Serre, Jean-Pierre (1988), Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-1035-1, ISBN 978-0-387-96648-9, MR 0918564
- Greenberg, Marvin J. (1969). Lectures on Forms in Many Variables. New York and Amsterdam: Benjamin. ASIN B0006BX17M. MR 0241358.
References
- Dolgachev, Igor V. (2001) [1994], "Witt vector", Encyclopedia of Mathematics, EMS Press
- Hazewinkel, Michiel (2009), "Witt vectors. I.", Handbook of algebra. Vol. 6, Amsterdam: Elsevier/North-Holland, pp. 319–472, arXiv:0804.3888, doi:10.1016/S1570-7954(08)00207-6, ISBN 978-0-444-53257-2, MR 2553661
- Witt, Ernst (1936), "Zyklische Körper und Algebren der Characteristik p vom Grad pn. Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassenkörper der Charakteristik pn", Journal für die Reine und Angewandte Mathematik (in German), 1937 (176): 126–140, doi:10.1515/crll.1937.176.126