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 formula_1 over the finite field of order "p" is the ring of formula_2-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 instead of using the standard formula_2-adic expansionformula_4to represent an element in formula_5, we can instead consider an expansion using the Teichmuller characterformula_6which sends each element in the solution set of formula_7 in formula_8 to an element in the solution set of formula_7 in formula_5. That is, we expand out elements in formula_5 in terms of roots of unity instead of as profinite elements in formula_12. We can then express a formula_2-adic integer as an infinite sumformula_14which gives a Witt vectorformula_15Then, the non-trivial additive and multiplicative structure in Witt vectors comes from using this map to give formula_1 an additive and multiplicative structure such that formula_17 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 "k" be a field containing a primitive "n"th root of unity. Kummer theory classifies degree "n" cyclic field extensions "K" of "k". Such fields are in bijection with order "n" cyclic groups formula_18, where formula_19 corresponds to formula_20.
But suppose that "k" has characteristic "p". The problem of studying degree "p" extensions of "k", or more generally degree "pn" extensions, may appear superficially similar to Kummer theory. However, in this situation, "k" cannot contain a primitive "p"th root of unity. If "x" is a "p"th root of unity in "k", then it satisfies formula_21. But consider the expression formula_22. By expanding using binomial coefficients we see that the operation of raising to the "p"th power, known here as the Frobenius homomorphism, introduces the factor "p" to every coefficient except the first and the last, and so modulo "p" these equations are the same. Therefore formula_23. Consequently Kummer theory is never applicable to extensions whose degree is divisible by the characteristic.
The case where the characteristic divides the degree is now 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 that no other such fields exist, they proved that degree "p" extensions of a field "k" of characteristic "p" were the same as splitting fields of "Artin–Schreier polynomials". These are by definition of the form formula_24 By repeating their construction, they described degree "p"2 extensions. Abraham Adrian Albert used this idea to describe degree "pn" 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 "pn". In the process of doing so, certain polynomials related to addition of formula_2-adic integers appeared. Witt seized on these polynomials. By using them systematically, he was able to give simple and unified constructions of degree "pn" field extensions and cyclic algebras. Specifically, he introduced a ring now called "Wn"("k"), the ring of "n"-truncated "p"-typical Witt vectors. This ring has "k" as a quotient, and it comes with an operator "F" which is called the Frobenius operator because it reduces to the Frobenius operator on "k". Witt observes that the degree "pn" analog of Artin–Schreier polynomials is
where formula_27. To complete the analogy with Kummer theory, define formula_28 to be the operator formula_29 Then the degree "pn" extensions of "k" are in bijective correspondence with cyclic subgroups formula_30 of order "pn", where formula_19 corresponds to the field formula_32.
Motivation.
Any formula_2-adic integer (an element of formula_34, not to be confused with formula_35) can be written as a power series formula_36, where the formula_37 are usually taken from the integer interval formula_38. 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 formula_39 is only one of many choices, and Hensel himself (the creator of formula_2-adic numbers) suggested the roots of unity in the field as representatives. These representatives are therefore the number formula_41 together with the formula_42 roots of unity; that is, the solutions of formula_43 in formula_34, so that formula_45. This choice extends naturally to ring extensions of formula_34 in which the residue field is enlarged to formula_47 with formula_48, some power of formula_2. Indeed, it is these fields (the fields of fractions of the rings) that motivated Hensel's choice. Now the representatives are the formula_50 solutions in the field to formula_51. Call the field formula_52, with formula_53 an appropriate primitive formula_54 root of unity (over formula_34). The representatives are then formula_41 and formula_57 for formula_58. 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 formula_47 of order formula_50 by taking residues modulo formula_2 in formula_52, and elements of formula_63 are taken to their representatives by the Teichmüller character formula_64. This operation identifies the set of integers in formula_52 with infinite sequences of elements of formula_66.
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: formula_67): given two infinite sequences of elements of formula_68 describe their sum and product as formula_2-adic integers explicitly. This problem was solved by Witt using Witt vectors.
Detailed motivational sketch.
We derive the ring of formula_2-adic integers formula_34 from the finite field formula_72 using a construction which naturally generalizes to the Witt vector construction.
The ring formula_34 of formula_2-adic integers can be understood as the projective limit of formula_75 Specifically, it consists of the sequences formula_76 with formula_77 such that formula_78 for formula_79 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 formula_80
The elements of formula_34 can be expanded as (formal) power series in formula_2
where formula_37 are usually taken from the integer interval formula_85 Of course, this power series usually will not converge in formula_86 using the standard metric on the reals, but it will converge in formula_87 with the formula_2-adic metric. We will sketch a method of defining ring operations for such power series.
Letting formula_89 be denoted by formula_90, one might consider the following definition for addition:
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 formula_92
Representing elements in Fp as elements in the ring of Witt vectors W(Fp).
There is a better coefficient subset of formula_34which does yield closed formulas, the Teichmuller representatives: zero together with the formula_42roots of unity. They can be explicitly calculated (in terms of the original coefficient representatives formula_95) as roots of formula_96 through Hensel lifting, the formula_2-adic version of Newton's method. For example, in formula_98 to calculate the representative of formula_99 one starts by finding the unique solution of formula_100 in formula_101 with formula_102; one gets formula_103 Repeat this in formula_104 with the conditions formula_100 and formula_106 gives formula_107 and so on; the resulting Teichmüller representative of formula_108, denoted formula_109, is the sequenceformula_110The existence of a lift in each step is guaranteed by the greatest common divisor formula_111 in every formula_112
This algorithm shows that for every formula_113, there is exactly one Teichmuller representative with formula_114, which we denote formula_115 Indeed, this defines the Teichmüller character formula_116 satisfying formula_117 if we denote formula_118 Note that formula_17 is "not" additive, as the sum need not be a representative. Despite this, if formula_120 in formula_87 then formula_122 in formula_123
Representing elements in Zp as elements in the ring of Witt vectors W(Fp).
Because of this one-to-one correspondence given by formula_17, one can expand every formula_2-adic integer as a power series in formula_2 with coefficients taken from the Teichmüller representatives. An explicit algorithm can be given, as follows. Write the Teichmüller representative as formula_127 Then, if one has some arbitrary formula_2-adic integer of the form formula_129 one takes the difference formula_130 leaving a value divisible by formula_2. Hence, formula_132. The process is then repeated, subtracting formula_133 and proceed likewise. This yields a sequence of congruences
So that 
and formula_136 implies: 
for 
Hence we have a power series for each residue of "x" modulo powers of "p", but with coefficients in the Teichmüller representatives rather than formula_139. It is clear that 
since 
for all formula_142 as formula_143 so the difference tends to 0 with respect to the formula_2-adic metric. The resulting coefficients will typically differ from the formula_37 modulo formula_146 except the first one.
Additional properties of elements in the ring of Witt vectors motivating general definition.
The Teichmuller coefficients have the key additional property that formula_147 which is missing for the numbers in formula_95. This can be used to describe addition, as follows. Since the Teichmüller character is not additive, formula_149 is not true in formula_34. But it holds in formula_151 as the first congruence implies. In particular, 
and thus
Since the binomial coefficient formula_154 is divisible by formula_2, this gives
This completely determines formula_157 by the lift. Moreover, the congruence modulo formula_2 indicates that the calculation can actually be done in formula_151 satisfying the basic aim of defining a simple additive structure.
For formula_160 this step is already very cumbersome. Write
Just as for formula_162 a single formula_2th power is not enough: one must take
However, formula_165 is not in general divisible by formula_166 but it is divisible when formula_167 in which case formula_168 combined with similar monomials in formula_169 will make a multiple of formula_170.
At this step, it becomes clear that one is actually working with addition of the form
This motivates the definition of Witt vectors.
Construction of Witt rings.
Fix a prime number "p". A Witt vector over a commutative ring formula_172 (relative to a prime number formula_2) is a sequence: formula_174 of elements of formula_172. Define the Witt polynomials formula_176 by 
and in general
The formula_181 are called the ghost components of the Witt vector formula_174, and are usually denoted by formula_183The ghost components can be thought of as an alternative coordinate system for the formula_172-module of sequences. 
The ring of Witt vectors formula_185 (relative to a prime formula_2) 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 formula_172 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. For example, 
These are to be understood as shortcuts for the actual formulas. If for example the ring formula_172 has characteristic formula_2, the division by formula_2 in the first formula above, the one by formula_170 that would appear in the next component and so forth, do not make sense. However, if the formula_2-power of the sum is developed, the terms formula_202 are cancelled with the previous ones and the remaining ones are simplified by formula_2, no division by formula_2 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 formula_205 is the elementformula_206Adding this element to itself gives a non-trivial sequence, for example in formula_207,formula_208sinceformula_209which is not the expected behavior, since it doesn't equal formula_210. But, when we reduce with the map formula_211, we get formula_212.
Note of we have an element formula_213 and an element formula_214 thenformula_215showing multiplication also behaves in a highly non-trivial manner.
Universal Witt vectors.
The Witt polynomials for different primes "p" are special cases of universal Witt polynomials, which can be used to form a universal Witt ring (not depending on a choice of prime "p"). Define the universal Witt polynomials "Wn" for "n" ≥ 1 by 
and in general
Again, formula_232 is called the vector of ghost components of the Witt vector formula_233, and is usually denoted by formula_234.
We can use these polynomials to define the ring of universal Witt vectors over any commutative ring "R" in much the same way as above (so the universal Witt polynomials are all homomorphisms to the ring "R").
Generating Functions.
Witt also provided another approach using generating functions.
Definition.
Let formula_235 be a Witt vector and define
For formula_237 let formula_238 denote the collection of subsets of formula_239 whose elements add up to formula_240. Then 
We can get the ghost components by taking the logarithmic derivative:
Sum.
Now we can see formula_243 if formula_244. So that 
if formula_246 are the respective coefficients in the power series formula_247. Then 
Since formula_249 is a polynomial in formula_250 and likewise for formula_251, we can show by induction that formula_252 is a polynomial in formula_253
Product.
If we set formula_254 then
But
Now 3-tuples formula_257 with formula_258 are in bijection with 3-tuples formula_259 with formula_260, via formula_261 (formula_262 is the least common multiple), our series becomes
So that 
where formula_265 are polynomials of formula_253 So by similar induction, suppose 
then formula_181 can be solved as polynomials of formula_269
Ring schemes.
The map taking a commutative ring "R" to the ring of Witt vectors over "R" (for a fixed prime "p") 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 formula_270 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 formula_172 to the set formula_272 is represented by the affine space formula_273, and the ring structure on formula_272 makes formula_273 into a ring scheme denoted formula_276. From the construction of truncated Witt vectors, it follows that their associated ring scheme formula_277 is the scheme formula_273 with the unique ring structure such that the morphism formula_279 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 formula_280. The analogue of this for fields of characteristic "p" 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 "p", any unipotent abelian connected algebraic group is isogenous to a product of truncated Witt group schemes.