An Introduction to Formal Algebra

David Dror Ben-Zvi

January, 1995

Contents

1 Formal Groups

A fundamental idea in geometry and analysis is the use of Taylor series approximations to study the infinitesimal behavior of a function, and to isolate local geometry from more global complications. The Taylor series itself, considered as a formal power series, contains all the higher order information about our function at a fixed point. Analytically, we would like our power series to have a nontrivial radius of convergence within which it agrees with our function. In such a case infinitesimal data and local geometry are compatible, and we are free to attend to cohomological, or local-to-global, problems. An important instance of this philosophy is prevalent in Lie group theory. The tangent space at the identity of Lie group comes equipped with the structure of a Lie algebra, which can be thought of as an infinitesimal approximation to the Lie multiplication. In fact the Lie algebra contains the first order information about the corresponding group, but we can naturally pass from it to the universal envelopping algebra, which is indeed the infinitesimal realization of the Lie group. Clasically the Lie algebra of a group comes with the exponential map, which allows us to take infinitesimal information and recover from it the geometry of the Lie group in some small neighborhood of the identity. One interprets this as saying that the Baker-Campbell-Hausdorff infinitesimal (or formal) multiplicative structure of the Lie group has a nontrivial radius of convergence.

In algebraic geometry, however, this interaction between the infinitesimal and the local becomes more subtle. Besides local being dangerously close to global, we encounter the issue of integrality - we are no longer allowed to introduce denominators with impunity. But the most significant difference between say Lie group geometry and its algebro-geometric analogues in number theory is the problem of characteristic - we cannot hope to create the exponential map over a field of characteristic p, and as a partial consequence the beautiful correspondence between Lie groups and Lie algebras malfunctions badly. As an attempt to bridge the growing gap between algebraic groups and their Lie algebras, S. Böchner introduced the notion of a formal group, which was greatly developed in the works of J. Dieudonné, M. Lazard, P. Cartier and others. A formal group is what remains of a group structure when we pass to a completion - in other words from elements of the group we pass to formal power series, dropping any convergence hypotheses. The resulting theory is a powerful and flexible one whose ensuing applications range far beyond (or perhaps to the side of) the original purpose.

As we survey some introductory topics in formal group theory, we shall attempt to keep in mind two essential issues which underlie much of the theory:

Integrality: Which power series constructions, typically involving denominators, can be performed for formal groups over A?

Realization: How do formal group laws relate to actual groups?

1.1 Formal Group Laws and Homomorphisms

A will denote throughout a commutative ring with unit. The notion [?, ?] of a formal group over A encapsulates the behavior of an algebraic group over A when restricted to smaller and smaller neighborhoods of the identity - that is, in its formal completion.

Definition 1.1 A one-dimensional formal group (F, F) defined over A is given by a power series F (X, Y) ∈ A[[X, Y]] satisfying^{1}:

i) F (X, Y) = X + Y +(terms of higher degree)

ii) F (X, (F (Y, Z)) = F (F (X, Y), Z)

1Similarly an n-dimensional formal group law is a power series F (X, Y) satisfying identical axioms, where by X, Y we now denote n-dimensional vectors and all our operations are componentwise.

We will refer to F or F/A as the formal group, and to F (X, Y) as the formal group law - one we would rather think of homomorphisms and so forth as being of “groups” than of “laws” - but more on that later. Note that we have apparently omitted the existence of an inverse, and the properties of the putative identity element 0. In fact properties i, ii together imply that F (X, 0) = X, F (0, Y) = Y as well as the existence of a unique power series i(X) = −X + … in A[[X]] such that F (X, i(X)) = F (i(X), X) = 0. To establish these identities we may study property ii, setting Y = Z = 0: by comparing the two sides of the resulting equation we discover inductively that all the coefficients of X^{n} in F (X, Y) vanish for n > 1 (and similarly for Y). The properties of the inverse are verified in a similar fashion. In fact it is not hard to see that for any power series f(X) = aX + … with a invertible in A we can find, by figuring out terms inductively, an inverse power series g(f(X)) = f(g(X)) = X. This is but a first indication of the power and flexibility of formal groups - we may have abandoned (for the moment) actual “group elements” to manipulate, but our substitute is explicit formal power series which we freely manipulate, ignoring tricky issues such as convergence.

Definition 1.2 A homomorphism between formal groups (F, F) and (G, G) over A is a series f(X) ∈ A[[X]] such that f(F (X, Y)) = G(f(X), f(Y)).

The formal additive group Ĝ_{a} over A is given by F (X, Y) = X + Y, and the formal multiplicative group Ĝ_{m} is given by F (X, Y) = X + Y + XY = (1 + X)(1 + Y) − 1. The fundamental examples of homomorphisms between formal groups are the logarithm, mapping Ĝ_{m} → Ĝ_{a}:

log(1 + X) = ∑_{n=1}^{∞}(−1)^{n+1} \frac{X^{n}}{n}

and the exponential Ĝ_{a} → Ĝ_{m}:

exp(X) = ∑_{n=1}^{∞} \frac{X^{n}}{n!}

We see that whenever these expressions make sense, the additive and multiplicative formal groups are isomorphic. The definition of the logarithm and exponential make essential use of denominators, and hence these formulas will only make sense over rings A where n = 2, 3, … are invertible. More generally, if the natural map A → A⊗ℚ is injective, (in which case we say A is of characteristic zero), then the additive and multiplicative formal groups are isomorphic over A⊗ℚ. Note that we can compare formal groups defined over different rings using maps of the underlying rings. If ψ : A → B is a ring homomorphism, we get an induced homomorphism ψ_{∗} : A[[X, Y]] → B[[X, Y]] by mapping coefficients via ψ. Thus for a formal group F/A we may define ψ_{∗}F to be the formal group defined by ψ_{∗}F (X, Y). As a general rule, as we map into increasingly divisible rings or increasingly “algebraically closed” ones, the number of isomorphism classes of formal groups will shrink as more and more operations become feasible over A.

We have seen that for ℚ-algebras, the additive and multiplicative formal groups are isomorphic. To prove that this is not the case when the characteristic is p > 0, we first introduce the useful “multiplication by m” map F → F for integers m. We procede inductively, starting from [0](X) = 0:

[m + 1](X) = F ([m](X), X) [m − 1](X) = F ([m](X), i(X))

We see immediately that [m](X) = mX+(higher order terms), and that in Ĝ_{a}, [m](X) = mx while in Ĝ_{m}, [m](X) = (1 + X)^{m} − 1. Since an isomorphism would intertwine the operations [p]Ĝ_{a} and [p]Ĝ_{m}, we deduce that if the characteristic is p > 0, then the additive and multiplicative formal groups over A cannot be isomorphic (X will be nilpotent in one and not in the other).

When the characteristic allows it, the construction of logarithms (namely, isomorphisms with the additive formal group) can be extended to a wide class of formal groups. One way to construct logarithms for one-dimensional formal groups is via the use of the invariant differential [?]. To proceed, let F/A be a formal group.

Definition 1.3 A differential on F/A is an expression ω(X) = P (X)dX in A[[X]]dX, satisfying the formal chain rule

ω ∘ f(X) = P (f(X))f′(X)dX

ω is said to be invariant if ω ∘ F (X, Y) = ω(X), namely^{2}

P (F (X, Y)) ∂_{1}F (X, Y) = P (X)

2Here by ∂_{1}F (X, Y) we mean the formal derivative with respect to the first variable.

As usual, invariant differentials on the formal additive and multiplicative groups are given by dX and

\frac{dX}{1 + X} = (1 − X + X^{2} − …)dX

respectively. In fact every F admits a unique invariant differential with P (X) = 1 + P_{1}X + …, given by

ω_{F}(X) = \frac{dX}{∂_{1}F (0, X)}

The invariance follows from differentiating the associative law

F (X, F (Y, Z)) = F (F (X, Y), Z)

with respect to X:

∂_{1}F (X, F (Y, Z)) = ∂_{1}F (F (X, Y), Z) ∂_{1}F (X, Y)

Setting X = 0 we see

\frac{1}{∂_{1}F (0, F (X, Y))}∂_{1}F (X, Y) = \frac{1}{∂_{1}F (0, X)}

yielding invariance. Uniqueness of the normalized differential follows from

P (X) ∂_{1}F (0, X) = P (0)

for an invariant differential P (X)dX. Furthermore, since ω_{F} ∘ f(X) is naturally an invariant differential on G for f : F → G, the Jacobian rule

ω_{F} ∘ f(X) = f′(0) ω_{G}

also carries over from ordinary calculus.

Now if A is of characteristic zero and we work over A ⊗ ℚ, we may define the logarithm

log_{F} = ∫ ω_{F}

This construction only works over A ⊗ ℚ since ∫ X^{n}dX introduces a denominator of \frac{1}{n+1}. Since

log_{F}(X) = X + \frac{P_{1}}{2} + …

we can also invert it to obtain a power series exp_{F}. In the case of the multiplicative group these indeed give the standard logarithm and differential. In general the invariance of ω_{F} implies that

log_{F} F(X, Y) = log_{F}(X) + f(Y)

for some power series f, which is found to be log_{F}(Y) upon setting X = 0. Hence we have established the following:

F (X, Y) = exp_{F}(log_{F}(X) + log_{F}(Y))

Moreover it follows that F (X, Y) = F (Y, X). We have now proven

Proposition 1.4 Any one-dimensional formal group law F over a ring A of characteristic zero is automatically commutative. F/A ⊗ ℚ is in fact isomorphic over A ⊗ ℚ to the additive formal group law.

1.2 Constructing Formal Groups

The construction of the logarithm suggests a useful procedure by which to construct formal group laws over a ring A [?]. Given a power series f(X) ∈ X + X(A ⊗ ℚ)[[𝕏]], we may attempt to define

F (X, Y) = f^{−1}(f(X) + f(Y))

if this makes sense as a power series in A[[X, Y]] - in other words, we posit that f be the logarithm of some thence uniquely determined formal group. This enables one to produce a host of formal group laws over A if one can prove the appropriate integrality lemmas for F (X, Y). The most important example of such a construction is that of the Lubin-Tate formal group laws [?, ?], which apply to the construction of extensions in local class field theory. Formal groups such as the Lubin-Tate groups can also be constructed via a similar technique - we are given an appropriate power series f(X), which we now take to be an endomorphism of an as yet undefined formal group. That is, we try to solve

f(F [X, Y]) = F (f(X), f(Y))

for F (X, Y) ∈ A[[X, Y]]. In the Lubin-Tate construction, A is a discrete valuation ring with residue field 𝔽_{q} and π is a uniformizer of A. Given any power series f(X) over A with

f(x) ≡ πX mod degree 2, f(X) = X^{q} mod (π)

One then proves that there is a unique formal group law F_{f}(X, Y) over A for which f(X) is an endomorphism, and in fact all of A acts as endomorphisms. The choice of f(X) is irrelevant up to isomorphism, but the choice of π is all-important and different choices give rise to a wide range of formal groups of great importance in local class field theory.

Yet another application of such functional equations is the construction of the Witt vectors, a central theme of this paper. Here we are given a sequence of polynomials w_{n}(X⃗) ∈ ℤ[𝕏_{⊮}, …, 𝕏_{⋉}] which we’d like to use to define a nonstandard addition on power series:

w_{n}(Σ⃗(X⃗, Y⃗)) = w_{n}(X⃗) + w_{n}(Y⃗)

for some hopefully integral polynomials

Σ⃗(X⃗, Y⃗) = (Σ_{0}(X_{0}, Y_{0}), …, Σ_{n}(X_{0}, …, X_{n}, Y_{0}, …, Y_{n}))

We see that general techniques for studying solutions to functional equations can be instrumental to understanding formal groups. One powerful such result is Hazewinkel’s functional equation lemma [?], which applies to the integrality of power series defined by certain recursion relations on its coefficients. In particular Hazewinkel’s result can be used to study problems such as the structure of generalized Lubin-Tate groups and Witt groups. In this paper we will not pursue these general techniques, but it is important to be aware that these and ideas produce a wealth of different formal groups, giving a scope and depth to the theory which will not be obvious from this survey.

We do feel obliged however to mention one important result that can be established by these means.

Definition 1.5 A one-dimensional commutative formal group law F_{U}/L is said to be universal if for every commutative formal group law G/A there is a unique homomorphism ψ : L → A such that ψ_{∗}F_{U} = G.

As usual for such universal objects, the ring L is uniquely determined. To construct L first observe that the ring ℤ[𝕏⃗] = ℤ[𝕏_{⊬}, 𝕏_{⊮}, …] represents the functor A → A[[X]], in other words Hom(ℤ[𝕏⃗], 𝔸) ≅ 𝔸[[𝕏]] as a set. We now build L from this universal object by imposing the minimal necessary restrictions. Let L̄ = ℤ[ℂ_{⊮⊮}, ℂ_{⊮⊭}, …, ℂ_{⊭⊮}, …, ℂ_{ℶℷ}, …]. We let

F (X, Y) = X + Y + ∑_{i,j≥1} C_{ij}X^{i}Y^{j}

and impose the associativity and commutativity relations by force: we mod out L̄ by the ideal generated by C_{ij} −C_{ji} and the polynomials P_{ijk}(C) defined by

F (F (X, Y), Z) − F (X, F (Y, Z)) = ∑_{i,j,k} P_{ijk}(C)X^{i}Y^{j}Z^{k}

This defines a ring L and a formal group F_{U} given by the image of F (X, Y) under the quotient map. It is not hard to see that F_{U}/L is indeed universal. The great difficulty lies in elucidating the structure of L. The fascinating result is that the ring which is represents power series rings in fact carries the universal formal group law:

Theorem 1.6 (Lazard) [?] The coefficient ring L for the universal one dimensional commutative formal group law F_{U} is isomorphic to ℤ[𝕋_{⊮}, 𝕋_{⊭}, …], the ring of polynomials in infinitely many indeterminates.

Similar results describe universal commutative n-dimensional formal group laws. The freeness of L is often very useful in the creation of formal group laws satisfying specific desired relations.

1.3 Groups and Group Schemes

So far in this paper, we have approached formal groups from the standpoint of formal group laws, as being primarily concerned with formal power series and their manipulations^{3}. However, we may also interpret formal groups as certain functors, taking algebras over our base ring A and assigning them actual groups - but with a novel law of addition. The idea is to single out elements where the formal power series converge, and then to use them to define a group structure.

Consider first the case of a nilpotent algebra B/A - that is, every b ∈ B satisfies b^{n} = 0 for n sufficiently large. Then the expression F (b⃗_{1}, b⃗_{2}), where (F, F) is an n-dimensional formal group law over A, is in fact a finite sum of elements of B^{n} and thus well defined. In other words, if B is any A-algebra and 𝔫(B) is the ideal of nilpotent elements, we may define a group F(B) whose underlying set is n-tuples in 𝔫(B) and whose group law is given by

n_{1} +_{F} n_{2} = F (n_{1}, n_{2})

3This is the viewpoint prevalent in our main referrence, [?].

We may thus consider F as a functor from the category Alg_{A} of commutative A-algebras with unit, to Group, factoring through the functor 𝔫. By studying the group law induced on the nilpotent algebras A[X⃗]/X⃗^{n} we may recover the coefficients of F (X, Y) from the functor F. Once we note that a homomorphism of formal groups over A induces a natural transformation of the corresponding functors, we see that we may equally well choose to consider the formal group law F (X, Y) or the functor F associated to it.

More generally, in order to construct a group operation out of a formal group law, it is not necessary to restrict to the ideal of nilpotent elements in an algebra. Rather what is required is some notion of convergence, which will allow one to consider infinite sums of powers as actual elements of an algebra. In other words, we need a topology with respect to which our algebra is complete. So let an algebra B/A be given, with an I-adic topology defined by some ideal I. We assume that this topology is complete and that ∩_{i}I^{i} = {0}, so it is also Hausdorff. Then given n-tuples x⃗, y⃗ in I the series F (x⃗, y⃗) is convergent to some n-tuple x +_{F} y in I, thereby producing a novel group structure on (I)^{\}. This leads to a functorial construction as in the case of the previous paragraph, where instead of the ideal of nilpotent elements we consider the ideal of topologically nilpotent elements for complete topological algebras over A.

The language of formal group laws now suggests a natural construction of a group associated to a formal group law. The ring of formal power series itself is a complete local ring, whose maximal ideal consists of series with no constant term. Thus we may apply the previous construction to the power series ring itself.

Definition 1.7 A curve in F is an n-tuple of formal power series γ(t) = t(a_{1} + a_{2}t + …) with a_{i} ∈ A. Given a formal group (F, F) of dimension n over A, the group of curves in F, denoted C(F) is the set of n-tuples of power series

γ(t) = (γ_{1}(t), …, γ_{n}(t)), γ_{i}(t) ∈ tA[[t]]

endowed with the group operation

γ(t) +_{F} γ′(t) = F (γ(t), γ′(t))

It will turn out that the group of curves in F, equipped with some natural algebraic operators, is sufficient to characterize F - a proposition that shouldn’t appear all that surprising given the naturality of the definition. The curves functor is also intimately related to the notion of Witt vectors.

Although we will mainly be concerned with the formal power series aspects of formal group theory, it is highly beneficial to keep in mind the more functorial and algebro-geometric side of the subject [?, ?]. In particular, a formal group can be considered as an example of the notion of a group object in a category. Let us first recall that a group scheme is the scheme theoretic analogue of an algebraic group or topological group. Namely, we have a scheme 𝔊 over our base SpecA and morphisms 𝔊 × 𝔊 → 𝔊, SpecA → 𝔊 and 𝔊 → 𝔊 corresponding to multiplication, identity and inverse respectively, such that the appropriate axioms hold. Identical definitions provide us with the notion of a group in any category (with a final object, the generalization of the one-point set.) A group scheme 𝔊/A also defines a functor Hom( , 𝔊) : Schemes_{A} → Groups via the natural multiplication of maps into a group object. We may in fact identify 𝔊 with this functor, and the category of group schemes becomes equivalent to a category of such (representable) functors. We may also interpret the construction of an (affine) group scheme SpecR in terms of the dual category consisting of commutative rings. The multiplication map SpecR × SpecR → SpecR is transformed into a comultiplication R → R × R, the unit becomes a counit R → A, and the inverse becomes an antipode R → R. Categorically, these data define a cogroup object in the category of A-algebras; these are Hopf algebras, or if we omit the antipode axiom, bialgebras. Now the associated functor is Hom(R, ).

A formal group may now be defined in several ways [?, ?, ?]. First of all, the notion of formal group should involve some type of completion or topology. Thus instead of considering functors from the category of all schemes to groups, as above, a formal functor should be a functor defined on some subcategory defined by complete topological algebras of some sort. Following [?], we take the appropriate category to be the “very finite” schemes over a complete local ring R, which are given as finite schemes SpecA with A a module of finite length over R. (In [?] one rather develops a category analogous to certain topological vector spaces.) To define a formal group scheme we need a notion of representability for such functors from very finite schemes to groups. For this we allow profinite rings, by which we mean inverse limits of very finite rings. Our hypothesis on R guarantees that any finite R-module is also profinite, as is the power series ring R[[X⃗]]. An (affine) profinite scheme SpecA defines a functor on very finite schemes SpecB (where B has a discrete topology) via

SpfA(B) = Hom_{R,cont}(A, B)

Finally we define a formal group scheme as a functor from very finites to groups which is representable by a profinite scheme. If SpecA is such a formal group scheme, then A is endowed with a coproduct

μ : A = lim_{←} A/I → A ˆ⊗A = lim_{←} (A/I) ⊗ (A/I)

The axioms for a formal group scheme now read much like those of an (affine) group scheme, with tensor products replaced by their profinite analog, the completed tensor product. (This is a manifestation of the problem with duality for infinite-dimensional vector spaces and their relatives - the completed tensor product is created as the dual of the ordinary tensor product.) The need for the completed tensor product also becomes apparent from the equation

R[[X_{1}, …, X_{n}]] ˆ⊗R[[Y_{1}, …, Y_{n}]] = R[[X_{1}, …, X_{n}, Y_{1}, …, Y_{n}]]

Since we have defined the formal functor SpfA as being continuous homomorphisms to a discrete ring, it follows that when A is a power series ring, these maps factor through the ideal of nilpotent elements 𝔫(𝔅). Thus in that case we recover the notion of the formal group functor discussed earlier in the chapter. Similarly we may extend the functor SpfA to profinite rings B, and this group functor produces the group structure on the ideal of topologically nilpotent elements as before. In particular Hom(SpecA[[X⃗]], G) in this cateogry is the group of curves in G.

Recall [?] that a formal scheme is a locally ringed space which is locally isomorphic to the completion of a scheme along a closed subscheme. If we are given a (regular) group scheme 𝔊 of dimension n, then the completion 𝔬̂_{e} of its local ring 𝔬_{e} at the identity element (the closed subscheme SpecA → 𝔊) is isomorphic to A[[T_{1}, …, T_{n}]]. The group operation induces a map

𝔬̂_{e} → 𝔬̂_{(e,e)} ≅ A[[X_{1}, …, X_{n}, Y_{1}, …, X_{n}]]

to the completed local ring of the product 𝔊 × 𝔊. Writing this map as

(T_{1}, …, T_{n}) ↦ G(X⃗, Y⃗)

we obtain a formal group law^{4} G over A. More generally if 𝔊 is not regular, the above construction will give a formal group scheme which is not of the type we have discussed - that is, given by a power series ring. Generally we will ignore such formal groups and concentrate for concreteness on formal group laws.

One can consider formal group laws as Hopf algebras in an appropriate category, or as completed group schemes. The axioms defining the multiplication and comultiplication on a Hopf algebra (such as the coordinate ring of an affine group scheme or formal group scheme) are remarkably symmetric and admit a beautiful duality known as Cartier duality, which interchanges these two operations. This is the scheme theoretic analog of the Pontrjagin duality of locally compact groups, which associates to such a topological group its group of characters. Finite groups are sent to finite groups, and discrete groups are interchanged with compact groups. Similarly one can consider Cartier duality on the category of finite group schemes, or one may allow the topology to intervene and obtain a duality between affine group schemes and formal group schemes. On the bialgebra level, one simply takes the duals (over A) of algebras involved. This flips the arrows and (mindful of topology) turns the completed tensor product in the formal group comultiplication into an ordinary tensor product defining the algebra underlying an affine group scheme. Similarly one takes continuous duals to obtain completed tensor products, and hence a formal group structure, on the dual of an affine group scheme. In terms of schemes, these constructions correspond to taking characters: Cartier duality is expressed as

D(F) = Hom(F, Ĝ_{m}) D(𝔊) = Hom(𝔊, G_{m})

As in all worthy dualities, the double dual becomes identified with the original scheme. The precise definitions and verifications are an exercise in general nonsense [?, ?, ?]. We will encounter Cartier duality again in the form of the Witt vectors 𝕎 and their formal counterpart 𝕎̂. (This particular case is also crucial in elucidating the (finite) Cartier duality between the p^{r}-th roots of unity and ℤ/∣^{∖}ℤ [?, ?].)

4The condition G(X⃗, Y⃗) = X⃗ + Y⃗ + … follows from the (co)unit axiom and the characterization of the maximal ideal of 𝔬 as the kernel of the resulting augmentation-i.e. the first order behavior of the law must agree with the identity.

1.4 Lie Algebras

In Lie theory [?], we find that to every Lie group is associated a Lie algebra structure on its tangent space at the identity. From this Lie algebra we can then reconstruct the original Lie group, at least in some neighborhood of the identity, via the exponential mapping. We sketch this theory in the context of Lie algebras over a ℚ-algebra A, in order to emphasize the purely formal nature of the construction [?].

Assume given an n-dimensional formal group (G, G), which we think of as coming from the formal structure near the identity of an algebraic group over A. We write G in the form

G_{i}(X⃗, Y⃗) = X_{i} + Y_{i} + ∑_{l,k=1}^{n} γ_{lk}^{i}X_{l}Y_{k} mod degree 3

or alternatively as

G(X⃗, Y⃗) = X⃗ + Y⃗ + Γ(X⃗, Y⃗) mod degree 3

where Γ is a vector of bilinear forms. To derive what the appropriate Lie algebra structure of A^{n} corresponding to G should be, we first reconstruct a group from G and then derive it in a standard fashion. Namely, let’s consider the group of curves C(G), whose elements are n-tuples of topologically nilpotent power series in A[[X⃗]]. For convenience we now denote the product of two curves X⃗, Y⃗ by X⃗ ⋄ Y⃗, and the ⋄-inverse of X⃗ by X⃗^{−1}. X⃗^{−1}, is found to be

X⃗^{−1} = −X⃗ + Γ(X⃗, X⃗) mod degree 3

A further calculation shows

X⃗ ⋄ Y⃗ ⋄ X⃗^{−1} = Y + Γ(X⃗, Y⃗) − Γ(Y⃗ , X⃗) mod degree 3

and

Y⃗^{−1} ⋄ X⃗ ⋄ Y⃗ = X + Γ(Y⃗ , X⃗) − Γ(X⃗, Y⃗) mod degree 3

so that we finally have

X⃗^{−1} ⋄ Y⃗^{−1} ⋄ X⃗ ⋄ Y⃗ = [X⃗, Y⃗]

where

[X⃗, Y⃗] = Γ(X⃗, Y⃗) − Γ(Y⃗ , X⃗)

This motivates the introduction of the Lie bracket

[x_{j}, x_{k}] = ∑_{i=1}^{n}(γ_{jk}^{i} − γ_{kj}^{i})x_{i}

on A^{n}. The anticommutativity of the bracket follows immediately from the definition, while the proof of the Jacobi identity follows in a standard fashion from the expression for the bracket as a commutator in the group C(G). We have thus created the Lie algebra 𝔤 associated to an algebraic group formally. Given a homomorphism α(X⃗) : F → G of formal groups, one differentiates it to find the induced homomorphism of Lie algebras. Namely, writing α(X⃗) = J · X⃗ mod degree 2, where J is the Jacobian matrix of α, one finds

α(X⃗)^{−1} ⋄ α(Y⃗)^{−1} ⋄ α(X⃗) ⋄ α(Y⃗) = [J · X⃗, J · Y⃗] mod degree 3

and

α(X⃗^{−1} ⋄ Y⃗^{−1} ⋄ X⃗ ⋄ Y⃗) = J · [X⃗, Y⃗] mod degree 3

so that the linearization J indeed induces a homomorphism of Lie algebras. Thus the association of a Lie algebra to a formal group is indeed a functor.

Our goal now is to sketch how one may, in favorable circumstances, recover the formal group law G from its Lie algebra 𝔤. In fact the following holds:

Theorem 1.8 [?] For A a ℚ-algebra, the functor associating a Lie algebra to an n-dimensional formal group over A is an equivalence of categories to the category of Lie algebra structures on A^{n}.

We content ourselves here with proving the bijectivity of this functor on sets - in other words, with the construction of the formal group G from 𝔤, via the Baker-Campbell-Hausdorff (BCH) formal multiplication. The comparison between Lie algebras and formal groups is best established on the bialgebra level by considering the structure of the universal envelopping algebra U𝔤 of 𝔤 as an object close in spirit to the bialgebra defined by a formal group law.

First recall that there is a natural diagonal map (which should correspond to a group multiplication) from 𝔤 to U𝔤 taking x ↦ 1 ⊗ x + x ⊗ 1. By universality this extends to a comultiplication Δ : U𝔤 → U𝔤 ⊗ U𝔤. (The counit of U𝔤 is given by its natural augmentation, the zeroth order term in a tensor.)

If we pick a basis x_{i} of 𝔤, and use boldface indices to denote multi-indices, then the Poincaré-Birkhoff-Witt theorem implies that the comultiplication is completely determined by the formula

Δ(x^{k}) = ∑_{i+j=k} \binom{k}{i} x^{i} ⊗ x^{j}

It is also known that for A a ℚ-algebra, 𝔤 is precisely the set of elements x of U𝔤 with Δx = x ⊗ 1 + 1 ⊗ x, also known as primitive elements.

The construction of a formal group law from 𝔤 now follows from some universal formal algebra. Our problem is to determine a universal rule by which, given elements x, y in a Lie algebra, to calculate a z so that exp(z) = exp(x) exp(y), where exp denotes a formal mapping into a multiplicative group. In other words, we want to know how the group structure on an exponentiated form of our Lie algebra should be reflected inside, whence we will be able formally to define such a group structure. To be more precise, we let S = {x, y} be a two-element set. We set L_{S} to be the free Lie algebra on S, UL_{S} its universal envelopping algebra and Ass_{S} free associative algebra on S. Ass_{S} has a natural Lie algebra structure, and hence a map L_{S} → Ass_{S} which further induces UL_{S} → Ass_{S}. We claim the latter map is an isomorphism - in fact this follows from the universal property of Ass_{S} and the inclusion S → UL_{S}. This now implies that UL_{S} is not only filtered, but graded by length of associative (noncommutative) words in x, y. To allow formal manipulations, we now complete UL_{S} with respect to the topology defined by the grading - we allow infinite sums of words of increasing (finite) length. In this completion ÛL_{S} we denote the closure of L_{S} by L̂_{S}, and we also extend the comultiplication of UL_{S} to the completed tensor product:

Δ : ÛL_{S} → ÛL_{S} ˆ⊗ÛL_{S}

The primitive elements of ÛL_{S} are then elements of L̂_{S} - this follows since by linearity and homogeneity of Δ, an infinite sum is primitive if and only if all its terms are.

As usual in our formal considerations, we set 𝔪̂_{S} ⊂ ÛL_{S} to be the ideal of topologically nilpotent elements - that generated by S itself. The natural additive and multiplicative groups which we are trying to bridge are now 𝔪̂_{S} and 1 + 𝔪̂_{S}, and there are as usual mutually inverse maps exp(t)^{5}, log(1 + t) between them.

Theorem 1.9 (Baker-Campbell-Hausdorff) There is a unique element z ∈ L̂_{S} satisfying exp(x) exp(y) = exp(z).

Proof: We first establish that exp takes the set of primitive elements in 𝔪̂_{S} onto the elements v ∈ 1 + 𝔪̂_{S} satisfying Δv = v ⊗ v. Δ commutes with the exponential map since it is a homomomorphism of algebras (hence commutes with all the terms of exp, and so we calculate for u primitive

Δ(exp(u)) = exp(Δ(u)) = exp(1 ⊗ u + u ⊗ 1)

Now for u, v ∈ 𝔪̂_{S} commuting elements, a term-by-term calculation establishes that exp(uv) = exp(u) exp(v), hence

Δ(exp(u)) = exp(1 ⊗ u) exp(u ⊗ 1) = exp(u) ⊗ exp(u)

and our first step is established. Since exp is bijective onto 1 + 𝔪̂_{S}, there exists a unique element z ∈ 𝔪̂_{S} with exp(z) = exp(x) exp(y). Since x, y ∈ L̂_{S} it follows that

Δ(exp(z)) = (exp(x) ⊗ exp(x))(exp(y) ⊗ exp(y)) = exp(z) ⊗ exp(z)

Thus z ∈ L̂_{S} as desired. □

By recursion we can calculate explicitly the coefficients of z:

z(x, y) = x + y + \frac{1}{2}[x, y] + \frac{1}{12}[x, [x, y]] + \frac{1}{12}[y, [y, x]] + …

In fact the BCH series z = z(x, y) defines a formal group law over ℚ (associativity follows from the definition), which encapsulates in a universal sense how a Lie algebra should be exponentiated. Given this information, we may proceed to construct a formal group G for a Lie algebra 𝔤. We first make 𝔤 look like a power series ring by tensoring with one: set R = A[[X_{1}, …, X_{n}, Y_{1}, …, Y_{n}]], which we identify with the underlying space^{6} of 𝔤 ⊗ A[[X, Y]], and which we consider a Lie algebra via this identification. We now map in the two-element free Lie algebra ϕ : L_{S} → R by sending x ↦ X⃗ and y ↦ Y⃗. Checking our topology yields that high order words in L_{S} map to topologically small elements in R, and so this map extends to the completion L̂_{S}. We now push forward our BCH formal group law to R via this map - in other words, set

5It is important to note at this stage that here in the expression for exp we do need the constant term, unlike in the previous case of formal groups where the form of the multiplicative group was not the standard multiplication s, t → st.

6Remember we are only considering Lie algebra structures on A^{n}.

G(X⃗, Y⃗) = ϕ(z(x, y)) = X + Y + \frac{1}{2}[X, Y] mod degree 3

It is straightforward from this expression that the Lie algebra associated to G is indeed 𝔤^{7}. We have now established a bijective correspondence between Lie algebras and formal groups.

2 Witt Vectors and Their Relatives

It is often the case in the theory of formal power series that certain expressions, which have universal and concise characterizations and meanings, have very complicated forms. We have already encountered an example of this in the Baker-Campbell-Hausdorff formal group law, for which we do not have any general expression for the coefficients. Another example is the power series expansion of \frac{t}{e^{t}−1}, which involves the Bernoulli numbers. In this chapter we will encounter a pervasive example of this phenomenon, namely the polynomials describing the addition of Witt vectors. Arising from the solution of a functional equation in terms of the more intuitive Witt polynomials, these mysterious functions turn out to have truly miraculous properties and applications ranging from number theory to algebraic topology. As we investigate some universal algebraic constructions related in various ways to the Witt vectors, we wish to emphasize the universal and functorial meanings of certain explicit polynomial and formal expressions. Thus the attraction for us of these series is not direct, in their formal beauty or wealth of examples, but in the way they express fundamental structures.

The Witt vectors are initially introduced in order to systematize the transition from the field of p elements to the p-adic integers, a complete local ring which we wish to understand as formal power series with finite field coefficients. The Witt vectors enable the lifting of characteristic p objects to characteristic zero, and this construction motivates their form. But upon further investigation, one discovers a close symbiotic connection between the Witt construction and the group of curves in a formal group. In various forms the Witt vectors embody the curve notion and provide an explicit approach to their analysis. In an exponentiated form, the Witt vectors become a universal object in the theory of λ-rings. This structure in turns embodies the behavior of symmetric functions, and especially contains the full algebraic structure assoicated with Chern classes, and as such is of fundamental interest in geometry. It is through this medium of formal power series that number theoretic questions (such as the denominators of Bernoulli numbers) have become most prominent in topology, and we close this chapter with some vague hints at this beautiful interaction.

7The factor of \frac{1}{2} comes from the discrepancy between the definitions of the γ_{ij}^{k} and the structure constants [x_{i}, x_{j}] = ∑_{k=1}^{n} c_{ij}^{k}x_{k} of 𝔤

2.1 The Witt Construction

In this section we define the Witt and p-typical Witt vectors and study some of their basic properties [?]. This slightly dull work will be amply justified later.

Consider the following collection of polynomials in increasingly many variables:

pw_{0}(X_{0}) = X_{0}
pw_{1}(X_{0}, X_{1}) = X_{0}^{p} + pX_{1}
…
pw_{n}(X_{0}, …, X_{n}) = X_{0}^{p^{n}} + pX_{1}^{p^{n−1}} + ⋯ + p^{n}X_{n}

Two key observations concerning these polynomials are:

I We may define the Witt polynomials recursively via the relation

pw_{n}(X⃗) = _{p}w_{n−1}(X⃗^{p}) + p^{n}X_{n}

II _{p}w_{n} increases the p-adic order of numbers by n. More precisely, if x⃗, y⃗ ∈ A^{n} are n-tuples in any ring A and x⃗ ≡ y⃗ modulo pA, then _{p}w_{n}(x⃗) ≡ pw_{n}(y⃗) modulo p^{n+1}A.

We interpret the Witt polynomials as defining a map

pw : 𝕎_{∣} = Specℤ[𝕏_{⊬}, 𝕏_{⊮}, …] → 𝔸^{∞} = Specℤ[𝕊_{⊬}, 𝕊_{⊮}, …]

via S_{n} ↦ _{p}w_{n}(X_{0}, …, X_{n}). Now 𝔸^{∞} is a ring scheme, with componentwise addition and multiplication. On the level of coordinate rings these operations correspond to

α : S_{i} ↦ S_{i} ⊗ 1 + 1 ⊗ S_{i}
μ : S_{i} ↦ S_{i} ⊗ S_{i}

Our aim now is to transform 𝕎_{∣} into a ring scheme, in such a way that _{p}w becomes a ring homomorphism. In other words, we wish to define a nonstandard ring structure on infinite-dimensional affine space, which is derived from the standard one by twisting by the Witt polynomials. In other words, in the language of Section 2.2, we need to construct polynomials

pΣ(X⃗, Y⃗) = (_{p}Σ_{1}(X_{0}, Y_{0}), …,_{p}Σ_{n}(X_{0}, …, X_{n}, Y_{0}, …, Y_{n}), …)

so as to satisfy the functional equation

pw_{n}(_{p}Σ(X⃗, Y⃗)) = _{p}w_{n}(X⃗) +_{p}w_{n}(Y⃗)

and similarly multiplication polynomials _{p}Π satisfying

pw_{n}(_{p}Π(X⃗, Y⃗)) = _{p}w_{n}(X⃗) _{p}w_{n}(Y⃗)

and inverse law polynomials _{p}ι,

pw_{n}(_{p}ι(X⃗)) = −_{p}w_{n}(X⃗)

Now if one is allowed to divide by p, it is easy to invert the polynomials pw_{n} to derive expressions for the X_{i} in terms of the S_{j} for j ≤ i. Thus over any ring in which p is invertible, the map of schemes _{p}w : 𝕎_{∣} → 𝔸^{∞} is an isomorphism, and any polynomial expression in the S_{j} can be transformed into a polynomial in the X_{i}. It follows that the _{p}Σ_{n}, _{p}Π_{n} and _{p}ι_{n} are indeed well defined as polynomials in ℤ[\frac{⊮}{∣}]. “And then a miracle occurs”: while_{p}w is certainly not invertible over ℤ, nevertheless the Witt ring operations in fact have integral coefficients, and thus define the desired ring scheme structure on 𝕎_{∣}.

Proposition 2.1 Let Φ ∈ ℤ[𝕏, 𝕐] be an integral polynomial in two variables. If the rational polynomials

ϕ⃗(X⃗, Y⃗) = (ϕ_{1}(X_{0}, Y_{0}), …, ϕ_{n}(X_{0}, …, X_{n}, Y_{0}, …, Y_{n}))

are defined by the functional equation

pw_{n}(ϕ⃗(X⃗, Y⃗)) = Φ(_{p}w_{n}(X⃗),_{p}w_{n}(Y⃗))

then it follows that the ϕ_{i} are integral.

Proof: From I it follows that

pw_{n−1}(ϕ⃗^{p}(X⃗, Y⃗)) + p^{n}ϕ_{n}(X⃗, Y⃗) = Φ(_{p}w_{n}(X⃗),_{p}w_{n}(Y⃗))

Since_{p}w_{n−1}(ϕ⃗^{p}(X⃗, Y⃗)) only depends on the ϕ_{i} for i < n, this gives a recursive expression for the ϕ’s. Integrality of ϕ_{n} will now follow inductively if we can establish that

Φ(_{p}w_{n}(X⃗),_{p}w_{n}(Y⃗)) ≡ _{p}w_{n−1}(ϕ⃗^{p}(X⃗, Y⃗)) mod p^{n}

Working mod p^{n} and using I again, we can rewrite the left hand side of the congruence as

Φ(_{p}w_{n−1}(X⃗^{p}),_{p}w_{n−1}(Y⃗^{p})) = _{p}w_{n−1}(ϕ⃗(X⃗^{p}, Y⃗^{p}))

the latter statement coming from the definition of ϕ. But we know that pw_{n−1} improves p-power congruences by II, so to prove the above congruence it suffices to show that

ϕ_{i}^{p}(X⃗, Y⃗) ≡ ϕ_{i}(X⃗^{p}, Y⃗^{p}) mod p

But this is an equation for polynomials 𝔽_{∣}, which follows immediately from the Frobenius automorphism. □

It follows from the proposition that addition, multiplication and inverse operations extend from 𝕎_{∣}⊗ℤ[\frac{⊮}{∣}] to 𝕎_{∣}, lnd since they satisfy the ring axioms on the latter dense subscheme, 𝕎_{∣} is now a ring scheme. To translate this into more concrete language, recall that the ring ℤ[𝕏_{⊮}, …] represents the functor associating to any ring A the set A^{∞} underlying the ring of formal power series. In scheme language, it follows that

Hom_{Scheme}(SpecA, 𝕎_{∣}) ≅ 𝔸[[𝕏]]

as sets. Thus one is led to the interpretation of the Witt scheme as associating to any ring A the ring of A-points of 𝕎_{∣}, which is a nonstandard ring structure on the set A[[X]], defined by the universal polynomials _{p}Σ and _{p}Π.

2.2 Interlude: Teichmüller and Witt

As a first application of Witt vectors [?], and to see what motivated the theory before we slide into generality, we demonstrate how to reconstruct the p-adic numbers functorially from their residue field 𝔽_{∣}. One can represent p-adic numbers x in the form

x = ∑_{i=0}^{∞} a_{i}p^{i}, a_{i} ∈ {0, 1, …, p − 1}

However this choice of representatives for 𝔽_{∣} = ℤ_{∣}/∣ℤ_{∣} as coefficients is clearly arbitrary, and behaves very poorly under arithmetic operations on ℤ_{∣}. The Teichmüller construction provides a canonical choice of coefficients to choose from, and in this section we will try to see how to do arithmetic with it.

We first study the dynamics of Frobenius. Let A be a ring such that A/p has no nilpotents, and assume furthermore that A is complete in its p-adic topology. The Frobenius, or p-th power map, F on A has the property of improving p-adic congruences: if a ≡ b modulo p, then a^{p} ≡ b^{p} modulo p^{2}, since (a + px)^{p} ≡ a^{p} modulo p^{2}. Similarly

a ≡ b mod p^{k} ⇒ a^{p} ≡ b^{p} mod p^{k+1}

However Frobenius is injective on A/p - if two elements are distinct modulo p, they won’t get any closer. Thus Frobenius shuffles the residue classes modulo p while shrinking them as we iterate. Thus, starting from some such residue class [a], F^{k}[a] makes sense as a well-defined residue class modulo p^{k+1}. Recalling our completeness hypothesis, it follows that the limit F^{∞}[a] is a uniquely defined element of A. Now restrict to the case A/p perfect - in other words, we require that Frobenius be an isomorphism of A/p onto itself. It follows that every residue class b modulo p contains a unique element in the image of F^{∞}, characterized as the unique element of b possessing p^{k}th roots in A for every k. Thus we have constructed a map

T : A/p → A, T : [a] ↦ [a] ∩ Im(F^{∞})

In fact, since the product of two elements with all p-power roots also possesses all such roots, it follows that T is the unique multiplicative homomorphism A/p → A which is a section of the quotient map. T is known as the Teichmüller mapping^{8}. In the case A = ℤ_{∣}, the image T(𝔽_{∣}) is necessarily the set of (p − 1)st roots of unity, as well as zero.

Equipped with a canonical embedding of A/p into A, one is tempted to attempt reconstruction of the entire ring A. We seek to represent elements of A as infinite sequences of elements of A/p via the Teichmüller construction - in other words as

x = ∑_{i=0}^{∞} T(a_{i})p^{i}, a_{i} ∈ A/p

The difficulty lies in figuring out how to translate back arithmetic operations on A in terms of the a_{i}. To solve this we note that in order to add or multiply in A/p^{n}, only knowledge of the a_{i} with i < n is necessary. Thus rather than proceed all at once it is better (in fact, essential) to construct canonical representations and their arithmetic term by term. In other words, the construction of canonical representatives should proceed in a coherent fashion over increasing powers of p, so that if we know the canonical representative of x modulo p^{n} the next step will give us a canonical expression modulo p^{n+1}. If we denote by W_{n} the map assigning to a⃗ ∈ (A/p)^{∞} an element in A correct modulo p^{n+1}, we will then need to solve inductively

W_{n}(a⃗) + W_{n}(b⃗) = W_{n}((a_{0} + b_{0}, Σ_{1}(a_{0}, a_{1}, b_{0}, b_{1}), …, Σ_{n}(a⃗, b⃗))

for the nth addition polynomial Σ_{n}, and similarly for the other arithmetic operations.

Hopefully by now the reader has begun to see the reason for this interlude. We constructed the Witt polynomials _{p}w_{n} precisely to fit the rather stringent requirements of the previous paragraph. Namely, observations I and II are precisely what is necessary to carry through the stepwise procedure outlined above^{9}: they enable a coherent induction on n, and furthermore the recipe I implies that the low order terms become higher and higher p-powers, and thus increasingly well represented. Moreover we have also established how to carry out all arithmetic operations on the Witt vectors.

8Not likely to be confused with the extremal quasiconformal map between Riemann surfaces also known by this name!

9It is possible to directly arrive at the recursive definition of the Witt polynomials from the setup above, a problem which we leave to the more energetic reader.

There are several different notational possibilities for the resulting Witt-Teichmüller construction. The map

(x_{0}, …, x_{n}) ↦ _{p}w_{n}(T(x_{0}), …, T(x_{n})) = T(x_{0})^{p^{n}} + ⋯ + p^{n}T(x_{n})

does not work as a reduction mod p^{n+1} of the map

(x_{0}, …, x_{n}, …) ↦ ∑_{i=0}^{∞} T(x_{i})p^{i}

Instead we may (and do) choose to relabel exponents around and define

τ : (x_{0}, …, x_{n}, …) ↦ ∑_{i=0}^{∞} T(x_{i}^{p^{−i}})p^{i}

which reduces modulo p^{n+1} to

T(x_{0}) + pT(x_{1}^{p^{−1}}) + ⋯ + p^{n}T(x_{n}^{p^{−n}}) ≡ _{p}w_{n}(T(x_{0}^{p^{−n}}, …, x_{n}^{p^{−n}}))

Theorem 2.2 There is a canonical isomorphism 𝕎_{∣}(𝔸/∣) ≅ 𝔸.

Proof: This will follow immediately once we establish, in the language of Proposition 3.1, that

Φ(τ(x⃗), τ(y⃗)) ≡ τ(ϕ_{0}(x_{0}, y_{0}), …, ϕ_{i}(x⃗, y⃗)) mod p^{n+1}

for all n. Denoting x_{i}^{p^{−n}}, y_{i}^{p^{−n}} by u_{i}, v_{i} and recalling that by definition _{p}w commutes Φ with the ϕ_{i}, we may rewrite this congruence as

pw_{n}(ϕ⃗(T(u⃗), T(v⃗))) ≡ _{p}w_{n}(T(ϕ⃗(u⃗, v⃗))) mod p^{n+1}

But T is the identity moduolo p, hence the congruence

T(ϕ_{i}(u⃗, v⃗)) ≡ ϕ_{i}(T(u⃗), T(v⃗)) mod p

automatically holds. The result now follows from the congruence-improving observation II. □

One consequence of this result (which could also be verified directly) is that the operation [p] of multiplication by p in the Witt ring ^{10} 𝕎_{∣}(𝔸/∣) is in fact equivalent to shifting once to the right and applying Frobenius:

[p](x_{0}, x_{1}, …) = (0, x_{0}^{p}, x_{1}^{p}, …)

10We should be sure to remember that though the Witt polynomials, when reduced modulo p, become fairly trivial, the corresponding addition and multiplication clearly do not - they are not defined via these reductions, but universally over ℤ.

2.3 The Anatomy of the Witt Ring

We now wish to de-specify the prime p in the construction of the ring scheme 𝕎_{∣}, to which we will henceforth refer as the p-typical Witt vectors [?, ?]. To do so we first relabel our former variables X_{0}, X_{1}, …, X_{n}, … by X_{1}, X_{p}, …, X_{p^{n}}, … Define

w_{1}(X_{1}) = X_{1}

w_{2}(X_{1}, X_{2}) = X_{1}^{2} + 2X_{2}
…

w_{n} = ∑_{d|n} dX_{d}^{n/d}

Observe that for any prime p, w_{p^{n}} agrees with _{p}w_{n} after relabeling coordinates. Again we interpret these maps scheme-theoretically as defining

w : 𝕎 = Specℤ[𝕏_{⊮}, 𝕏_{⊭}, …] → 𝔸^{∞} = Specℤ[𝕊_{⊮}, 𝕊_{⊭}, …]

Again, working over ℚ this map is invertible, thus transforming 𝕎 ⊗ Specℚ into a ring scheme with ring structure defined by polynomials Σ_{n}, Π_{n} and ι_{n}. (Instead of depending on all X_{i} for i < n, these polynomials now depend only on those X_{i} with i|n.) To make 𝕎 itself into a ring scheme, one makes recourse to our previous integrality result for all primes at once. The resulting ring scheme will again be interpreted as providing a novel ring structure on the set of ∞-tuples of elements of an arbitrary ring A.

Proposition 2.3 Let Φ ∈ ℤ[𝕏, 𝕐] be an integral polynomial in two variables. If the rational polynomials

ϕ⃗(X⃗, Y⃗) = (ϕ_{1}(X_{0}, Y_{0}), …, ϕ_{n}(X_{0}, …, X_{n}, Y_{0}, …, Y_{n}))

are defined by the functional equation

w_{n}(ϕ⃗(X⃗, Y⃗)) = Φ(w_{n}(X⃗), w_{n}(Y⃗))

then it follows that the ϕ_{i} are integral.

Proof: We proceed prime by prime: for each p we need to show that p does not divide the denominators of ϕ_{n}. The equivalent of the key observation II is now the statement that w_{n} increases the p-adic order of expressions by ord_{p}(n). In other words, if p^{k}|n and x ≡ y modulo p then w_{n}(x) ≡ w_{n}(y) modulo p^{k+1}. The proof now proceeds by induction, assuming the result known for all proper divisors of n. Let k = ord_{p}(n), and n = p^{k}m. The analogue of observation I is

w_{n}(X⃗) = w_{n/p}(X⃗^{p}) + nX_{n} + polynomials in lower X_{i}

(where w_{n/p} = 0 if k = 0), which results directly from separating the \frac{n}{p} term in the definition of w_{n}. Copying the former proposition, this allows us to reduce the question to a congruence

Φ(w_{n}(X⃗), w_{n}(Y⃗)) ≡ w_{n/p}(ϕ⃗^{p}(X⃗, Y⃗)) mod p^{k}

Once again this follows by the equivalent of II and the Frobenius in 𝔽_{∣}. □

The relationship between the universal Witt scheme 𝕎 and its p-typical relative can now be extended to construct an entire family of truncated, or restricted, Witt ring schemes. These correspond to subsets 𝔑 of the positive integers which contain all the divisors of their elements. Given such a subset there is a truncation 𝕎 → 𝕎_{𝔑} = Specℤ[𝕏_{⋉}]_{⋉∈𝔑} and maps

𝕎_{𝔑} → 𝔸^{𝔑}

which as for the p-typical and ordinary Witt vectors defines a ring scheme structure on 𝕎_{𝔑}. Of course in this notation the p-typical Witt vectors 𝕎_{∣} = 𝕎_{{∣,∣^{⊭},...}}, and 𝕎 = 𝕎_{ℕ}. We can now define a general Teichmüller mapping T : A → 𝕎_{𝔑}(𝔸) for an arbitrary ring A by sending

T : a ↦ (a, 0, 0, …)

Examining the Witt multiplication shows that T is multiplicative:

(a, 0, 0, …)(b, 0, 0, …) = (ab, 0, 0, …)

The Teichmüller mapping is not additive, but its behavior under addition is not difficult to describe. If we write

(a, 0, 0, …) + (b, 0, 0, …) = (r_{1}(a, b), …, r_{i}(a, b), …)

with only the indices in 𝔑 appearing, then the r_{i} are clearly some universal integral polynomials. In particular, applying the Witt polynomials, we see that

w_{n}(a, 0, 0, …) + w_{n}(b, 0, 0, …) = a^{n} + b^{n}

and so the r_{i} are the solutions of the functional equations

a^{n} + b^{n} = ∑_{d|n} dr_{d}(a, b)^{\frac{n}{d}}

We now wish to probe deeper into the algebraic structures associated with the Witt ring. For this purpose we first introduce a close relative of the Witt scheme, the λ-ring scheme Λ [?]. The points Λ(A) correspond to infinite sequences

Λ(A) = {(1 + a_{1}t + a_{2}t^{2} + …), a_{i} ∈ A}

with constant term one. Thus as a mere scheme Λ looks identical with our other ring schemes 𝕎 and 𝔸 (we suppress for now the burning question of the relationship of Λ with curves). However we note that the natural group operation to consider on the above set is the multiplication of power series. This converts Λ into a group scheme, which is different from 𝔸 or 𝕎 under the apparent map. To relate the group structure of Λ with 𝔸 consider the following map:

ψ : 𝔸 ⊗ Specℚ → Λ ⊗ Specℚ

ψ(s_{1}, s_{2}, …) = exp(− ∑ \frac{s_{m}}{m} t^{m})

This rather messy definition becomes more pleasant when viewed under the composite^{11} E = ψ ∘ w : 𝕎 → Λ:

∑ a_{i}t^{i} = exp(− ∑ \frac{s_{m}}{m} t^{m}) = exp(− ∑_{m} \frac{1}{m} ∑_{nd=m} nx_{n}^{d} t^{m}) =

= exp(− ∑_{n} ∑_{d} \frac{(x_{n}t^{n})^{d}}{d}) = exp(∑_{n} log(1 − x_{n}t^{n})) = ∏_{n}(1 − x_{n}t^{n})

11This map E is a form of the Artin-Hasse exponential.

It now follows that we can recover the a_{i} and x_{i} from each other via polynomials with integral coefficients: to find the a_{i} expand out the product, and to find the x_{i} set x_{1} = −a_{1}, x_{2} = −a_{2}, x_{3} = −a_{3} − x_{1}x_{2} and in general

x_{n} = −a_{n} + ∑ ; i_{1},...,i_{l} distinct, ∑_{k} i_{k}=n(−1)^{l+1} ∏ x_{i_{k}}

This proves that E : 𝕎 → Λ is an isomorphism of schemes (over ℤ).

Moreover, ψ is an exponential and hence maps the additive group of 𝔸 ⊗ Specℚ homomorphically to the multiplicative group Λ ⊗ Specℚ. We can also see this by noting that the inverse of ψ is given by the (shifted) logarithmic derivative:

−t\frac{d}{dt} log(∏_{n}(1 − x_{n}t^{n})) = −t\frac{d}{dt}(− ∑_{m} \frac{s_{m}}{m} t^{m}) = ∑_{m} s_{m}t^{m}

We thus think of the s_{m} as functions on Λ(A) just as (via the Witt polynomials) they are functions on 𝕎(𝔸). Since the Witt addition was defined in terms of 𝔸, it follows that E is a group homomorphism, albeit on a dense subset of 𝕎. But since we know that the group operations and the mapping extend to all of 𝕎, it follows by continuity that E is indeed an isomorphism of group schemes E : 𝕎 ≅ Λ. Viewed less scheme-theoretically, we first proved that E_{A} is an isomorphism, induced by integral polynomials, between the groups 𝕎(𝔸) and Λ(A) where A is a ℚ-algebra. This immediately implies the result for rings of characteristic zero, by inclusion and functoriality. Finally any ring admits a surjection from a characteristic zero ring, and the resulting commutative diagram yields the statement over any ring. This technique of using denominators as a crutch to proving integral results is prevalent in the study of Witt vectors and their relatives.

In order to see what ring scheme structure Λ inherits from its isomorphism with 𝕎, we prevue some relations of Λ with symmetric functions. Given finite collections of indeterminates ζ_{i} and η_{i}, we can write the products

∏_{i}(1 − ζ_{i}) = 1 + X_{1}t + X_{2}t^{2} + …

∏_{i}(1 − η_{i}) = 1 + Y_{1}t + Y_{2}t^{2} + …

where the X_{i} and Y_{i} are the elementary symmetric functions in the ζ_{i} and η_{i}, respectively. The fundamental theorem on symmetric functions asserts that any symmetric expression in the ζ_{i} can be written as a polynomial in the X_{i}. This fact can be utilized to construct various operations on the scheme Λ. Writing

∏_{i}(1 − ζ_{i}^{n}t) = 1 + ν_{1}t + ν_{2}t^{2} + …

it follows that we can express the ν_{i} as ν_{i} = ν_{i}(X_{1}, …, X_{ni}) in terms of the elementary symmetric functions. Moreover ν_{i} is independent of the number of indeterminates ζ_{i} as long as there are at least ni of them, and thus are defined formally when we allow the number of ζ_{i} to go to infinity. Similarly the expression

∏_{i,j}(1 − ζ_{i}η_{j}t) = 1 + P_{1}t + P_{2}t^{2} + …

defines the P_{i} = P_{i}(X_{1}, …, X_{i}, Y_{1}, …, Y_{i}) as polynomials in the elementary symmetric polynomials which are stable in terms of the number of indeterminates ζ_{i}, η_{i}. We may now ignore the existence of these auxiliary indeterminates and use the polynomials P_{i} to define the multiplication on Λ(A):

(1 + a_{1}t + a_{2}t^{2} + …) ∗ (1 + b_{1}t + b_{2}t^{2} + …) =
= 1 + P_{1}(a_{1}, b_{1}) + P_{2}(a_{1}, a_{2}, b_{1}, b_{2})t^{2} + …

To verify that this is indeed the ring operation on Λ(A) induced by the Witt multiplication, we resort to the functions s_{n} on Λ(A) describing the coefficients of the map

−t\frac{d}{dt} log : Λ(A) ⊗ Specℚ → 𝔸 ⊗ Specℚ

Thus we wish to show that for f(t), g(t) ∈ Λ(A) we have

s_{n}(f(t) ∗ g(t)) = s_{n}(f(t))s_{n}(g(t))

In terms of auxiliary variables (interpreting the coefficients formally as symmetric functions)

f(t) = ∏_{i=1}^{∞}(1 − ζ_{i}t) g(t) = ∏_{i=1}^{∞}(1 − η_{i}t)

the coefficients s_{n} become the n-th power sum symmetric functions:

s_{n}(f(t)) = ∑_{i=1}^{∞} ζ_{i}^{n} s_{n}(g(t)) = ∑_{i=1}^{∞} η_{i}^{n}

On the other hand, we calculate

∑_{i=1}^{∞} s_{i}(f(t) ∗ g(t))t^{i} = −t\frac{d}{dt} log( ∏_{i,j=1}^{∞}(1 − ζ_{i}η_{j}t)) =

= ∑_{i,j=1}^{∞} \frac{ζ_{i}η_{j}}{1 − ζ_{i}η_{j}t} = ∑_{i,j=1}^{∞}(ζ_{i}η_{j})^{i}t^{i}

from which it follows that the multiplications on 𝕎 and Λ agree.

Equipped with the perspective afforded by Λ, we may endow the Witt ring with a significantly expanded algebraic structure. The most natural of the additional structures is provided by the homothety, shift and Frobenius homothety operators which we now define, postponing the study of λ-rings to a later point. The simplest of these are the homothety, or stretch, operators ⟨c⟩ (for c ∈ A). We define these on Λ(A) by ⟨c⟩^{Λ}f(t) = f(ct), or

⟨c⟩^{Λ}(1 + a_{1}t + a_{2}t^{2} + …) = 1 + a_{1}ct + a_{2}c^{2}t^{2} + …

This implies the relation

i. ⟨c⟩⟨d⟩ = ⟨cd⟩

Converting to 𝔸(𝔸),

−t\frac{d}{dt} log⟨c⟩^{Λ}(∏_{i=1}^{∞}(1 − ζ_{i}t)) = −t\frac{d}{dt} log(∏_{i=1}^{∞}(1 − ζ_{i}ct)) =

= ∑_{i=1}^{∞} \frac{ct}{1 − ζ_{i}ct} = ∑_{i=1}^{∞} ∑_{k=1}^{∞} c^{k}ζ_{i}^{k}t^{k}

and so, recalling that the s_{m} are the m-th power sums on Λ, we read off s_{m}⟨c⟩^{Λ} = c^{m}s_{m}.

The Frobenius operators F_{n} are generalizations of the ordinary p-typical Frobenius F = F_{p} which we discussed in the previous section. In general however it is far more convenient to define F_{n} on Λ and then transfer them to 𝕎. Denoting by τ_{i} the formal n-th root of the indeterminate t, define

F_{n}^{Λ}(f(t)) = ∏_{τ_{i}^{n}=t} f(τ_{i})

This makes sense as an expression in Λ(A) since the resulting polynomial is Galois-invariant under the permutations of the τ_{i} and hence can be written purely in terms of t. In fact, in terms of auxiliary variables

F_{n}^{Λ}(∏_{i}(1 − ζ_{i}t)) = ∏_{i}(1 − ζ_{i}^{n}t)

so that in fact we may directly define the Frobenius operator by

F_{n}^{Λ}(1 + a_{1}t + a_{2}t^{2} + …) = 1 + ν_{1}(a⃗)t + ν_{2}(a⃗)t + …

in terms of the operators defined above. Changing to 𝔸(𝔸),

−t\frac{d}{dt} log(F_{n}^{Λ}(∑_{i=1}^{∞} a_{i}t^{i})) = −t\frac{d}{dt} log(∏_{i=1}^{∞}(1 − ζ_{i}^{n}t)) =

= ∑_{i=1}^{∞} \frac{ζ_{i}^{n}t}{1 − ζ_{i}^{n}t} = ∑_{i=1}^{∞} ∑_{k=1}^{∞}(ζ_{i}^{n}t)^{k}

The coefficient of t^{m} in this expression gives the mn-th power sum of the ζ_{i} - in other words the function s_{mn}. Thus

F_{m}^{𝔸}(s_{1}, s_{2}, …) = (s_{m}, s_{2m}, …)

In this form it is clear that

ii. F_{m} ∘ F_{n} = F_{mn}

as well as that the Frobenii are in fact ring homomorphisms, since the ring operations are componentwise in the s_{i}. In general F_{m} does not assume a very simple form on 𝕎. However, recalling the Frobenius map F : 𝕎_{∣} → 𝕎_{∣} raising each coordinate to the p-th power, and composing with the map _{p}w to 𝔸, we see the coordinates (s_{1}, s_{p}, s_{p^{2}}, …) (in our new notation) replaced by (s_{p}, s_{p^{2}}, s_{p^{3}}, …), so indeed F agrees with F_{p}. More generally truncated Frobenius operators may be defined on the truncated Witt schemes

F_{n} : 𝕎_{𝔑} → 𝕎_{𝔑/𝔫}

whenever this makes sense.

The final operator we introduce is known as the Verschiebung, or shift. The m-th Verschiebung V_{m} : 𝕎 → 𝕎 is defined on the coordinate ring as follows: V_{m}^{∗}(X_{n}) = X\frac{n}{m} for m|n and V_{m}(X_{n}) = 0 otherwise. On A-valued points,

V_{m}(x_{1}, x_{2}, …) = (\underbrace{0, 0, …, x_{1}}_{m}, \underbrace{0, 0, …, x_{2}}_{m}, …)

Thus the shift operator V_{m} contracts by a factor of m on coordinates and spreads out by a factor of m on points. This suggests that V_{m} should be some form of dual to F_{m} (this is in fact accomplished under Cartier duality). On 𝔸, V_{m} takes S_{m} to mS\frac{n}{m} if m|n and to 0 otherwise, while points are spread out as in 𝕎 with each of the coordinates multiplied by m. Finally we see that on Λ V_{m} takes the form

V_{m}^{Λ} : 1 + a_{1}t + a_{2}t^{2} + … ↦ 1 + a_{1}t^{m} + a_{2}t^{2}m + …

It is clear that

iii. V_{n} ∘ V_{m} = V_{nm}

Moreover V_{m} maps 𝕎 into the kernel of the truncation map 𝕎 → 𝕎_{ℕ−⋗ℕ}. (It is in fact an isomorphism onto this kernel, as can be seen by checking A-valued points.) Corresponding Verschiebung operators can also be constructed on the truncated Witt rings^{12} as the kernels

V_{m} : 𝕎_{𝔑/𝔪} → 𝕎_{𝔑} → 𝕎_{𝔑−𝔪𝔑}

There is a collection of fundamental relations between the operators we have just introduced. Inspecting the form our operators take on 𝔸, we may read off several of these directly:

iv. ⟨1⟩ = V_{1} = F_{1} = id
v. ⟨a⟩V_{n} = V_{n}⟨a^{n}⟩

12The Verschiebungs V_{m} are defined on any truncation 𝕎_{𝔑} but are zero unless m ∈ 𝔑.

vi. F_{n}⟨a⟩ = ⟨a^{n}⟩F_{n}
vii. F_{n} ∘ V_{m} = V_{m} ∘ F_{n} when (n, m) = 1

At the end of the section on the Teichmüller construction, we established another one:

(0, x_{0}^{p}, x_{1}^{p}, …) = [p](x_{0}, x_{1}, …)

which translates to F_{p} ∘ V_{p} = [p]. More generally, since on 𝔸(𝔸) the Frobenius contracts by a factor of n and the Verschiebung acts by expanding by a factor of n while multiplying coordinates by n, it follows that

viii. F_{n} ∘ V_{n} = [n]

Before introducing our final relation, we return to the Teichmüller construction for the last time. Our construction of the p-adics generalizes using Verschiebungs to the construction of canonical infinite sum expressions in 𝕎(𝔸) from Teichmüller representatives. First observe that if we add two Witt vectors a⃗, b⃗ which are such that for every index i either a_{i} or b_{i} vanishes, then the sum a⃗ + b⃗ is indeed the componentwise sum (by direct calculation). Since on 𝕎_{𝔑} the V_{n} act by shifting coordinates, we have the following result:

Proposition 2.4 (V-basis) Every element a = (a_{1}, …, a_{d}, …) ∈ 𝕎_{𝔑} can be written uniquely as a sum

a = ∑_{d∈𝔑}V_{d}(T(a_{d}))

Of course to properly interpret the above infinite sum, we must have an appropriate notion of convergence. The topology on 𝕎 is given by the family of subgroups V_{n}𝕎 of Witt vectors whose first n − 1 entries vanish. 𝕎 is complete and Hausdorff in this topology, and sums such as above now make sense as Witt vectors.

In a similar spirit we may now describe the sum of two homotheties:

ix. ⟨a⟩ + ⟨b⟩ = ∑_{m=1}^{∞} V_{m}⟨r_{m}(a, b)⟩F_{m}

where the r_{i} are the universal polynomials describing addition of two Teichmüller representatives,

a^{n} + b^{n} = ∑_{d|n} dr_{d}(a, b)^{\frac{n}{d}}

That such a relation should hold becomes plausible upon recalling that, mapping to 𝔸,

w_{n}(⟨a⟩ + ⟨b⟩) = (a^{n} + b^{n})w_{n}

Explicitly, we find

w_{n}(∑_{m=1}^{∞} V_{m}⟨r_{m}(a, b)⟩F_{m}) = ∑_{n=md} mw_{d}⟨r_{m}(a, b)⟩F_{m} =

= ∑_{n=md} mr_{m}(a, d)^{d}w_{d}F_{m} = ( ∑_{n=md} mr_{m}(a, b)^{d})w_{n}

completing the list of relations.

2.4 Formal Groups and Witt Vectors

We now return to the study of formal groups and their realiztion through the group of curves. The similarity between the group of curves C(F) in a formal group F/A and the underlying group of Λ suggests a construction of natural operators on C(F) analogous to the Frobenius, Verschiebung and homothety operators. In fact these operators satisfy the same relations as those of the previous section, and define a structure known as a Cartier^{13} module on C(F). Cartier, in the course of his detailed study of formal groups, established that this structure in fact classifies formal groups - the assignment F ↦ C(F) is in fact an equivalence of categories between that of formal groups and a certain category of topological groups with Cartier-module structure. The theory is held together by the realization of the Witt vectors as the universal embodiment of the structure of Cartier module. This relation can be seen in several different guises: as an isomorphism between C(Ĝ_{m}) and the underlying additive group of 𝕎; as isomorphism between C(F) and a certain “twisted” version 𝕎^{F} of the Witt group; and, most satisfyingly, in the guise of an infinite-dimensional formal group 𝕎̂ which is constructed from 𝕎 via Cartier duality. In this section, we attempt to sketch these intriguing algebraic constructions, with barely a hint at their concrete applications to the study and classification of formal groups.

13C(F) is also known as the Cartier-Dieudonné module of the formal group F.

We assume henceforth that F/A is a commutative^{14} formal group. Recall that C(F; A), the group of curves (with coefficients in A) in an n-dimensional formal group F/A, is the set of n-tuples γ(t) of power series in A[[X⃗]] with vanishing constant term, γ(0) = 0. Its group structure is defined via the formal group law F of F as

γ_{1}(t) ⋄ γ_{2}(t) = F (γ_{1}(t), γ_{2}(t))

We topologize the group of curves as follows: let C^{\}(F) denote curves which vanish to order n at zero. Thus

C(F) = C′(F) ⊃ ⋯ ⊃ C^{\}(F) ⊃ ⋯

Moreover, C^{\}(F) is a subgroup of C(F) and this filtration defines a topology on C(F) in which it is complete. This topology is analogous to the natural one on 𝕎(𝔸) defined by the subgroups of vectors whose first n components vanish.

We now define the homothety operator ⟨a⟩ for a ∈ A as

⟨a⟩γ(t) = γ(at)

and the m-th Verschiebung as

V_{m}γ(t) = γ(at)

We would also like to define the Frobenius F_{m} on C(F) as

F_{m}(γ(t)) = γ(ζ_{m}t^{\frac{1}{m}} ) ⋄ γ(ζ_{m}^{2}t^{\frac{1}{m}} ) ⋄ ⋯ ⋄ γ(ζ_{m}^{m}t^{\frac{1}{m}} ) = ∑_{τ^{m}=t}^{C(F)}γ(τ)

To make sense of this over an arbitrary ring A we need to exercise some caution - this is done rigorously in terms of symmetric functions as for the Frobenii on Λ. These operators, by their very scheme-theoretic nature, commute with the operations of change of rings

ϕ : A → B, ϕ_{∗} : C(F, A) → C(ϕ_{∗}F, B)

14Much of the theory can be extended to noncommutative formal groups. This however requires more emphasis on the Lie-theoretic picture, which we ignore for simplicity.

They are also functorial under homomorphisms of formal groups: if α(X) : F → G is a homomorphism, we may define an induced map α_{⋄} on curves via

α_{⋄}(γ(t)) = α(γ(t)) ∈ C(G)

since α(X) has no constant term by definition of homomorphisms. This induced map is readily seen to be a group homomorphism, and to commute with Verschiebungs and Frobenii. It is also continuous in the topology we have defined on curves.

One further useful property of curves, in analogy with Witt vectors again, is the V-basis. Denote by δ_{i}(f(t)) the curve

δ_{i}(t) = (\underbrace{0, 0, …, f(t)}_{i}, 0, …, 0) ∈ C(F, A)

for some power series f(t). Then ⟨a⟩δ_{i}(t) = d_{i}(at) and V_{m}δ_{i}(t) = δ_{i}(t^{m}) and so we expect that we can write any curve directly in terms of the various δ_{i}(t^{m}) basic series. This construction is known as the V-basis:

Proposition 2.5 [?] Every curve γ(t) ∈ C(F, A) in the n-dimensional formal group F can be written uniquely in the form

γ(t) = ∑_{m=1}^{∞}^{C} ∑_{i=1}^{n}^{C}V_{m}⟨a_{mi}⟩δ_{i}(t)

(The proof procedes by induction, constructing the necessary a_{mi} modulo C^{∥+∞} given the expression for γ(t) modulo C^{∥}.)

We now wish to show that the Frobenius, Verschiebung and homothety operators on C(F) satisfy the relations i through ix derived for the Witt vectors. The logarithmic derivative which we used to pass from Λ to 𝔸 is here replaced by the logarithm of the formal group F. For a general ndimensional formal group, we may interpret the logarithm in relation to the Lie algebra of F, but since we are assuming F commutative this is none other than an isomorphism with the n-dimensional additive formal group. By passing to the additive group, we simplify significantly the form of our operators, in particular the Frobenius. Writing log_{F}(X⃗) for the logarithm of F, and

log_{F}(γ(t)) = ∑_{i=1}^{∞} x_{i}t^{i}

(with the x_{i} ∈ A^{m}), the Frobenius becomes

log_{F}(F_{m}γ(t)) = log_{F}(γ(ζ_{m}t^{\frac{1}{m}} ) ⋄ γ(ζ_{m}^{2}t^{\frac{1}{m}} ) ⋄ ⋯ ⋄ γ(ζ_{m}^{m}t^{\frac{1}{m}} )) =

= log_{F}(γ(ζ_{m}t^{\frac{1}{m}})) + ⋯ + log_{F}(γ(ζ_{m}^{m}t^{\frac{1}{m}})) = ∑_{i=1}^{∞} x_{i}t^{\frac{i}{m}} ∑_{j=1}^{m} ζ_{m}^{ij} =

= ∑_{m|i} mx_{i}t^{\frac{i}{m}} = ∑_{i=1}^{∞} mx_{im}t^{i}

As was the case with the logarithmic derivative, the logarithm is only strictly defined over ℚ-algebras. For the most part, we may reduce checking relations to this case by the usual scheme tricks. The one tricky part is the problem of lifting to characteristic zero: given any ring A there is a ring B of characteristic zero mapping onto A. The problem is finding a formal group G/B lifting F. To do so, we resort to the universal formal group in n dimensions and its analog of Lazard’s theorem. Namely, there is a unique homomrphism ϕ : ℤ[𝕌⃗] → 𝔸 with ϕ_{∗}F_{U} = F. We create a map ϕ^{0} : ℤ[𝕌⃗] → 𝔹 by lifting the images of the generators to B, which we can accomplish thanks to the freeness of ℤ[𝕌⃗], which is Lazard’s theorem. We may now define G = ϕ_{∗}^{0}F_{U} and verify that indeed G lifts F. This is an important example of how we may use the structure of the universal formal group to improve our techniques of studying concrete formal groups. As a result we can now reduce the verification of relations on arbitrary groups of curves to the curves in the n-dimensional additive formal group Ĝ_{a}^{n}, or if we wish (via the exponential map) to the multiplicative formal group Ĝ_{m}^{n}.

Proposition 2.6 The group of curves C(Ĝ_{m}) in the one-dimensional multiplicative formal group is naturally isomorphic to the underyling group 𝕎^{+} of the Witt vectors. Furthermore the operators ⟨a⟩, F_{m} and V_{m} defined for both groups correspond under this isomorphism.

Proof: The isomorphism is more readily seen between the Witt vectors and the formal group Ĝ_{m}^{−} defined by F^{−}[X, Y] = X + Y − XY. The latter group has logarithm

log^{−} = ∑ \frac{x^{n}}{n}

and is itself isomorphic to Ĝ_{m} via the homomorphism

f(X) : Ĝ_{m}^{−} → Ĝ_{m}, f(X) = X + X^{2} + X^{3} + … = \frac{1}{1 − X} − 1

(which is the composite of log^{−} and the standard exponential):

\frac{1}{1 − X − Y + XY} − 1 =

= (\frac{1}{1 − X} − 1) + (\frac{1}{1 − Y} − 1) + (\frac{1}{1 − X} − 1)(\frac{1}{1 − Y} − 1)

Now given two curves γ_{1}, γ_{2}(t) ∈ C(Ĝ_{m}^{−}, A) we calculate

γ_{1}(t) ⋄ γ_{2}(t) = 1 − (1 − γ_{1}(t))(1 − γ_{2}(t))

So if we assign γ(t) ↦ 1−γ(t) we obtain an isomorphism from the (additive) group C(Ĝ_{m}^{−}, A) to the (multiplicative) group Λ(A). Combined with the identification Λ(A) ≅ 𝕎(𝔸), this becomes β : C(Ĝ_{m}^{−}, A) ≅ 𝕎^{+}(𝔸).

Now from the form of the homotheties and Verschiebungs on Λ, it is clear that

β ∘ V_{n} = V_{n} ∘ β, β ∘ ⟨a⟩ = ⟨a⟩ ∘ β

Using the formal definitions

F_{m}^{C}(γ(t)) = ∑_{τ^{m}=t}^{C(F)}γ(τ) F_{m}^{Λ}(f(t)) = ∏_{τ^{m}=t} f(τ)

we can also see that the Frobenii agree. Somewhat more explicitly, we have calculated (in characteristic zero) that

log_{F} F_{m}^{C}(γ(t)) = ∑_{i=1}^{∞} mx_{im}t^{i}

which again agrees with our calculations for Λ. □

Now the multiplicative formal group laws of Ĝ_{m}^{n} are defined componentwise in terms of their one-dimensional counterpart Ĝ_{m}. It follows immediately from the above that the Cartier relations must now hold for Ĝ_{m}^{n} as well (since they too are defined componentwise). But we had reduced the general verification of the relations to the case of the multiplicative groups, so we have now established:

Corollary 2.7 The operators ⟨a⟩, F_{m} and V_{m} defined on the group of curves C(F, A) in any commutative formal group satisfy the Cartier relations i through ix of the previous section. □

Now that we have described the multiplicative formal group Ĝ_{m} in terms of the Witt vectors, and furthermore used this realization to establish a general result on formal groups, it is tempting to ask whether there is a more direct general link between Witt-like constructions and formal groups. The first such construction which we briefly survey is an attempt to ape the Witt procedure directly in the context of a one-dimensional commutative formal group F over a characteristic zero ring A. First write (over A ⊗ ℚ) the logarithm of F:

log_{F}(T) = T + a_{2}T^{2} + …

We now define Witt polynomials

w̄_{n}^{F}(X⃗) = ∑_{d|n} a_{n/d}X_{d}^{n/d}

In the case of the formal group Ĝ_{m}^{−}, we read off from the form of log^{−} that the normal Witt polynomials become w_{n} = nw̄Ĝ_{m}^{−} ; n.

We now define the addition polynomials Σ_{n}^{F} using these new Witt polynomials:

w̄_{n}^{F}(Σ_{n}^{F}(X⃗, Y⃗)) = w̄_{n}^{F}(X⃗) + w̄_{n}^{F}(Y⃗)

Lemma 2.8 [?] The Σ_{n}^{F} have their coefficients in A.

Thus we may define a Witt group scheme 𝕎^{F} over A whose B-points are infinite vectors in B and whose addition is defined in terms of the polynomials Σ_{n}^{F}. To construct an isomorphism 𝕎^{F} ≅ C(F) one modifies the exponential map E : 𝕎 → ≱ as follows (remembering that there is a factor of n discrepancy between our new and old w’s):

E^{F}(X⃗) = log_{F}^{−1}(∑_{1}^{∞} w_{n}^{F}(X⃗)t^{n}

More explicitly, in terms of B-valued points, we may define

E(b_{1}, b_{2}, …) = ∑^{F}b_{i}t^{i}

directly in terms of curves and using the V-basis of C(F), which establishes the isomorphism directly. These two definitions naturally agree:

log_{F}(∑_{i}^{F}b_{i}t^{i}) = ∑_{i} log_{F}(b_{i}t^{i}) =

= ∑_{i} ∑_{j} a_{j}b_{i}^{j}t^{ij} = ∑_{k} w_{k}^{F}(b_{1}, b_{2}, …)t^{k}

A deeper connection between the Witt vectors and the curves functor exists however. The formal group 𝕎̂ which is the Cartier dual of the Witt group scheme turns out to represent the functor of curves in a formal group, as we now describe. The construction of 𝕎̂ from 𝕎 is direct and simple; the fact that the two are Cartier duals is less obvious. We may first define 𝕎̂ as the subfunctor of 𝕎 which assigns to a ring B the subgroup

𝕎̂(B) = {(b_{1}, b_{2}, …) ∈ 𝔫(𝔅) almost all zero}

This defines an infinite dimensional formal group, a ‘formalization’ of the Witt functor, whose group law can be given explicitly in terms of the power series Σ_{n} (which are in fact polynomials) as

F 𝕎̂(X⃗, Y⃗) = (Σ_{1}(X⃗, Y⃗), Σ_{2}(X⃗, Y⃗), …)

(We will not get involved in the precise definition of an infinite dimensional formal group law [?]. The ring of power series in infinitely many variables involves only finite monomials, and we further require a quasi-finiteness condition which guarantees that every particular coefficient in F (X⃗, Y⃗) is given as a finite sum of terms, and is thus well defined.) We are particularly interested in the group of curves C(𝕎̂, A) associated to the Witt formal group. Such a curve may be written as an ∞-tuple

(f_{1}, f_{2}, …) = ∑ 𝕎̂δ_{i}(f_{i}), f_{i} ∈ A[[t]]

Of particular interest is the power series γ𝕎̂(t) = δ_{1}(t) consisting of a t in the first position and nothing elsewhere. As for any group of curves, we have V_{m}γ𝕎̂(t) = δ_{1}(t^{m}). Assuming momentarily (as usual this assumption will have no consequence) that A is such that we may speak of the distinct m-th roots of unity over A, we calculate the Frobenius of γ𝕎̂(t):

F_{m}γ𝕎̂(t) = γ𝕎̂(ζ_{m}t^{\frac{1}{m}} ) ⋄ ⋯ ⋄ γ𝕎̂(ζ_{m}^{m}t^{\frac{1}{m}} )

Applying the Witt polynomials, we derive

w_{i}(F_{m}γ𝕎̂(t)) = (ζ_{m}^{i} + ⋯ + ζ_{m}^{mi})t^{\frac{i}{m}} = \{ \begin{matrix}mt^{i/m}, & if m divides i; \\ 0, & otherwise. \end{matrix}
= w_{i}(δ_{m}(t))

Thus the Frobenius operator transforms δ_{1}(t) to δ_{m}(t).

It is amusing to note that the curves in 𝕎̂ look remarkably like vectors in 𝕎(𝔸[[≈]]), with the curve γ𝕎̂(t) corresponding to the vector t⃗ = (t, 0, …). However, the Frobenius F_{m} on 𝕎 sends t⃗ to (t^{m}, 0, …) and the corresponding Verschiebung sends t⃗ to (0, 0, …, t, 0, …). Thus the roles of V and F are reversed! This is one manifestation of the (Cartier) duality between 𝕎̂ and 𝕎, which interchanges Frobenii and Verschiebungs. One proof of this duality is direct and uses the exponential map E : 𝕎 → Λ and the Witt multiplication. Considering 𝕎̂ as a subfunctor of 𝕎 and given vectors a⃗ ∈ 𝕎(𝔸) and b⃗ ∈ 𝕎̂(A), then a⃗b⃗ is again in 𝕎̂(A) - all entries are nilpotent and almost all are zero. Taking E(a⃗b⃗) we obtain a polynomial with leading term one all of whose other coefficients are nilpotent. Thus substituting t = 1 we obtain a well defined element 1 + u (u nilpotent) of the multiplicative group G_{m}(A). This is the desired pairing 𝕎̂ ⊗ 𝕎 → G_{m}. A slightly tedious verification now establishes that these two schemes indeed become Cartier duals under this pairing^{15}. This duality will also fall out as an immediate consequence of Cartier’s first theorem.

Returning to curves, the V-basis of C(𝕎̂, A) now takes on the pleasant form

Proposition 2.9 Every curve γ(t) ∈ C(𝕎̂, A) can be written uniquely as

γ(t) = ∑_{m=1}^{∞}^{C} ∑_{i=1}^{∞}^{C}V_{m}⟨a_{mi}⟩F_{i}γ𝕎̂(t)

for some a_{mi} ∈ A. □

15The p-typical form of this pairing, namely the Artin-Hasse truncated exponential, is also the one used [?] to establish the Cartier duality between p^{r}-th roots of unity and ℤ/∣^{∖}ℤ, expressed as the group-scheme kernel of the Artin-Schreier map on 𝕎_{∣} [?].

The simple structure of C(𝕎̂, A) suggests that it might possess some freeness properties as a group of curves. Any homomorphism of formal groups induces a (continuous) homomorphism of the corresponding group of curves, which commutes with the natural operators. The curve γ𝕎̂(t) generates C(𝕎̂, A) as above, and moreover the only obvious linear relation it satisfies is that the shift V_{m} scales the exponent of t by m, which holds for any group of curves. The Frobenii of γ𝕎̂(t) are linearly independent of it, and thus in a homomorphism we may force them to map to the Frobenii of the image of γ𝕎̂. This appparent freeness is the subject of Cartier’s first theorem:

Theorem 2.10 (Cartier’s First Theorem, [?]) The formal group 𝕎̂(A) represents the functor C( , A). In other words, let γ(t) ∈ C(F, A) be a curve in a (commutative) formal group F/A. Then there exists a unique homomorphism α : 𝕎̂(A) → F with α_{⋄}(γ𝕎̂(t)) = γ(t). Moreover the ensuing (set) isomorphism

C(F, A) ≅ Hom_{A}(𝕎̂(A), F)

is an isomorphism of topological groups.

We will not go into the full details of the proof, though the proof is very much in the spirit of other results we have discussed. The uniqueness is straightforward since the map α_{⋄} must commute with Frobenii and Verschiebungs, and γ𝕎̂(t) generates the curves in 𝕎̂ under these standard algebraic operations. In terms of power series rings and the formal group law F 𝕎̂, it follows formally that knowing how to map curves tells one how to map formal group laws. Seen from the perspective of coordinate rings, we have

α((X_{1}, 0, …)) = γ(X_{1})

which forces

α((X_{1}, X_{2}, …)) = ∑_{i=1}^{∞} F_{i}γ(X_{i})

establishing both uniqueness and the statement that the induced map α_{⋄} must be an isomorphism of topological groups. As for existence, the proof in [?] uses the universal formal group law and the functional equation lemma in a typical combination to first create a map and then establish the necessary integrality. We leave the details to the writer.

The representation theorem has many powerful consequences, and inexorably unites the study of formal groups with Witt vectors. First of all we may ask is there any interesting structure of the functor C that we have missed? Suppose τ is a functor endomorphism of C( , A) (that is, a natural transformation from the curves functor to itself), merely as a set-valued functor. Then it follows immediately from the representability theorem that τ is completely determined (by naturality) by its value on the curve γ𝕎̂(t). Moreover, we can send γ𝕎̂(t) by fiat to any curve in C(𝕎̂, A) we wish, and the corresponding τ is now forced to be additive and continuous. Thus we have:

Corollary 2.11 Operators on curves (endomorphisms τ of C( , A)) are in one-to-one correspondence with curves in C(𝕎̂, A). □

Using the V-basis on curves in 𝕎̂ and the Cartier relations i through ix discussed previously, we may now put together our operators V_{m}, F_{m} and ⟨a⟩ into a ring of operators whose structure is completely understood:

Definition 2.12 The Cartier ring Cart(A) is defined as the ring of all operators written as formal sums

∑_{m,n} V_{m}⟨a_{mn}⟩F_{n}

(where for fixed m only finitely many a_{mn} are nonzero), subject to the relations i through ix. Cart(A) contains all operators on C( , A).

Since (by representability) endomorphisms of 𝕎̂(A) correspond to curves, it follows that Cart(A) is also the ring End𝕎̂(A). Reconsidering 𝕎̂ as a subfunctor of 𝕎 we note that 𝕎̂ has induced endomorphisms V_{m} and F_{m} identical to those in 𝕎. However, as we have already discovered, these are reversed in their roles: under the identification between endomorphisms of 𝕎̂ and the Cartier ring, Verschiebungs and Frobenii switch roles. In fact Cartier duality for 𝕎̂ and 𝕎 is an immediate consequence of the representability theorem. Consider formal group homomorphisms Hom(𝕎̂, Ĝ_{m}). This group is isomorphic to the group of curves C(Ĝ_{m}). But as we know the latter group is isomorphic to the underlying additive group scheme of 𝕎. Thus we have a cheap proof of the duality Proposition 2.13 The formal group scheme 𝕎̂ and the group scheme 𝕎 are Cartier duals, namely 𝕎 ≅ Hom(𝕎̂, Ĝ_{m}) and 𝕎̂ ≅ Hom(𝕎, G_{m}). □

We conclude this discussion by citing Cartier’s second and third theorems, whose proofs are based on the use of 𝕎̂ and are not very difficult at this stage. At this stage we are willing to accept that the curves functor is faithful for formal groups over A, and the question becomes what conditions on a group make it a group of curves, besides the action of the Cartier operators. Since Cart(A) inherits a natural topology, we may consider C(F, A) as a topological Cart(A)-module, which furthermore satisfies

I. If (x_{i}) is a sequence in Cart(A) converging to zero and (γ_{i}) is any sequence in C(F, A), then ∑ x_{i}γ_{i} converges in C(F, A).

II. The topology of C(F, A) is defined by the subgroups C^{\}(F, A), which are the sum of all shifts V_{m}C(F, A) with m ≥ n.

III. The shift V_{m} induces a bijection

C(F, A)/C^{∈}(F, A) → C^{⇕}(F, A)/C^{⇕+∞}(F, A)

IV. C(F, A)/C^{∈}(F, A) is a free A-module (via the action of ⟨a⟩).

Any topological Cart(A)-module satisfying the above four conditions is called reduced. Now Cartier’s theorems state:

Theorem 2.14 (Cartier’s Second and Third Theorems, [?]) The functor F ↦ C(F, A) induces an equivalence between the category of formal groups over A and the category of reduced Cart(A)-modules.

These powerful results allow one to study formal groups directly in terms of Witt vectors and constructions over them. Alas there is no room here to pursue this study further and unlock the secrets of the world of formal groups, but at least we now have laid the groundwork for such a worthy endeavor.

2.5 Postlude: λ-Rings

In this final section, we introduce the theory of λ-rings, which are rings equipped with certain exterior power operations. λ-rings are the underlying algebraic structure describing symmetric functions [?], and in geometry [?, ?] embody the behavior of Chern classes [?]. The primary algebraic example of a λ-ring is the ring Λ(A), the ‘multiplicative’ form of the Witt vectors. This ring plays a central role in the theory of λ-rings in general, and as such provides a connection between the formal theory of this paper and algebraic topology. Considered as an enhancement of the algebraic structure of 𝕎, this theory provides a fitting conclusion for this survey.

The motivating example for the theory is the K-ring K(X) of a topological space, constructed from formal differences of vector bundles, whose addition corresponds to Whitney sum of bundles, multiplication to tensor product. The exterior product of vector bundles provides an additional operation on K(X) which will be the prototype of a λ-operation. In particular, we have Λ^{1}V = V and

Λ^{n}(U ⊕ V) = ⊕_{i=0}^{n}(Λ^{i}U ⊗ Λ^{n−i}V)

A particularly pleasant example is K(pt) ≅ ℤ since vector bundles over a point are classified by their dimension. In this ring exterior power corresponds to taking binomial coefficients:

Λ^{d}V = \binom{dimV}{d} , V ∈ K(pt)

We axiomatize these properties into the following definition [?]:

Definition 2.15 A pre-λ-ring is a commutative ring A equipped with operations λ^{i} : A → A for i ∈ ℕ satisfying λ^{0}(x) = 1, λ^{1}(x) = x and

λ^{n}(x + y) = ∑_{i=0}^{n} λ^{i}(x)λ^{n−i}(y)

The relation with Λ(A) becomes clear once we write the generating function of the λ^{i}:

λ_{t}(x) = 1 + λ^{1}(x)t + λ^{2}(x)t^{2} + …

and the addition axiom becomes λ_{t}(x + y) = λ_{t}(x)λ_{t}(y). Thus a pre-λ-ring structure on A is simply a homomorphism λ_{t} from the additive group of A to that of Λ(A), with the additional requirement that modulo degree two terms x ↦ 1 + x. Those elements of A for which indeed λ_{t}(x) = 1 + x are of special interest. These are known as degree one elements. In terms of Λ(A), these correspond to our indeterminates 1 + ζ_{i} in terms of which we defined the multiplication and Frobenius on Λ. If an element x ∈ A can be written as a sum of elements of degree 1,

x = x_{1} + ⋯ + x_{q}, λ_{t}(x_{i}) = 1 + x_{i}

then the exterior powers of x become the elementray symmetric functions of the x_{i}:

λ_{t}(x) = 1 + (x_{1} + ⋯ + x_{q})t + (x_{1}x_{2} + ⋯ + x_{q−1}x_{q})t^{2} + …

For such sums of degree one elements λ_{t} is automatically multiplicative, λ_{t}(xy) = λ_{t}(x)λ_{t}(y). We can similarly work out in this case to which symmetric functions in the x_{i} corresponds λ^{m}(λ^{n}(x)). The answer works out to be the coefficient Q_{m,n}(x_{1}, …, x_{mn}) of t^{n} in the product

∏_{1≤i_{1}<···<i_{m}≤q}(1 + x_{i_{1}}x_{i_{2}} ⋯ x_{i_{m}}t)

This motivates the use of these polynomials to define a pre-λ-ring structure on Λ(A) by setting

λ^{i}(1 + a_{1}t + a_{2}t^{2} + …) = 1 + Q_{1,i}(a⃗) + Q_{2,i}(a⃗) + …

It is straightforward to verify (in terms of indeterminates say) that these operations indeed make Λ(A) into a pre-λ-ring.

We may now define a true λ-ring to be one in which these identities hold for every element, not necessarily of degree one - in other words, one in which the voodoo of appealing to the indeterminates x_{i} = ζ_{i} to decompose λ_{t}(x) makes sense:

Definition 2.16 A λ-ring A is a pre-λ-ring for which λ_{t} : A → Λ(A) is a homomorphism of pre-λ-rings. That is, λ_{t}(1) = 1 + t, λ_{t}(xy) = λ_{t}(x) ∗ λ_{t}(y), and

λ^{m}(λ^{n}(x) = Q_{m,n}(λ^{1}x, …, λ^{mn}x)

Using this definition, it is easy to establish that a pre-λ-ring in which λ_{t}(1) = 1 + t, every element is a finite sum of elements of degree one and the product of two elements of degree one is likewise, is indeed a λ-ring [?]. Using this, or even more directly writing elements as infinite formal products of indeterminates ζ_{i} as we have done [?], it is then straightforward to verify that Λ(A) is itself a λ-ring.

The structure of λ-rings is closely related with the structure of the symmetric functions. To make this more explicit we can construct a free λ-ring on one generator, which is the formal sum of infinitely many indeterminates ζ_{i}. This ring is then the free polynomial algebra on the symmetric functions of the ζ_{i}, and one sets the n-th exterior power of the generator ∑ ζ_{i} to be the n-th symmetric function of the ζ_{i}. Any element of this ring is a symmetric function in the ζ_{i}, which is the same as a polynomial in the elementary symmetric functions. Thus any such element defines an operation on any λ-ring by taking x to the corresponding polynomial in the λ^{i}x.

For our purposes the most interesting such operations are the Adams operations, which correspond to the n-th power sums s_{n} = ∑ ζ_{i}^{n}. Thus we define, for an element V ∈ A in a λ-ring Ψ^{n}(V) = s_{n}(V). But we are already familiar with these power-sums: the functions s_{n} are precisely those defining the map s : Λ(A) → 𝔸(𝔸), and thus

−t\frac{d}{dt} log λ_{t}(a) = ∑_{n=1}^{∞} Ψ^{n}(a)t^{n}

It follows immediately that, when characteristic problems don’t interfere, it is practically equivalent on a ring to be given the exterior power operations λ^{i} satisfying λ-axioms, or the Adams operations Ψ^{i}, satisfying the appropriate modified addition and multiplication axioms. In other words, we may express the entire theory of λ-rings in terms of the Ψ-operations. This is an expression of the fact that over ℚ the power sums give an alternative basis for the space of symmetric functions. An additional note of interest for Λ(A) (with A arbitrary) is that the Adams operations Ψ^{n} on it agree with the Frobenii F_{n}. This too is immediate, once we express formally elements in Λ(A) as infinite products of terms 1−ζ_{i}t. We defined the Frobenius as corresponding to the operation ζ_{i} ↦ ζ_{i}^{n}, which transforms our variable from the sum of the indeterminates ζ_{i} to their n-th power sum as desired.

We will not be able, in the confines of this paper, to delve further into the applications of the algebraic structure of λ-rings to K-theory and the study of characteristic classes, developed for the general Grothendieck-Riemann-Roch theorem [?]. The Grothendieck splitting-principle approach to the definition of Chern classes [?] motivates the symmetric-functions-in-indeterminates flavor of λ-ring theory, which in turn facilitates the introduction of gadgets such as the Adams operations. The role of Hirzebruch’s multiplicative sequences [?] in index theory also becomes much clearer when viewed in terms of homomorphisms of λ-type rings. Sequences such as the Todd polynomials and the Hirzebruch L-series are singled out as special for their algebraic properties and their role in bridging different λ-structures (the Todd polynomials help correct the Chern character relating K-theory and cohomology Chern classes, both of which are equipped with natural exterior power operations.) The resulting appearnce of number theoretic quantities such as the Bernoulli numbers in topology is less surprising in view of these connections with formal group theory. The Witt vectors themselves appear directly in Chern class theory in the form of the cohomology ring H^{∗}(BU, ℤ) which turns out to be isomorphic to the bialgebra associated with 𝕎̂ [?]. Another link to formal groups is that assigning to a generalized complex cohomology theory the one-dimensional formal group constructed from the behavior of its Chern classes under tensoring of line bundles. Ordinary cohomology is assigned the additive formal group Ĝ_{a}, K-theory the multiplicative Ĝ_{m}, while the gem of the theory is the theorem that complex cobordism is assigned the universal formal group law F_{U}(X, Y) [?]... but all these scattered observations must await another time to be examined and finally set straight.

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