TITLE: Ghost components of a Witt vector - Motivation
QUESTION [14 upvotes]: I'd like to know if anyone has a good explanation for where the ghost components that are used to define Witt vectors come from.  A lot of sources I've read take the ghost components for their starting point.
The closest I've seen to an explanation was in Joe Rabinoff's notes on Witt vectors.  This is my paraphrasing what I understood from them.  Consider $p$-typical Witt vectors and let $k$ be a perfect field (or ring) of characteristic $p$.  We have a multiplicative Teichmüller map $[\ ]: k \rightarrow W(k)$ which is a section to the projection map.  Every element of $W(k)$ can be uniquely written as a series $$\sum_{i=0}^{\infty} p^i[a_i],$$ with $a_i \in k$.  If we know how to add witt vectors, then we will know how to multiply them as well, because the Teichmüller map is multiplicative.  
The series expression for $[a_0] + [b_0]$ is already not obvious.  Write $\sum p^i [c_i]$ for this expression.  We have $c_0 = a_0 + b_0$, but to find $c_1$ we need the following trick.  Because our field is perfect, there is no harm in replacing $a_0$ and $b_0$ by $\alpha^p$ and $\beta^p$.  Then we want to compute $[\alpha^p] + [\beta^p] = [\alpha]^p + [\beta]^p.$  We know $[\alpha] + [\beta] \equiv [\alpha + \beta] \mod p$.  Raising both sides to the $p$th power, we can find a new expression for $[\alpha]^p + [\beta]^p \mod p^2$, and this gives us $c_1$.  
The formulas that are used to find $c_1$, $c_2$, etc. look similar to the ghost polynomials.  [They're a little different because the Witt vector corresponding to a series as above is $(c_0, c_1^p, c_2^{p^2}, \ldots)$, and so we haven't found the usual Witt vector components.  On the other hand, the usual ghost components don't involve any $p$th roots, which is what $\alpha$ and $\beta$ are.  Finally, the ghost map tells us not just how to add $[a_0]$ and $[b_0]$, but how to add any two series $\sum p^i[a_i]$ and $\sum p^i[b_i]$.]
I'm willing to believe that we can find precisely the usual ghost components this way (although I haven't actually succeeded in doing this).  So it seems natural that the ghost map $w: W(k) \rightarrow k^{\mathbb{N}}$ (where the ring operations on $k^{\mathbb{N}}$ are componentwise) is an additive map.  What I find surprising is that it's also multiplicative; it seems like we didn't do anything to guarantee this. 
Am I correct that this is one way to find the ghost components?  Is this the most natural path to them?  Should it be obvious that the ghost map is multiplicative?
I'd be happy to hear how you think of the ghost components.

REPLY [3 votes]: I might probably add some hint to the above motivations. I'm trying to provide some relations, using the exponential map on power series, that allow to see the definition of the ghost map as natural.
It is actually easier to begin by motivating the "big" Witt vectors $\mathbb{W}$.
Let $A$ be a commutative ring with unit element. Everything starts with   $\Lambda(A) = 1+ t A[[t]]$. It is a group under multiplication of power series.
Now, let $f(t)$ be a power series in $\Lambda(A)$. For every $f\in\Lambda(A)$ there are (at least) two ways to write it

$f(t) = 1 + \sum_{n\geq 1} b_n t^n$ with $b_n\in A$

$f(t) = \prod_{n=1}^{\infty}(1-a_nt^n)^{-1}$, with $a_n\in A$


The first sequence $(b_n)_n$ is often what we want to really understand.
The sequence $(a_n)_n$ is the big Witt vector associated to $f$. Denote by $\mathbb{W}(A)$ the family of sequence $(a_n)_n$ as above.
Now, if $A$ is a $\mathbb{Q}$-algebra there is a third way to write such a power series

$f(t) = \exp(\sum_{n\geq 1}\phi_n\frac{t^n}{n})$

The sequence $(\phi_n)_n$ is the ghost vector associated to the big Witt vector $(a_n)_n$. Calculating $\frac{d}{dt}(\log(f(t))$ provides the relation
$$ \phi_n\;=\;\sum_{d|n}d\cdot a_d^{n/d}\;. $$
When $A$ is a $\mathbb{Q}$-algebra, the multiplication rule of $\Lambda(A)$ induces a multiplication rule on the sequences $(a_n)_n$ which is the sum in $\mathbb{W}(A)$. One proves that this addition law is defined by polynomials with coefficients in $\mathbb{Z}$ and therefore it defines a scheme $\mathbb{W}$, and hence a group law on the Witt vectors $\mathbb{W}(A)$ for every ring $A$.
The fact that every power series in $\Lambda(A)$ can be written as in 1. and 2. translates the idea that for all ring $A$ one has $\mathbb{W}(A)\cong\Lambda(A)$.
These relations can be a starting point to justify the definition of the ghost map. When $n=p^m$, you can recognize the expression
$$\phi_{p^m}\;=\;\sum_{d|p^m}d\cdot a_d^{p^m/d}\;=\;\sum_{k=0}^m p^k\cdot a_{p^k}^{p^{m-k}}$$
which define the $p$-typical ghost map associated with the $p$-typical Witt vector $(a_1,a_{p},a_{p^2},\ldots)$. This expression shows that the $p$-typical Witt vectors $W(A)$ is a sub-group of $\mathbb{W}(A)$ but unfortunately it is not a sub-ring. The $p$-typical Witt vectors $W(A)$ is actually a quotient of the ring $\mathbb{W}(A)$, and the ring $\mathbb{W}(A)$ can be expressed as product of copies of the ring $W(A)$. It is hence relatively complicated to describe it easily as above. Allow me to quote the ADDENDUM 15 of the nice survey https://www.math.nagoya-u.ac.jp/~larsh/papers/s03/wittsurvey.pdf by Lars Hesselholt, which provide an exact relation between $\mathbb{W}(A)$ and $W(A)$ (seen as $\mathbb{W}_{S}(A)$, with $S=\{1,p,p^2,p^3,\ldots\}$ using the notations of Hesselholt's paper). Unfortunately, it does not mention explicitly the relations with the exponentials. More informations using the exponentials can be found in the Exercices of Bourbaki Alg. Comm. Chapter 9.
However, let me try to sketch out a similar approach using Artin-Hasse exponential at the place of exponential (these are my personal computations since I'm unable to find references). Again, from now on $A$ is a $\mathbb{Q}$-algebra (as for the global Witt vectors, the rationality of the coefficients has to be treated in a second moment). Let us rewrite every series $f(t)\in\Lambda(A)$ as follows. First we observe that $\exp(-\log(1-a_nt^n)) = (1-a_nt^n)^{-1}$ and then 2. can be rewritten as

$f(t) \;=\; \prod_{n\geq 1} \exp(-\log(1-a_nt^n))\;=\;\prod_{n\geq 1}\exp(\sum_{k\geq 1}\frac{(a_nt^{n})^k}{k})\;=\;\prod_{n\geq 1}\varepsilon(a_nt^n)$

where
$$\varepsilon(t) \;=\; \exp(-\log(1-t)) \;=\;\exp(\sum_{n\geq 1}\frac{t^n}{n})\;=\; \sum_{m\geq 0}t^m\;.$$
If we consider only the terms that are power of $p$, we obtain the Artin-Hasse exponential
$$E(t)\;=\;\exp(\sum_{k\geq 0}\frac{t^{p^k}}{p^k})\;=\;\exp(t+\frac{t^p}{p}+\frac{t^{p^2}}{p^2}+\cdots)$$
We may then start from the expression 3. and consider the following re-summation

$f(t)=\exp(\sum_{(n,p)=1}\sum_{m\geq 0}\phi_{np^m}\frac{t^{np^m}}{np^m})\; =\; \prod_{(n,p)=1}\exp(\sum_{m\geq 0}\phi_{np^m}\frac{t^{np^m}}{p^m})^{\frac{1}{n}}\;.$

For $n=1$, the coefficients $(\phi_{p^m})_{m\geq 0}$ are the phantom components of the $p$-typical Witt vector $(a_{p^m})_{m\geq 0}$ and it is easy to prove the relation
$$\exp(\sum_{m\geq 0}\phi_{p^m}\frac{t^{p^m}}{p^m})\;=\; \prod_{m\geq 0} E(a_{p^m}t^{p^m})$$
which is reminiscent of the expression $f(t)=\prod_{n\geq 1}\varepsilon(a_nt^n)$ in item 4. On the other hand, we have a similar relation for evert $(n,p)=1$.