# Witt vectors and maps of $\lambda$-rings

Consider the ring $W(\mathbb{F}_p)$ of big Witt vectors of $\mathbb{F}_p$. This has a natural structure of a $\lambda$-ring (in the strong sense) since rings of big Witt vectors always do.

$\mathbb{Z}_p$ also has the structure of a $\lambda$-ring, given by the maps $\lambda^n(x)= \binom{x}{n}$.

There is a canonical quotient map (of rings) $W(\mathbb{F}_p) \to \mathbb{Z}_p$ from the big Witt vectors to the classical $p$-adic Witt vectors. Is this a $\lambda$-ring homomorphism? More generally, when (if ever) do we get a $\lambda$-ring structure on the smaller $p$-adic Witt vectors?

I know this quotient map admits an additive section. We could potentially use this to define $\lambda$ operators on the ring of $p$-adic Witt vectors, but it is not clear to me that this would actually give the structure of a $\lambda$-ring. And in my particular case, it is unclear if this would agree with the $\lambda$-ring structure I already have.

I think the answer is No'' for $p$ odd. Stronger yet, there exists no $\lambda$-ring structure on $\mathbb{Z}_{p}$ under which the canonical quotient map $W\left( \mathbb{F}_{p}\right) \rightarrow\mathbb{Z}_{p}$ becomes a $\lambda$-ring morphism. However, due to the manifold occasions for mistakes and misunderstandings in my argument, please check it thoroughly.

Let $p$ be an odd prime. Let $\Lambda$ be the Hopf algebra of symmetric functions over $\mathbb{Z}$ (in the indeterminates $x_{1},x_{2},x_{3},\ldots$). It is known that $\Lambda$ carries further structures, such as plethysm, inner multiplication and inner comultiplication. I am using the material of Michiel Hazewinkel's Witt vectors, part 1 here (although not always his notations: my $\Lambda$ is his $\mathbf{Symm}$, and my $x_{1},x_{2} ,x_{3},\ldots$ are his $\xi_{1},\xi_{2},\xi_{3},\ldots$).

Fix a commutative ring $\mathbf{k}$. The $\lambda$-ring of big Witt vectors $W\left( \mathbf{k}\right)$ is defined as the set $\operatorname*{Ring} \left( \Lambda,\mathbf{k}\right)$ of all ring homomorphisms from $\Lambda\rightarrow\mathbf{k}$, endowed with an addition that comes from the comultiplication of $\Lambda$, an additive inverse that comes from the antipode of $\Lambda$, a multiplication that comes from the inner comultiplication of $\Lambda$, and a $\lambda$-ring structure that comes from the plethysm of $\Lambda$.

We let $p_{1},p_{2},p_{3},\ldots$ be the power sum symmetric functions in $\Lambda$ (that is, $p_{n}=\sum_{i\geq1}x_{i}^{n}$). I am sorry for using the letter $p$ which already stands for a prime, but this is a standard (and the $p$ in $p_{1},p_{2},p_{3},\ldots$'' always comes with a subscript).

We let $w_{1},w_{2},w_{3},\ldots\in\Lambda$ be the Witt vector coordinate'' symmetric functions in $\Lambda$; these are recursively defined by the power-series equality

$\prod_{d\geq1}\left( 1-w_{d}t^{d}\right) ^{-1}=\prod_{i\geq1}\dfrac {1}{1-x_{i}t}\in\mathbb{Z}\left[ \left[ x_{1},x_{2},x_{3},\ldots\right] \right] \left[ \left[ t\right] \right]$.

(These $w_{1},w_{2},w_{3},\ldots$ are called $x_{1},x_{2},x_{3},\ldots$ in §9.63 of Hazewinkel's notes. For a quick overview of their properties, see Exercise 2.79 in Victor Reiner and YHS, Hopf Algebras in Combinatorics, arXiv:1409.8356v3.) Let $\Lambda^{\prime}$ be the subring of $\Lambda$ generated by $w_{p^{0}},w_{p^{1}},w_{p^{2}},\ldots$. Then, the restriction map $\operatorname*{Ring}\left( \Lambda,\mathbf{k}\right) \rightarrow \operatorname*{Ring}\left( \Lambda^{\prime},\mathbf{k}\right)$ is (equivalent to) the canonical quotient map $W\left( \mathbf{k}\right) \rightarrow W_{p}\left( \mathbf{k}\right)$ from the big Witt vectors $W\left( \mathbf{k}\right)$ to the $p$-typical Witt vectors $W_{p}\left( \mathbf{k}\right)$.

It is well-known that $p_{n}=\sum_{d\mid n}dw_{d}^{n/d}$ for every positive integer $n$. In particular, $p_{p^{k}}=\sum_{i=0}^{k}p^{i}w_{p^{i}}^{p^{k-i}}$ for every $k\in\mathbb{N}$. From this, it is easy to see that for every $k\in\mathbb{N}$,

(1) there exists a positive integer $N$ such that $Nw_{p^{k}}$ is a polynomial in $p_{p^{0}},p_{p^{1}},\ldots,p_{p^{k}}$ with no constant term.

We now define two ring homomorphisms $\alpha:\Lambda\rightarrow\mathbf{k}$ and $\beta:\Lambda\rightarrow\mathbf{k}$ by setting

$\alpha\left( f\right) =f\left( 1,0,0,0,\ldots\right)$ and $\beta\left( f\right) =f\left( -1,0,0,0,\ldots\right)$ for every $f\in\Lambda$

(where the $1$ and the $-1$ are taken in $\mathbf{k}$). (Incidentally, $\alpha$ is the internal counit of $\Lambda$.)

Now, $\alpha+\beta$ is the ring homomorphism $\Lambda\rightarrow\mathbf{k}$ satisfying

$\left( \alpha+\beta\right) \left( f\right) =f\left( 1,-1,0,0,0,\ldots \right)$ for every $f\in\Lambda$.

Thus, it is easy to see that $\left( \alpha+\beta\right) \left( p_{i}\right) =0$ for every odd $i\geq1$. In particular, $\left( \alpha +\beta\right) \left( p_{p^{i}}\right) =0$ for every $i\in\mathbb{N}$. Thus, from (1) we see that $\left( \alpha+\beta\right) \left( w_{p^{k} }\right) =0$ for every $k\in\mathbb{N}$ as long as all positive integers are invertible in $\mathbf{k}$. But we can remove the all positive integers are invertible in $\mathbf{k}$'' condition (because proving $\left( \alpha+\beta\right) \left( w_{p^{k}}\right) =0$ for $\mathbf{k}=\mathbb{Z}$ is sufficient to ensure that $\left( \alpha +\beta\right) \left( w_{p^{k}}\right) =0$ holds for every $\mathbf{k}$). Hence, we obtain $\left( \alpha+\beta\right) \left( w_{p^{k}}\right) =0$ for every $k\in\mathbb{N}$. In other words, the canonical quotient map $W\left( \mathbf{k}\right) \rightarrow W_{p}\left( \mathbf{k}\right)$ sends $\alpha+\beta$ to the zero of the ring $W_{p}\left( \mathbf{k}\right)$.

On the other hand, recall that the $\lambda$-ring structure on $W\left( \mathbf{k}\right)$ comes from plethysm on $\Lambda$. More precisely,

$\left( \lambda^{n}\gamma\right) \left( f\right) =\gamma\left( e_{n}\left[ f\right] \right)$ for every $n\geq1$, $\gamma\in W\left( \mathbf{k}\right)$ and $f\in\Lambda$,

where $e_{n}\left[ f\right]$ denotes the plethysm of $f$ into $e_{n}$ (denoted by $e_{n}\circ f$ in Hazewinkel's notes). Applying this to $n=2$, $\gamma=\alpha+\beta$ and $f=p_{1}$, we obtain

$\left( \lambda^{2}\left( \alpha+\beta\right) \right) \left( w_{1}\right) =\left( \alpha+\beta\right) \left( \underbrace{e_{2}\left[ w_{1}\right] }_{=e_{2}}\right) =\left( \alpha+\beta\right) \left( e_{2}\right)$

$=e_{2}\left( 1,-1,0,0,0,\ldots\right) =-1$.

Assume now that $\mathbf{k}\neq0$. Thus, $\left( \lambda^{2}\left( \alpha+\beta\right) \right) \left( w_{1}\right) =-1\neq0$. Hence, the canonical quotient map $W\left( \mathbf{k}\right) \rightarrow W_{p}\left( \mathbf{k}\right)$ sends $\lambda^{2}\left( \alpha+\beta\right)$ to a nonzero element of the ring $W_{p}\left( \mathbf{k}\right)$.

Thus, the canonical quotient map $W\left( \mathbf{k}\right) \rightarrow W_{p}\left( \mathbf{k}\right)$ sends $\alpha+\beta$ to zero, but $\lambda^{2}\left( \alpha+\beta\right)$ to a nonzero element. Therefore, there exists no $\lambda$-ring structure on $W_{p}\left( \mathbf{k}\right)$ under which this canonical quotient map $W\left( \mathbf{k}\right) \rightarrow W_{p}\left( \mathbf{k}\right)$ becomes a $\lambda$-ring morphism. Now, applying this to $\mathbf{k}=\mathbb{F}_{p}$ and recalling that $W_{p}\left( \mathbb{F}_{p}\right) \cong\mathbb{Z}_{p}$, we see that there exists no $\lambda$-ring structure on $\mathbb{Z}_{p}$ under which the canonical quotient map $W\left( \mathbb{F}_{p}\right) \rightarrow \mathbb{Z}_{p}$ becomes a $\lambda$-ring morphism. As I said, this is all under the assumption that $p$ is odd, but I would not expect even $p$ to behave much different.

• Hi Darij. In fact, your original statement is true for all p, including 2. The big Witt ring W(F_p) can be expressed as a product of copies of Z_p where the index set is the set of integers n>1 and prime to p, the projection to Z_p being the projection onto the first component. Further the n-th Adams operation takes the n-th component to the first one. s if Z_p has a lambda-structure, then it has an – JBorger Oct 2 '15 at 18:26
• n-th Adams operator in particular. If the projection from W(F_p) is a lambda-ring map, then it has to commute with the n-th Adams operator. Looking at the effect on the idempotent describing the n-th component, you see the n-th Adams operator on Z_p would have to send 1 to 0 (if n>1!), which is impossible. – JBorger Oct 2 '15 at 18:30
• @JBorger: Nice, and thanks for reminding me of the direct-product description! (Though I think your n>1 should be $n\geq 1$, and you mean "0 to 1" instead of "1 to 0".) – darij grinberg Oct 2 '15 at 19:33
• Interesting. Follow-up question: Suppose I have a map of rings from a $\lambda$-ring $R$ to $\mathbb{F}_p$, under what circumstances does it lift to a map of $\lambda$-rings from $R$ to $\mathbb{Z}_p$? For example if $R$ is $\mathbb{Z}$ and I take the quotient mod $p$ map to $\mathbb{F}_p$ it lifts to the inclusion of $\mathbb{Z}$ in $\mathbb{Z}_p$ which is a map of $\lambda$-rings, what is it about this example that gives me this property? – Nate Oct 2 '15 at 20:45
• A map from such an $R$ to $\mathbb{F}_p$ lifts to a unique $\lambda$-map from $R$ to $W(\mathbb{F}_p)$. Since you can identify $\mathbb{Z}_p$ with the subring of $W(\mathbb{F}_p)$ on which all prime-to-$p$ Adams operators are trivial, the $\lambda$-ring map $R\to W(\mathbb{F}_p)$ has image in $\mathbb{Z}_p$ if and only if the map $f:R\to\mathbb{F}_p$ satisfies $f(\psi_n(r))=f(r)$ for all $n$ prime to $p$. For example, in the case $R=\mathbb{Z}$, all Adams operations are the identity; so it does indeed work there. – JBorger Oct 4 '15 at 19:20