Witt coordinates vs Joyal coordinates on the ring of Witt vectors

Question

I am learning about Witt vectors following [K, Ch. 3], but I am having trouble with the presentation of his material (see (Q1) below).

Kedlaya's definition for ring $W(A)$ of the Witt vectors on a commutative unital ring $A$ is the image of $A$ under the right adjoint of the forgetful functor from $\delta$-rings to rings ($\delta$-rings are defined in [K, 2.1.1]). This right adjoint exists by formal reasons. From this definition, it follows that: \begin{align} W(A)&=\operatorname{Hom}_{\mathbf{Ring}}(\mathbb{Z}[y], W(A)) \\ &=\operatorname{Hom}_{\mathbf{Ring}_\delta}(\mathbb{Z}\{y\}, W(A))\\ &=\operatorname{Hom}_{\mathbf{Ring}}\left(\mathbb{Z}\left[y_0, y_1, \ldots\right], A\right) \\ &=A \times A \times \cdots\tag{1}\label{joy} \end{align} Here $\mathbb{Z}\{y\}$ denotes the free $\delta$-ring on one variable [K, 2.4.5]. We define the $y$-coordinates or Joyal coordinates of an element of $W(A)$ as its image $(y_0,y_1,\dots)\in A\times A\times\cdots$ under the isomorphism \eqref{joy} [K, 3.1.1]. Reading off the isomorphism \eqref{joy}, the $y$-coordinates are given by the map $(\varepsilon\circ\delta^n)_{n\geq 0}:W(A)\to A\times A\times\cdots$ (here $\varepsilon:W(A)\to A$ is the counit). We have used that the map $\mathbb{Z}\{y\}\to W(A)$ associated to an element of $W(A)$ is a morphism of $\delta$-rings. Since \eqref{joy} is a bijection, the map $(\varepsilon\circ\delta^n)_{n\geq 0}$ is a bijection. Since \eqref{joy} is a natural bijection, the $y$-coordinates are natural in the following sense: Let $f:A\to B$ be a ring morphism. If $v\in W(A)$ is written in $y$-coordinates as $(y_0,y_1,\dots)$, then $W(f)(v)$ expressed in Joyal coordinates of $W(B)$ is $(f(y_0),f(y_1),\dots)$. In other words, $(\varepsilon\circ\delta^n)_{n\geq 0}$ is an isomorphism of functors $\mathbf{Ring}\to\mathbf{Set}$.

According to [K, 3.1.1], there is another set of coordinates $(x_0,x_1,\dots)$ with each $x_n\in A$ called the $x$-coordinates or Witt coordinates. My question is:

(Q1). How are the $x$-coordinates defined?

Surprisingly, it happens that Kedlaya never defines the $x$-coordinates anywhere in [K, Ch. 3]. I am aware these are just the coordinates in which other sources typically define $W(A)$ as the cartesian power $A^\mathbb{N}$ equipped with an exotic addition and multiplication [K, 3.2.1, 3.2.2]. But I don't want to go that way, instead I would like to stick to Kedlaya's way to see if I can understand his approach. To achieve the definition of the $x$-coordinates, I think one requires the Lemma below. Before stating it, I will introduce some notations.

Let $R$ be a ring. Given elements $r_i\in R$, $i\in I$, we denote $[r_i\mid i\in I]$ to the non-unital subring generated by $r_i$, $i\in I$. That is, $[r_i\mid i\in I]$ is the $\mathbb{Z}$-submodule generated by products of the $r_i$'s. Given non-unital subrings $S,T\subset R$, we write $ST=[st\mid s\in S,t\in T]$.

With notations of [K, 3.1.1], denote $S_0=S_1=0$ the trivial non-unital subring of $\mathbb{Z}\{y\}$ and for $n\geq 2$, denote $$ S_n=[y_1,\dots,y_{n-1}](\mathbb{Z}[y_0,\dots,y_{n-1}])=[y_1y_0^i,\dots,y_{n-1}y_0^i\mid i\geq 0]. $$ Then $S_n\subset S_{n+1}$ for $n\geq 0$.

[K, Lemma 3.1.3]. With notation as in [K, 3.1.1], in the ring $\mathbb{Z}\{y\}$ there exist unique elements \begin{equation} \label{eq:x_n-y_n_in_Y_n} x_n\in y_n+S_n \quad(n=0,1, \ldots) \end{equation} such that \begin{equation} \label{eq:phi_n_x_n} \phi^n\left(x_0\right)=x_0^{p^n}+p x_1^{p^{n-1}}+\cdots+p^n x_n \quad(n=0,1, \ldots). \end{equation}

(Where $\phi$ is the Frobenius lift of $\mathbb{Z}\{y\}$ [K, 2.1.3].) I introduced two refinements to the original statement:

1. I added the strengthening unique for the $x_n$'s. Indeed, on the one hand, from the statement, necessarily $x_0=y_0$ and $x_1=y_1$. On the other hand, if $x_0,\dots,x_{n-1}$ are the unique elements satisfying the required conditions, then one sees that the $x_n$ obtained in the proof of [K, Lemma 3.1.3] is uniquely determined by the $x_0,\dots,x_{n-1}$ thanks to $\mathbb{Z}\{y\}$ being $p$-torsion free.
2. I changed Kedlaya's “$(y_1,\dots,y_{n-1})\mathbb{Z}[y_0,\dots,y_{n-1}]$” to “$[y_1,\dots,y_{n-1}](\mathbb{Z}[y_0,\dots,y_{n-1}])$.” This is actually what happens in the proof if one checks it carefully.

After this Lemma, my trouble begins. I get lost at the proof of [K, 3.1.5]. This result asserts that $W(F)=\phi$ for the Frobenius map $F:A\to A$ of a characteristic $p$ ring $A$, i.e., $W(F):W(A)\to W(A)$ is the Frobenius lift associated to the $\delta$-structure on $W(A)$. In the proof of [K, 3.1.5], it is said:

For a general ring $A$, each of the elements $x_0, x_1, \cdots \in \mathbb{Z}\{x\}$ defines a function $W(A) \rightarrow A$ which is natural in $A$. Similarly, every element of $\mathbb{Z}\{x\}$ can be viewed as a "polynomial function" on $W(A)$ valued in $A$ which is again natural in $A$; that is, we have a map of sets $h: \mathbb{Z}\{x\} \rightarrow \operatorname{Hom}_{\mathbf{Set}}(W(A),A)$.

My other question is:

(Q2). What is $h$?

Assuming we knew what $h$ is, I came up with the following:

Tentative Definition. The $x$-coordinates of an element $v\in W(A)$ are $(h(x_n)(v))_{n\geq 0}$.

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References

[K] K. S. Kedlaya, Notes on Prismatic Cohomology https://kskedlaya.org/prismatic/frontmatter-1.html

Answer 1

The coordinates $y_n$ correspond to the operations $\delta^n$ (iterates of $\delta$), whereas the coordinates $x_n$ correspond to the operations $\delta_n$, defined by

$$ \phi^n(f) = \sum_{k=0}^n p^k \delta_k(f)^{p^{n-k}}. $$

To relate these, it might be easier to introduce a third operation $\theta_n$, defined by

$$ \phi(f^{p^{n-1}}) = f^{p^n} + p^n \theta_n(f)$$

so e.g. $\theta_1=\delta$, and $\phi(f^p)=f^{p^2}+p^2\theta_2(f)$. (A reference for the existence of these operations is Manam - On the existence of the Drinfeld formal group, Appendix B, although they've been known for much much longer than that.) Then

\begin{align*} \require{cancel} \phi^2(f) &= \phi(f^p) + p\delta(f)^\phi\\ &= f^{p^2} + p\theta_1(f)^\phi + p^2 \theta_2(f)\\ \cancel{f^{p^2}} + p\delta_1(f)^p + p^2\delta_2(f) &= \cancel{f^{p^2}} + p\theta_1(f)^\phi + p^2\theta_2(f)\\ \delta_2(f) &= \delta_1(\delta_1(f)) + \theta_2(f) \end{align*} so we get $$ \delta_2 = \delta^2 + \theta_2. $$ You can work out $\theta_2$ in terms of just $\delta_\bullet$ or just $\delta^\bullet$ but it'll be nasty.

Answer 2

I think I can answer (Q1) and (Q2). I say ‘think’ because I don't know what Kedlaya actually has on mind when speaking of $x$-coordinates and of the map $h$. The following definition proposal for the $x$-coordinates is coherent with Yuri Sulyma's answer.

Let $A$ be a $\delta$-ring. For $a\in A$, we write $\tilde{a}:\mathbb{Z}\{y\}\to A$ for the unique $\delta$-ring morphism sending $y_0$ to $a$. For each $t\in\mathbb{Z}\{y\}$, define a map of sets \begin{align*} H_A(t):A&\to A\\ a&\mapsto \tilde{a}(t) \end{align*} This map is natural, i.e, $H_B(t)\circ f=f\circ H_A(t)$ for a morphism $f:A\to B$ of $\delta$-rings. This follows from the fact that $$ \tag{1}\label{tilde} \widetilde{f(a)}=f\circ\tilde{a}, $$ as one may check by evaluating these expressions at $y_0$.

For any set $S$, we have that $\operatorname{Hom}_\mathbf{Set}(S,A)$ is a $\delta$-ring with pointwise addition and multiplication and pointwise evaluation of the $\delta$ map. Define a map $H_A:\mathbb{Z}\{y\}\to\operatorname{Hom}_\mathbf{Set}(A,A)$ sending $t\in\mathbb{Z}\{y\}$ to $H_A(t)$. Then $H_A$ is a $\delta$-ring morphism (use that $\tilde{a}$ is a $\delta$-ring morphism).

Now, given a ring $A$, if $\varepsilon:W(A)\to A$ is the counit, then we define $$ h=\varepsilon_*\circ H_{W(A)}:\mathbb{Z}\{y\}\to\operatorname{Hom}_\mathbf{Set}(W(A),A), $$ which is a ring homomorphism (since $\varepsilon_*$ is a ring homomorphism). For each $t\in\mathbb{Z}\{y\}$, the map $h(t)$ is natural in $A$, i.e., $$ \tag{2}\label{nat} f\circ h(t)=h(t)\circ W(f) $$ for a ring morphism $f:A\to B$ (use naturality of $\varepsilon$ and of $H(t)$).

The $x$-coordinates are then defined to be \begin{align*} X=(h(x_n))_{n\geq 0}:W(A)&\to A^\mathbb{N}\\ v&\mapsto((\varepsilon\circ \tilde{v})(x_n))_{n\geq 0}\tag{3}\label{xcoor} \end{align*} These coordinates are natural since $h(x_n)$ is natural. Note that the $y$-coordinates coincide with the map given by changing $x_n$ by $y_n$ in \eqref{xcoor}. We shall show that $X$ is a bijection.

Suppose that $v,v'\in W(A)$ have $x$-coordinates $(a_n)_{n\geq 0}$ and $(a_n')_{n\geq 0}$, resp., and that $(a_n)=(a_n')$. Let $(b_n)$, $(b_n')$ be the $y$-coordinates of $v,v'$, resp. Showing $v=v'$ amounts to showing that $(b_n)=(b_n')$. We show this by induction on $n$. For $n=0,1$ this is true (since the $x$ and $y$-coordinates coincide in degrees $n=0,1$). Assume $n\geq 2$ and that $b_k=b_{k}'$ for all $k< n$. Since $x_n-y_n$ is a polynomial in $y_0,\dots,y_{n-1}$, say $x_n-y_n=g_n(y_0,\dots,y_{n-1})$, we have that $a_n-b_n=g_n(b_0,\dots,b_{n-1})=g_n(b_0',\dots,b_{n-1}')=a_n'-b_n'$. Since $a_n=a_n'$, it follows that $b_n=b_n'$. This shows that \eqref{xcoor} is injective. Now let $(a_n)\in A^\mathbb{N}$. We shall find $v\in W(A)$ whose $x$-coordinates are given by $(a_n)$. We will describe $v$ by means of its $y$-coordinates. Define $b_0=a_0$, $b_1=a_1$. Suppose $n\geq 2$ and that we are given $b_0,\dots,b_{n-1}\in A$. Then we define $b_n=a_n-g_n(b_0,\dots,b_{n-1})$ (where $g_n=x_n-y_n\in\mathbb{Z}[y_0,\dots,y_{n-1}]$ is the same as before). By induction, we construct $(b_n)\in A^\mathbb{N}$ such that $b_n=a_n-g_n(b_0,\dots,b_{n-1})$ for all $n\geq 0$. Let $v\in W(A)$ be the Witt vector with $y$-coordinates $(b_n)$. Then its $x$-coordinates are $(a_n)$. Thus \eqref{xcoor} is onto.

Even though we already know that the $x$-coordinates are natural from the work done above, one could deduce this from the knowledge that the $y$-coordinates are natural: Let $f:A\to A'$ be a morphism of rings. Let $v\in W(A)$. Let $(a_n)$, $(a_n')$ (resp., $(b_n)$, $(b_n')$) respectively be the $x$-coordinates (resp., the $y$-coordinates) of $v$ and of $W(f)(v)$. Then we know $a_0'=f(a_0)$, $a_1'=f(a_1)$ (since $x_0=y_0$, $x_1=y_1$ and by naturality of the $y$-coordinates). Assume $n\geq 2$ and that $a_k'=f(a_k)$ for all $k<n$. Then \begin{gather} a'_n-b'_n=g_n(b'_0,\dots,b'_{n-1})=g_n(f(b_0),\dots,f(b_{n-1}))\\=f(g_n(b_0,\dots,b_{n-1}))=f(a_n-b_n)=f(a_n)-f(b_n)=f(a_n)-b_n', \end{gather} whence $a_n'=f(a_n)$.

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Finally, even though this was not part of the original question, I will provide a proof of the fact that for a ring $A$ of characteristic $p$, denoting $F:A\to A$ to the Frobenius homomorphism, we have that $W(F)=\phi$ is the Frobenius lift of $W(A)$ (this is more or less what [K, 3.1.5] states¹).

First, we show that $$ \label{h_Frob}\tag{4} h\circ\phi=F_*\circ h, $$ where $\phi$ is the Frobenius lift of $\mathbb{Z}\{y\}$. For $t\in\mathbb{Z}\{y\}$, \begin{align*} h(\phi(t)) &=\varepsilon_*(H(\phi(t)))\\ &=\varepsilon_*(\phi(H(t))) &\text{since }H\text{ is a }\delta\text{-ring morphism}\\ &=\varepsilon_*(H(t)^p+p\delta(H(t)))\\ &=\varepsilon_*(H(t))^p+p\varepsilon(\delta(H(t))) &\text{since }\varepsilon_*\text{ is a ring homomorphism}\\ &=\varepsilon_*(H(t))^p &\text{for }\operatorname{char}A=p\\ &=h(t)^p\\ &=F_*(h(t)), \end{align*} as desired. Showing that $W(F)=\phi$ amounts to showing that $Y_n\circ W(F)=Y_n\circ\phi$ for every $n\geq 0$, where $Y_n=h(y_n)$ is the $n$-th $y$-coordinate function. More generally, we will show that $T\circ W(F)=T\circ\phi$, where $T=h(t)$ and $t\in\mathbb{Z}\{y\}$. On the one hand, we have: \begin{align*} T\circ W(F) &=F\circ T &\text{by \eqref{nat}}\\ &=(F_*\circ h)(t)\\ &=(h\circ\phi)(t) &\text{by \eqref{h_Frob}.} \end{align*} Thus, for $v\in W(A)$, \begin{align*} (T\circ W(F))(v) &=h(\phi(t))(v)\\ &=\varepsilon[\tilde{v}(\phi(t))]\\ &=\varepsilon[\phi(\tilde{v}(t))]\\ &=\varepsilon[\widetilde{\phi(v)}(t)] &\text{by \eqref{tilde}}\\ &=h(t)(\phi(v))\\ &=(T\circ\phi)(v). \end{align*}

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¹In the proof of [K, 3.1.5], there seems to be a typo. There it is claimed that $$ h(\phi(t))(u)=\phi(h(t)(u)),\quad t\in\mathbb{Z}\{y\},\;u\in W(A), $$ for an arbitrary ring $A$. This expression does not have meaning, since $\phi$ is not defined in $A$, which is just a ring. I think what Kedlaya meant is that in the case of $A$ being of characteristic $p$, then $$ h(\phi(t))(u)=F(h(t)(u)),\quad t\in\mathbb{Z}\{y\},\;u\in W(A), $$ where $F:A\to A$ is the Frobenius homomorphism. That is, \eqref{h_Frob} holds.