Other forms of Witt vectors

Today we’ll discuss two other flavors of the ring of Witt vectors, which have some pretty neat applications to computing Cartier duals of group schemes. The ring we’ve constructed, {W(A)}, is sometimes called the ring of generalized Witt vectors. You can construct a similar ring associated to a prime, {p}.

Recall that the functor {W} was the unique functor {\mathrm{Ring}\rightarrow\mathrm{Ring}} that satisfies {W(A)=\{(a_1, a_2, \ldots): a_j\in A\}} as a set and for {\phi:A\rightarrow B} a ring map we get {W(\phi)(a_1, a_2, \ldots )=(\phi(a_1), \phi(a_2), \ldots)} and the previously defined {w_n: W(A)\rightarrow A} is a functorial homomorphism.

We can similarly define the Witt vectors over {A} associated to a prime {p} as follows. Define {W_{p^\infty}} to be the unique functor {\mathrm{Ring}\rightarrow \mathrm{Ring}} satisfying the following properties: {W_{p^\infty}(A)=\{(a_0, a_1, \ldots): a_j\in A\}} and {W_{p^\infty}(\phi)(a_0, a_1, \ldots )=(\phi(a_0), \phi(a_1), \ldots)} for any ring map {\phi: A\rightarrow B}. Now let {w_{p^n}(a_0, a_1, \ldots)=a_0^{p^n}+pa_1^{p^{n-1}}+\cdots + p^na_n}, then {W_{p^\infty}} also has to satisfy the property that {w_{p^n}: W(A)\rightarrow A} is a functorial homomorphism.

Basically we can think of {W_{p^\infty}} as the generalized Witt vectors where we’ve relabelled so that our indexing is actually {(a_{p^0}, a_{p^1}, a_{p^2}, \ldots)} in which case the {w_n} are the {w_{p^n}}. There is a much more precise way to relate these using the Artin-Hasse map and the natural transformation {\epsilon_p: W(-)\rightarrow W_{p^\infty}(-)} which maps {\epsilon_p (a_1, a_2, \ldots)\mapsto (a_{p^0}, a_{p^1}, a_{p^2}, \ldots)}.

Notice that when we defined {W(A)} using those formulas (and hence also {W_{p^\infty}(A)}) the definition of adding, multiplying, and additive inverse were defined for the first {t} components using only polynomials involving the first {t} components.

Define {W_t(A)} to be the set of length {t} “vectors” with elements in {A}, i.e. the set {\{(a_0, a_1, \ldots, a_{t-1}): a_j\in A\}}. The same definitions for multiplying and adding the generalized Witt vectors are well-defined and turns this set into a ring for the same exact reason. We also get for free that the truncation map {W(A)\rightarrow W_t(A)} by {(a_0, a_1, \ldots)\mapsto (a_0, a_1, \ldots, a_{t-1})} is a ring homomorphism.

For instance, we just get that {W_1(A)\simeq A}. These form an obvious inverse system {W_n(A)\rightarrow W_m(A)} by projection when {m|n} and we get that {W(A)\simeq \lim W_t(A)} and that {W_{p^\infty}(A)\simeq \lim W_{p^t}(A)}.

Today we’ll end with a sketch of a proof that {W_{p^\infty}(\mathbb{F}_p)\simeq \mathbb{Z}_p}. Most of these steps are quite non-trivial, but after next time when we talk about valuations, we’ll be able to prove much better results and this will fall out as a consequence of one of them.

Consider the one-dimensional formal group law over {\mathbb{Z}} defined by {F(x,y)=f^{-1}(f(x)+f(y))} where {f(x)=x+p^{-1}x^p+p^{-2}x^{p^2}+\cdots}. Then for {\gamma(t)\in \mathcal{C}(F; \mathbb{Z}_p)} (the honest group of power series with no constant term defined from the group law considered on {\mathbb{Z}_p}), there is a special subcollection {\mathcal{C}_p(F; \mathbb{Z}_p)} called the {p}-typical curves, which just means that {\mathbf{f}_q\gamma(t)=0} for {q\neq p} where {\mathbf{f}_q} is the frobenius operator.

Now one can define a bijection {E:\mathbb{Z}_p^{\mathbb{N}\cup \{0\}}\rightarrow \mathcal{C}_p(F;\mathbb{Z}_p)}. This can be written explicitly by {(a_0, a_1, \ldots)\mapsto \sum a_it^{p^i}} and moreover we get {w_{p^n}^FE=w_{p^n}} where {w_{p^n}^F(\gamma(t))=p^n}(coefficient of {t^{p^n}} in {f(\gamma(t))}). Now we put a commutative ring structure on {\mathcal{C}_p(F;\mathbb{Z}_p)} compatible with the already existing group structure and having unit element {\gamma_0(t)=t}.

There is a ring map {\Delta: \mathbb{Z}_p\rightarrow \mathcal{C}_p(F; \mathbb{Z}_p)} defined by {\Delta(a)=f^{-1}(af(t))}. Also, the canonical projection {\mathbb{Z}_p\rightarrow \mathbb{F}_p} induces a map {\rho: \mathcal{C}_p(F;\mathbb{Z}_p)\rightarrow \mathcal{C}_p(F; \mathbb{F}_p)}. It turns out you can check that the compostion {\rho\circ \Delta} is an isomorphism, which in turn gives the isomorphism {\mathbb{Z}_p\stackrel{\sim}{\rightarrow} W_{p^\infty}(\mathbb{F}_p)}.

Likewise, we can also show that {W_{p^\infty}(\mathbb{F}_p^n)} is the unique unramified degree {n} extension of {\mathbb{Z}_p}.

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