A Mind for Madness

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Other forms of Witt vectors

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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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Author: hilbertthm90

I write about math, philosophy, literature, music, science, computer science, gaming or whatever strikes my fancy that day.

One thought on “Other forms of Witt vectors

  1. Pingback: A Quick User’s Guide to Dieudonn√© Modules of p-Divisible Groups | A Mind for Madness

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