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, , is sometimes called the ring of *generalized* Witt vectors. You can construct a similar ring associated to a prime, .

Recall that the functor was the unique functor that satisfies as a set and for a ring map we get and the previously defined is a functorial homomorphism.

We can similarly define the Witt vectors over associated to a prime as follows. Define to be the unique functor satisfying the following properties: and for any ring map . Now let , then also has to satisfy the property that is a functorial homomorphism.

Basically we can think of as the generalized Witt vectors where we’ve relabelled so that our indexing is actually in which case the are the . There is a much more precise way to relate these using the Artin-Hasse map and the natural transformation which maps .

Notice that when we defined using those formulas (and hence also ) the definition of adding, multiplying, and additive inverse were defined for the first components using only polynomials involving the first components.

Define to be the set of length “vectors” with elements in , i.e. the set . 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 by is a ring homomorphism.

For instance, we just get that . These form an obvious inverse system by projection when and we get that and that .

Today we’ll end with a sketch of a proof that . 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 defined by where . Then for (the honest group of power series with no constant term defined from the group law considered on ), there is a special subcollection called the -typical curves, which just means that for where is the frobenius operator.

Now one can define a bijection . This can be written explicitly by and moreover we get where (coefficient of in ). Now we put a commutative ring structure on compatible with the already existing group structure and having unit element .

There is a ring map defined by . Also, the canonical projection induces a map . It turns out you can check that the compostion is an isomorphism, which in turn gives the isomorphism .

Likewise, we can also show that is the unique unramified degree extension of .

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