## Sheaf of Witt Vectors 2

Recall last time we talked about how we can form the sheaf of Witt vectors over a variety ${X}$ that is defined over an algebraically closed field ${k}$ of characteristic ${p}$. The sections of the structure sheaf form rings and we can take ${W_n}$ of those rings. The functoriality of ${W_n}$ gives us that this is a sheaf that we denote ${\mathcal{W}_n}$. For today we’ll be define ${\Lambda}$ to be ${W(k)}$.

Recall that we also noted that ${H^q(X, \mathcal{W}_n)}$ makes sense and is a ${\Lambda}$-module annihilated by ${p^n\Lambda}$ (recall that we noted that Frobenius followed by the shift operator is the same as multiplying by ${p}$, and since Frobenius is surjective, multiplying by ${p}$ is just replacing the first entry by ${0}$ and shifting, so multiplying by ${p^n}$ is the same as shifting over ${n}$ entries and putting ${0}$‘s in, since the action is component-wise, ${p^n\Lambda}$ is just multiplying by ${0}$ everywhere and hence annihilates the module).

In fact, all of our old operators ${F}$, ${V}$, and ${R}$ still act on ${H^q(X, \mathcal{W}_n)}$. They are easily seen to satisfy the formulas ${F(\lambda w)=F(\lambda)F(w)}$, ${V(\lambda w)=F^{-1}(\lambda)V(w)}$, and ${R(\lambda w)=\lambda R(w)}$ for ${\lambda\in \Lambda}$. Just by using basic cohomological facts we can get a bunch of standard properties of ${H^q(X, \mathcal{W}_n)}$. We won’t write them all down, but the two most interesting (of the very basic) ones are that if ${X}$ is projective then ${H^q(X, \mathcal{W}_n)}$ is a finite ${\Lambda}$-module, and from the short exact sequence we looked at last time ${0\rightarrow \mathcal{O}_X\rightarrow \mathcal{W}_n \rightarrow \mathcal{W}_{n-1}\rightarrow 0}$, we can take the long exact sequence associated to it to get ${\cdots \rightarrow H^q(X, \mathcal{O}_X)\rightarrow H^q(X, \mathcal{W}_n)\rightarrow H^q(X, \mathcal{W}_{n-1})\rightarrow \cdots}$

If you’re like me, you might be interested in studying Calabi-Yau manifolds in positive characteristic. If you’re not like me, then you might just be interested in positive characteristic K3 surfaces, either way these cohomology groups give some very good information as we’ll see later, and for a Calabi-Yau’s (including K3′s) we have ${H^i(X, \mathcal{O}_X)=0}$ for ${i=1, \ldots , n-1}$ where ${n}$ is the dimension of ${X}$. Using this long exact sequence, we can extrapolate that for Calabi-Yau’s we get ${H^i(X, \mathcal{W}_n)=0}$ for all ${n>0}$ and ${i=1, \ldots, n-1}$. In particular, we get that ${H^1(X, \mathcal{W})=0}$ for ${X}$ a K3 surface where we just define ${H^q(X, \mathcal{W})=\lim H^q(X, \mathcal{W}_n)}$ in the usual way.

## Sheaf of Witt Vectors

I was going to go on to prove a bunch of purely algebraic properties of the Witt vectors, but honestly this is probably only interesting to you if you are a pure algebraist. From that point of view, this ring we’ve constructed should be really cool. We already have the ring of ${p}$-adic integers, and clearly ${W_{p^\infty}}$ directly generalizes it. They have some nice ring theoretic properties, especially ${W_{p^\infty}(k)}$ where ${k}$ is a perfect field of characteristic ${p}$.

Unfortunately it would take awhile to go through and prove these things, and it would just be tedious algebra. Let’s actually see why algebraic geometers and number theorists care about the Witt vectors. First, we’ll need a few algebraic facts that we haven’t talked about. For today, we’re going to fix a prime ${p}$ and we have an ${\mathbf{important}}$ notational change: when I write ${W(A)}$ I mean ${W_{p^\infty}(A)}$, which means I’ll also write ${(a_0, a_1, \ldots)}$ when I mean ${(a_{p^0}, a_{p^1}, \ldots)}$ and I’ll write ${W_n(A)}$ when I mean ${W_{p^n}(A)}$. This shouldn’t cause confusion as it is really just a different way of thinking about the same thing, and it is good to get used to since this is the typical way they appear in the literature (on the topics I’ll be discussing).

There is a cool application by thinking about these functors as representable by group schemes or ring schemes, but we’ll delay that for now in order to think about cohomology of varieties in characteristic ${p}$ and hopefully relate it back to de Rham stuff from a month or so ago.

In addition to the fixed ${p}$, we will assume that ${A}$ is a commutative ring with ${1}$ and of characteristic ${p}$.

We have a shift operator ${V: W_n(A)\rightarrow W_{n+1}(A)}$ that is given on elements by ${(a_0, \ldots, a_{n-1})\mapsto (0, a_0, \ldots, a_{n-1})}$. The V stands for Verschiebung which is German for “shift”. Note that this map is additive, but is not a ring map.

We have the restriction map ${R: W_{n+1}(A)\rightarrow W_n(A)}$ given by ${(a_0, \ldots, a_n)\mapsto (a_0, \ldots, a_{n-1})}$. This one is a ring map as was mentioned last time.

Lastly, we have the Frobenius endomorphism ${F: W_n(A)\rightarrow W_n(A)}$ given by ${(a_0, \ldots , a_{n-1})\mapsto (a_0^p, \ldots, a_{n-1}^p)}$. This is also a ring map, but only because of our necessary assumption that ${A}$ is of characteristic ${p}$.

Just by brute force checking on elements we see a few relations between these operations, namely that ${V(x)y=V(x F(R(y)))}$ and ${RVF=FRV=RFV=p}$ the multiplication by ${p}$ map.

Now on to the algebraic geometry part of all of this. Suppose ${X}$ is a variety defined over an algebraically closed field of characteristic ${p}$, say ${k}$. Then we can form the sheaf of Witt vectors on ${X}$ as follows. Notice that all the stalks of the structure sheaf ${\mathcal{O}_x}$ are local rings of characteristic ${p}$, so it makes sense to define the Witt rings ${W_n(\mathcal{O}_x)}$ for any postive ${n}$. Now just form the natural sheaf ${\mathcal{W}_n}$ that has as its stalks ${(\mathcal{W}_{n})_x=W_n(\mathcal{O}_x)}$.

Note that forgetting ring structure and thinking only as a sheaf of sets we have that ${\mathcal{W}_n}$ is just ${\mathcal{O}^n}$, and when ${n=1}$ it is actually isomorphic as a sheaf of rings. For larger ${n}$ the addition and multiplication is defined in that strange way, so we no longer get an isomorphism of rings. Using our earlier operations and the isomorphism for ${n=1}$, we can use the following sequences to extract information.

When ${n\geq m}$ we have the exact sequence ${0\rightarrow \mathcal{W}_m\stackrel{V}{\rightarrow} \mathcal{W}_n\stackrel{R}{\rightarrow}\mathcal{W}_{n-m}\rightarrow 0}$. If we take ${m=1}$, then we get the sequence ${0\rightarrow \mathcal{O}_X\rightarrow \mathcal{W}_n\rightarrow \mathcal{W}_{n-1}\rightarrow 0}$. This will be useful later when trying to convert cohomological facts about ${\mathcal{O}_X}$ to ${\mathcal{W}}$.

We could also define ${H^q(X, \mathcal{W}_n)}$ as sheaf cohomology because we can think of ${\mathcal{W}_n}$ just as a sheaf of abelian groups. Let ${\Lambda=W(k)}$, then since ${\mathcal{W}_n}$ are ${\Lambda}$-modules annihilated by ${p^n\Lambda}$, we get that ${H^q(X, \mathcal{W}_n)}$ are also ${\Lambda}$-modules annihilated by ${p^n\Lambda}$. Next time we’ll talk about some other fundamental properties of the cohomology of these sheaves.

## 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 ${\mathbb{Z}_p}$.

## Formal Witt Vectors

Last time we checked that our explicit construction of the ring of Witt vectors was a ring, but in the proof we noted that ${W}$ actually was a functor ${\mathrm{Ring}\rightarrow\mathrm{Ring}}$. In fact, since it exists and is the unique functor that has the three properties we listed, we could have just defined the ring of Witt vectors over ${A}$ to be ${W(A)}$.

We also said that ${W}$ was representable, and this is just because ${W(A)=Hom(\mathbb{Z}[x_1, x_2, \ldots ], A)}$. We can use our ${\Sigma_i}$ to define a (co)commutative Hopf algebra structure on ${\mathbb{Z}[x_1, x_2, \ldots]}$.

For instance, define the comultiplication ${\mathbb{Z}[x_1, x_2, \ldots ]\rightarrow \mathbb{Z}[x_1,x_2,\ldots]\otimes \mathbb{Z}[x_1,x_2,\ldots]}$ by ${x_i\mapsto \Sigma_i(x_1\otimes 1, \ldots , x_i\otimes 1, 1\otimes x_1, \ldots 1\otimes x_i)}$.

Since this is a Hopf algebra we get that ${W=\mathrm{Spec}(\mathbb{Z}[x_1,x_2,\ldots])}$ is an affine group scheme. The ${A}$-valued points on this group scheme are by construction the elements of ${W(A)}$. In some sense we have this “universal” group scheme keeping track of all of the rings of Witt vectors.

Another thing we could notice is that ${\Sigma_1(X,Y)}$, ${\Sigma_2(X,Y)}$, ${\ldots}$ are polynomials and hence power series. If we go through the tedious (yet straightfoward since it is just Witt addition) details of checking, we will find that they satisfy all the axioms of being an infinite-dimensional formal group law. We will write this formal group law as ${\widehat{W}(X,Y)}$ and ${\widehat{W}}$ as the associated formal group.

Next time we’ll start thinking about the length ${n}$ formal group law of Witt vectors (truncated Witt vectors).

## Witt Vectors Form a Ring

Today we’ll check that the ring of Witt vectors is actually a ring. Let ${A}$ be a ring, then ${W(A)}$ as a set is the collection of infinite sequences of ${A}$. Recall that our construction involves lots of various polynomials and a strange definition addition and multiplication. I won’t rewrite those, since it was the entirety of the last post.

Now there is a nice trick to prove that ${W(A)}$ is a ring when ${A}$ is a ${\mathbb{Q}}$-algebra. Just define ${\psi: W(A)\rightarrow A^\mathbb{N}}$ by ${(a_1, a_2, \ldots) \mapsto (w_1(a), w_2(a), \ldots)}$. This is a bijection and the addition and multiplication is taken to component-wise addition and multiplication, so since this is the standard ring structure we know ${W(A)}$ is a ring. Also, ${w(0,0,\ldots)=(0,0,\ldots)}$, so ${(0,0,\ldots)}$ is the additive identity, ${W(1,0,0,\ldots)=(1,1,1,\ldots)}$ which shows ${(1,0,0,\ldots)}$ is the multiplicative identity, and ${w(\iota_1(a), \iota_2(a), \ldots)=(-a_1, -a_2, \ldots)}$, so we see ${(\iota_1(a), \iota_2(a), \ldots)}$ is the additive inverse.

We can actually get this idea to work for any characteristic ${0}$ ring by considering the embedding ${A\rightarrow A\otimes\mathbb{Q}}$. We have an induced injective map ${W(A)\rightarrow W(A\otimes\mathbb{Q})}$. The addition and multiplication is defined by polynomials over ${\mathbb{Z}}$, so these operations are preserved upon tensoring with ${\mathbb{Q}}$. We just proved above that ${W(A\otimes\mathbb{Q})}$ is a ring, so since ${(0,0,\ldots)\mapsto (0,0,\ldots)}$ and ${(1,0,0,\ldots)\mapsto (1,0,0,\ldots)}$ and the map preserves inverses we get that the image of the embedding ${W(A)\rightarrow W(A\otimes \mathbb{Q})}$ is a subring and hence ${W(A)}$ is a ring.

Lastly, we need to prove this for positive characteristic rings. Choose a characteristic ${0}$ ring that surjects onto ${A}$, say ${B\rightarrow A}$. Then since the induced map again preserves everything and ${W(B)\rightarrow W(A)}$ is surjective, the image is a ring and hence ${W(A)}$ is a ring.

So where does all this formal group stuff we started with come into play? Well, notice that what we were really implicitly using is that ${W:\mathbf{Ring}\rightarrow\mathbf{Ring}}$ is a functor. It takes a ring ${A}$ and gives a new ring ${W(A)}$. If ${\phi: A\rightarrow B}$ is a ring map, then ${W(\phi): W(A)\rightarrow W(B)}$ by ${(a_1, a_2, \ldots)\mapsto (\phi(a_1), \phi(a_2), \ldots)}$ is still a ring map. We also have ${w_n:W(A)\rightarrow A}$ by ${a\mapsto w_n(a)}$ are ring maps for all ${n}$.

Some people think it is cleaner to define the ring of Witt vectors as the unique functor ${W}$ that satisfies these three properties. From a functorial point of view it turns out that ${W}$ is representable. The representing ring via the ring axioms gives a Hopf algebra structure, and hence we get an affine group scheme out of it. Then as in the formal group discussion, we can complete this to get a formal group. This will be the discussion of next time.

## Witt Vectors 2

Today we’re going to accomplish the original goal we set out for ourselves. We will construct the ring of generalized Witt vectors. First, let ${x_1, x_2, \ldots }$ be a collection of indeterminates. We can define an infinite collection of polynomials in ${\mathbb{Z}[x_1, x_2, \ldots ]}$ using the following formulas:
${w_1(X)=x_1}$

${w_2(X)=x_1^2+2x_2}$

${w_3(X)=x_1^3+3x_3}$

${w_4(X)=x_1^4+2x_2^2+4x_4}$

and in general ${\displaystyle w_n(X)=\sum_{d|n} dx_n^{n/d}}$.

Now let ${\phi(z_1, z_2)\in\mathbb{Z}[z_1, z_2]}$. This just an arbitrary two variable polynomial with coefficients in ${\mathbb{Z}}$.

We can define new polynomials ${\Phi_i(x_1, \ldots x_i, y_1, \ldots y_i)}$ such that the following condition is met ${\phi(w_n(x_1, \ldots ,x_n), w_n(y_1, \ldots , y_n))=w_n(\Phi_1(x_1, y_1), \ldots , \Phi_n(x_1, \ldots , y_1, \ldots ))}$.

In short we’ll notate this ${\phi(w_n(X),w_n(Y))=w_n(\Phi(X,Y))}$. The first thing we need to do is make sure that such polynomials exist. Now it isn’t hard to check that the ${x_i}$ can be written as a ${\mathbb{Q}}$-linear combination of the ${w_n}$ just by some linear algebra.

${x_1=w_1}$, and ${x_2=\frac{1}{2}w_2+\frac{1}{2}w_1^2}$, etc. so we can plug these in to get the existence of such polynomials with coefficients in ${\mathbb{Q}}$. It is a fairly tedious lemma to prove that the coefficients ${\Phi_i}$ are actually in ${\mathbb{Z}}$, so we won’t detract from the construction right now to prove it.

Define yet another set of polynomials ${\Sigma_i}$, ${\Pi_i}$ and ${\iota_i}$ by the following properties:

${w_n(\Sigma)=w_n(X)+w_n(Y)}$, ${w_n(\Pi)=w_n(X)w_n(Y)}$ and ${w_n(\iota)=-w_n(X)}$.

We now can construct ${W(A)}$, the ring of generalized Witt vectors over ${A}$. Define ${W(A)}$ to be the set of all infinite sequences ${(a_1, a_2, \ldots)}$ with entries in ${A}$. Then we define addition and multiplication by ${(a_1, a_2, \ldots, )+(b_1, b_2, \ldots)=(\Sigma_1(a_1,b_1), \Sigma_2(a_1,a_2,b_1,b_2), \ldots)}$ and ${(a_1, a_2, \ldots )\cdot (b_1, b_2, \ldots )=(\Pi_1(a_1,b_1), \Pi_2(a_1,a_2,b_1,b_2), \ldots )}$.

Next time we’ll actually check that this is a ring for any ${A}$. To show you that this isn’t as horrifyingly strange and arbitrary as it looks, it turns out that all these rules boil down to just the ${p}$-adic integers when ${A}$ is the finite field of order ${p}$, i.e. ${W(\mathbb{F}_p)=\mathbb{Z}_p}$. It also turns out that there is a much cleaner construction of this than element-wise if all you care about are the existence and certain properties.

## Formal Groups 4

Let’s back up from all these definitions for now and see one situation in which all of these things just pop out. It also will show us where these names came from. Suppose ${G}$ is a smooth algebraic affine group scheme over ${k}$. Then ${G}$ is represented by a finitely generated Hopf algebra, say ${A}$. If ${I}$ is the augmentation ideal (i.e. ${I=\text{ker}(\epsilon^*: A\rightarrow k)}$), then by smoothness we get that ${I/I^2}$ is free on the generators ${x_1, \ldots, x_r}$ where ${r}$ is the number of generators of ${A}$ over ${k}$.

Likewise ${I^n/I^{n+1}}$ is free and generated by monomials ${x_1^{m_1}\cdots x_r^{m_r}}$, thus if we take the completion with respect to the ${I}$-adic topology, we get ${\widehat{A}=\lim (A/I^n)\simeq k[[x_1, \ldots x_r]]}$. Now notice that ${\Delta (I)\subset I\otimes A+A\otimes I}$, and this is just the maximal ideal defining ${(e,e)}$ in the product. Thus the maps pass to the successive quotients, and we get an induced map on completions.

This is just ${\widehat{\Delta}:k[[x_1, \ldots , x_r]] \rightarrow k[[x_1', \ldots, x_r', x_1'', \ldots , x_r'']]}$. But any such map is completely described by where each of the ${x_i}$ get sent to. But they are sent to power series in ${2r}$ variables! So we get ${r}$ power series in ${2r}$ variables, namely ${\widehat{\Delta}(x_i)=F_i(X,Y)}$. Notice these came from maps of Hopf algebras (not merely ring maps, otherwise this wouldn’t work), so tracing the coassociativity axiom we get precisely the associativity we need to say that these power series form a formal group law of dimension ${r}$.

Even more importantly, we could have chosen different generators originally, and this would have changed the construction and given a different formal group law, but they are isomorphic. The isomorphism is just given by a change of variables. So given any smooth affine algebraic group scheme over ${k}$ we get a unique formal group up to isomorphism, and moreover (given appropriate other technical conditions we won’t discuss) there is actually an (anti)equivalence of categories between formal groups and Hopf algebras.

This is exactly how the formal group laws ${\widehat{\mathbb{G}}_a}$ and ${\widehat{\mathbb{G}}_m}$ were formed. We take the affine algebraic group scheme ${\mathbb{G}_a}$ which is represented by ${k[z]}$, then ${\Delta(z)=z\otimes 1+1\otimes z}$. The completion is clearly ${k[[z]]\rightarrow k[[x,y]]}$ and the map ${\widehat{\Delta}(z)=x+y}$. Thus our one-dimensional additive formal group (scheme) is appropriately named, and it is just as easy to go through and check the multiplicative one gives the right law as well.

Now we see that these definitions of formal groups aren’t just arbitrary isolated things. They actually arise in practice. We will also see another important way in which they appear next time (or the time after) in defining new rings with particularly nice properties.

Lastly, we need to do one more thing that doesn’t really fit anywhere nicely. We started with one-dimensional formal groups, and then moved to arbitrary finite dimension, but really we can just keep going and define infinite dimensional in exactly the same way. If we have an index set ${I}$, then take a collection of indeterminates ${(x_i)}$ indexed by ${i}$. Define the formal power series ring ${A[[x_i]]}$ to be all formal sums ${\sum c_nx^n}$ where ${n}$ runs through functions ${n:I\rightarrow \mathbb{N}\cup \{0\}}$ with finite support, and ${x^n=\prod x_i^{n(i)}}$.

Now an infinite dimensional formal group law is a collection of elements ${F_i(X, Y)\in A[[X_i, Y_i]]_{i\in I}}$ which we write ${\displaystyle F_i(X,Y)=\sum_{m,n} c_{m,n}(i)X^mY^n}$, and the usual conditions ${F_i(X,Y)=X_i+Y_i+}$(higher degree) and ${F_i(F(X,Y),Z)=F_i(X, F(Y,Z))}$ and we need one extra finiteness condition that ${c_{m,n}(i)\neq 0}$ for only finitely many ${i}$.

This may seem implausible to appear in practice, but there is a really simple example where you get this occuring. Take any ring ${A}$, then we can form a new ring ${A[X_i, Y_i]}$ an infinite-dimensional polynomial algebra over ${A}$. Then take just a single variable power series over this to get ${A[X_i, Y_i][[t]]}$. Taking any two arbitrary elements with constant coefficient ${1}$, we can multiply them: ${(1+\sum X_it^i)(1+\sum Y_it^i)=1+\sum F_i(X,Y)t^i}$. We know the product has this form, and the ${F_i(X,Y)}$ form an infinite dimensional formal group law.

Reversing this process, we can define a multiplication given an infinite dimensional group law, and this is how we’ll define our new rings, including the Witt vectors.

## Formal Groups 3

Today we move on to higher dimensional formal group laws over a ring ${A}$. Side note: later on we’ll care about the formal group attached to a Calabi-Yau variety in positive characteristic which is always one-dimensional, but to talk about Witt vectors we’ll need the higher dimensional ones.
An ${n}$-dimensional formal group law over ${A}$ is just an ${n}$-tuple of power series, each of ${2n}$ variables and no constant term, satisfying certain relations. We’ll write ${F(X,Y)=(F_1(X,Y), F_2(X,Y), \ldots , F_n(X,Y))}$ where ${X=(x_1, \ldots , x_n)}$ and ${Y=(y_1, \ldots , y_n)}$ to simplify notation.

There are a few natural guesses for the conditions, but the ones we actually use are that ${F_i(X,Y)=x_i+y_i+}$(higher degree) and for all ${i}$ ${F_i(F(X,Y),Z)=F_i(X, F(Y,Z))}$. We call the group law commutative if ${F_i(X,Y)=F_i(Y,X)}$ for all ${i}$.

We still have our old examples that are fairly trivial ${\widehat{\mathbb{G}}_a^n(X,Y)=X+Y}$, meaning the i-th one is just ${x_i+y_i}$. For a slightly less trivial example, let’s explicitly write a four-dimensional one

${F_1=x_1+y_1+x_1y_1+x_2y_3}$

${F_2=x_2+y_2+x_1y_2+x_2y_4}$

${F_3=x_3+y_3+x_3y_1+x_4y_3}$

${F_4=x_4+y_4+x_3y_2+x_4y_4}$

It would beastly to fully check even one of those associative conditions. I should probably bring this to your attention, but the condition ${F_i(F(X,Y), Z)}$ has you input as the first four variables those four equations, so in this case checking the first condition amounts to ${F_1(F_1(X,Y),F_2(X,Y),F_3(X,Y),F_4(X,Y),z_1,z_2,z_3,z_4)}$

${=F_1(x_1,x_2,x_3,x_4, F_1(Y,Z),F_2(Y,Z),F_3(Y,Z),F_4(Y,Z))}$.

But on the other hand, the fact that ${\widehat{\mathbb{G}}_a^n}$ satisfies it is trivial since you are only adding everywhere which is associative.

Now we can do basically everything we did with the one-dimensional case now. For one thing we can form the set ${\mathcal{C}(F)}$ of ${n}$-tuples of power series in one indeterminant and no constant term. This set has an honest group structure on it given by ${\gamma_1(t)+_F \gamma_2(t)=F(\gamma_1(t), \gamma_2(t))}$.

We can define a homomorphism between an ${n}$-dimensional group law ${F}$ and ${m}$-dimensional group law ${G}$ to be an ${m}$-tuple of power series in ${n}$-indeterminants ${\alpha(X)}$ with no constant term and satisfying ${\alpha(F(X,Y))=G(\alpha(X), \alpha(Y))}$.

From here we still have the same inductively defined endomorphisms for any ${n}$ (not just the dimension) from the one-dimensional case ${[0]_F(X)=0}$, ${[1]_F(X)=X}$ and ${[n]_F(X)=F(X, [n-1]_F(X))}$. That’s a lot of information to absorb, so we’ll end here for today.

## Formal Groups 2

We have a lot to cover and could spend forever just on one-dimensional formal groups, so I think I’ll have to force myself to end our discussion on these today and next time move on to higher dimensional formal groups. Last time we defined a formal group law on a ring ${A}$ to be a power series (with no constant term) satisfying associativity ${F\in A[[x,y]]}$.

These things should form a category, so what is a morphism between two of them? Suppose ${F, G}$ are two formal group laws over ${A}$. Then a homomorphism is defined to be a power series (with no constant term) ${\alpha(x)=b_1x+b_2x^2+\cdots}$ such that ${\alpha(F(x,y))=G(\alpha(x),\alpha(y))}$.

Recall that we made an honest group out of the formal group laws. This condition on ${\alpha}$ is just the condition that makes the map ${\mathcal{C}(F)\rightarrow \mathcal{C}(G)}$ an honest group homomorphism. Using the general categorical yoga we want an isomorphism to be a homomorphism ${\alpha: F\rightarrow G}$ such that there is another morphism ${\beta: G\rightarrow F}$ with the property that ${\alpha(\beta(x))=\beta(\alpha(x))=x}$ the composition is the identity.

It turns out that ${\alpha: F(x,y)\rightarrow G(x,y)}$ is an isomorphism if and only if ${\alpha(x)=b_1x+b_2x^2+\cdots}$ has the property that ${b_1}$ is a unit.

I’ve said it in the past, we’re mostly concerned with positive characteristic, but just a quick note is that over ${\mathbb{Q}}$ our earlier examples ${\widehat{\mathbb{G}}_m}$ and ${\widehat{\mathbb{G}}_a}$ are isomrophic using the standard power series representations of the exponential map and log maps. But if we take ${k}$ to be a field of positive characteristic, then it turns out that they are not isomorphic.

Let’s prove this. We can inductively define a power series using a formal group law ${F}$ for any ${n}$ called ${[n]_F(x)}$ by defining ${[0]_F(x)=0}$, ${[1]_F(x)=x}$, and ${[n]_F(x)=F(x, [n-1]_F(x))}$. For instance, ${[n]_{\widehat{\mathbb{G}}_a}(x)=nx}$ by an easy check. Also, ${[n]_{\widehat{\mathbb{G}}_m}(x)=(1+x)^n-1}$ which is not any harder, but maybe takes a little more to check.

Now let’s suppose that ${\widehat{\mathbb{G}}_a}$ and ${\widehat{\mathbb{G}}_m}$ are isomorphic over our field ${k}$. Then there is some power series ${\alpha(x)=b_1x+b_2x^2+\cdots}$ with ${b_1\neq 0}$ that gives the isomorphism. But these ${[n]}$ commute with the homomorphism by construction, so ${0=[p](\alpha(x))=\alpha([p](x))=\alpha(x^p)}$ (with the subscripts suppressed). This is a contradiction. Thus ${\widehat{\mathbb{G}}_a}$ is not isomorphic to ${\widehat{\mathbb{G}}_m}$ over a field of characteristic ${p}$. This might be a little surprising, since they are over ${\mathbb{Q}}$.

Recall in the last post that we produced a non-commutative formal group law. The last thing we’ll discuss today is that in the one-dimensional case we actually have rather nice necessary and sufficient conditions to know when we can have non-commutative formal group laws.

Here is the theorem: Every one-dimensional formal group law is commutative over ${A}$ if and only if ${A}$ has no non-zero elements that are simultaneously nilpotent and torsion (i.e. ${na=a^m=0}$ for some positive integers ${n,m}$). Recall we had to form a torsion nilpotent ${k[\epsilon]/(\epsilon^2)}$ by making ${p\epsilon=\epsilon^2=0}$. The proof of this statement is rather unenlightening involving huge manipulations of formal power series, so we won’t do it, but by exactly the same construction as last post we can always use that torsion nilpotent, say ${c}$ to make a non-commutative formal group law ${F(x,y)=x+y+cxy^q}$, where ${q}$ is chosen to make everything work.

Next time we’ll up the dimension. I probably skipped something we’ll need in the future, so we may return to this topic (I also feel bad for not pointing out why these formal group laws are called what they are, but if you care enough you’ve probably already figured out the connection with completions of affine group schemes).

## Formal Groups 1

Today we’ll start our long journey on the definition of the ring of Witt vectors. Our first step will be to think about formal group laws, since this will be useful to us for things I have planned later on (the formal groups attached to varieties in positive characteristic).

Let ${A}$ be commutative ring with ${1}$. Then a one-dimensional formal group law over ${A}$ is just a formal power series in two variables ${F(x,y)\in A[[x,y]]}$ of the form ${F(x,y)=x+y+\sum c_{i,j}x^iy^j}$ that satisfies “associativity”. This means that ${F(x, F(y,z))=F(F(x,y),z)}$.

A key thing is that the power series has no constant term. We call the ring ${A}$ equipped with this ${F}$ a (one-dimensional) formal group. This terminology makes some sense considering it is sort of giving a group operation on ${A}$, but since these are just formal series, we may not have convergence when plugging actual elements of ${A}$ in.

The formal group is called commutative if ${F(x,y)=F(y,x)}$.

Here are two easy examples to see that this is really quite concrete. The additive formal group is using ${F(x,y)=x+y}$ and the multiplicative formal group uses ${F(x,y)=x+y+xy}$. They are both commutative. We’ll suggestively write ${\widehat{\mathbb{G}}_a(x,y)=x+y}$ and ${\widehat{\mathbb{G}}_m(x,y)=x+y+xy}$.

Let’s not shy away from positive characteristic, since that will be our main usage of formal groups in the future. We have a nice non-commutative formal group on ${A=k[\epsilon]/(\epsilon^2)}$ where ${\text{char}(k)=p>0}$ given by ${f(x,y)=x+y+\epsilon xy^p}$.

Now we can get honest groups out of our formal groups. Suppose ${F}$ is a formal group law on ${A}$, then we can consider power series in one variable with no constant term. We take this just as a set, so given ${\gamma_1(t), \gamma_2(t)\in tA[t]}$, it makes sense to do ${F(\gamma_1(t), \gamma_2(t))}$, so we’ll define ${\gamma_1(t)+_F \gamma_2(t)=F(\gamma_1(t), \gamma_2(t))}$. This turns our set into an honest group which we denote ${\mathcal{C}(F)}$.

But why are there inverses? We need some sort of lemma that says: given any formal group law ${F}$ over ${A}$ there is a power series ${i(x)=-x+b_2x^2+\cdots}$ such that ${F(x,i(x))=0}$. This is just a special case of what is known as the “Formal Implicit Function Theorem”.

With an eye towards our goal of the Witt vectors ${W(A)}$, we’ll just say here that ${\mathcal{C}(\widehat{\mathbb{G}_m})}$ is the underlying additive group of the ring of Witt vectors over ${A}$.