# 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.

The formula for $w_n(X)$ should be $w_n(X)=\sum\limits_{d|n}dx_{d}^{n/d}$ (with a $d$ in the subscript of $x$ rather than $n$.