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.

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