$\mathbb{Z}_p$, $\mathbb{Q}_p$, and the ring of Witt vectors

No.12.extra:

PROPOSITION 12.1   Let $a,b\in A$. Assume $n,m \in\mathbb{Z}_{>0}$ such that $\gcd(n,m)=d$, $\operatorname{lcm}(n,m)=l$. Then:

1. $(1- T^n)_W(1- T^m)_W= d \cdot (1- T^{l})_W$
2. $(1-a T^n)_W(1- b T^m)_W= (1-a^{l/n} b^{l/m} T^{l})^{d}$

PROOF.. We will firstly prove the proposition when $A=\mathbb{C}$. unity in $\mathbb{C}$. (1) Let $\zeta_n$ be a primitive root of unity in $\mathbb{C}$. Then we have:

$$(1- T^n)_W(1- T^m)_W = \sum_{k=0}^{n-1} (1-\zeta_n^k T)_W (1-T^m)_W = \sum_{k=0}^{n-1} (1-\zeta_n^{km} T^m)_W .$$

Knowing that $\zeta_n^m$ is a primitive $n'$-th root of unity, we get the desited result.

(2)

$$(1-a T^n)_W(1- b T^m)_W$$

$$= (1-a^{1/n}T)_W(1-T^n)_W\cdot(1- b^{1/m}T)_W(1- T^{l})_W.$$

By functoriality, we see that the proposition is also valid over the polynomial ring $\mathbb{Z}[a,b]$. Then by functoriality we see that the result is also true for any ring $A$.

$\qedsymbol$

LEMMA 12.2 (=Proosition 8.9)   Let $n$ be a positive integer. If $n$ is invertible in $A$, then it is also invertible in $\Lambda(A)$.

PROOF.. Let us define $\alpha_1=(1-\frac{1}{n} T)$. Then we have

$$\alpha_1^n=(1-\frac{1}{n}T)^n=(1-T) \pmod {T^2}.$$

Let us now assume that for a postive integer $k$, we have an polynomial $\alpha_k$ such that

$$\alpha_k^n=(1-T) \pmod {T^{k+1}}$$

holds. Then there exists an element $c_k \in A$ such that

$$\alpha_k^n=(1-T)+c_{k} T^{k+1} \pmod {T^{k+2}}.$$

Let us put $\alpha_{k+1}=\alpha_k-\frac{1}{n} c_{k} T^{k+1}$.

$$\alpha_{k+1}^n\equiv \alpha_k^n- c_{k} T^{k+1} \equiv 1 \pmod{T^{k+2}}.$$

The statement now follows by the induction.

$\qedsymbol$

PROPOSITION 12.3   Let $n$ be a positive integer which is invertible in $A$. The range $e_n \Lambda(R)$ of the idempotent $e_n$ is isomorphic to $(1+T^n A[[T^n]])$ via $V_n$

Ring of Witt vectors

Let $A$ be a ring. Then

$$A^{\mathbb{Z}_{>0}}\ni (a_1,a_2,\dots) \mapsto \sum_{j=1}^\infty (1-a_j T^j)_W \in \Lambda(A)$$

is a bijection. In other words, $\{a_j\}$ plays the role of a coordinate of $\Lambda(A)$. We call the ring $\Lambda(A)$ with the coordinate given this way the ring of Witt vectors. In this lecture, we do not distinguish too much between $W(A)$ and $\Lambda(A)$.

Verschiebung and Frobenius map.

DEFINITION 12.4   We define:

1. Verschiebung. $V_n: \Lambda(A)\ni (f(T))_W \mapsto (f(T^n))_W\in \Lambda(A)$
2. Frobenius map. $F_n: (1-a T )_W\mapsto (1-a^n T)_W$

PROPOSITION 12.5   Let $A$ be a ring. Let $n$ be a positive integer such that it is invertible in $A$.Then $e_n=\frac{1}{n}(1-T^n)_W$ is an idempotent in $\Lambda(A)$. $e_n \Lambda(A)$ is equal to the image $\operatorname{Image}(V_n)$ of the Verschiebung map. In other words, it is isomorphic to $\Lambda(A)$ itself via the non-unital isomorphism $V_n$.