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title: Preliminaries and Notations.
tags: [research]

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tags: research
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# Preliminaries and Notations.

### Witt ring scheme

**Definition** The Witt ring scheme is a ring scheme with underlying space

$$
W = \mathrm{Spec}\left(\mathbb{Z}[T_1,T_2,\cdots,T_n,\cdots]\right)\cong \mathbb{A}^\infty_{\mathbb{Z}}
$$

the ring scheme structure is obtained from a certain pull back map of natural ring structure on $\mathbb{A}_{\mathbb{Z}}^\infty$, the map is defined by the following (called goast map).

$$
(w_1,\cdots,w_n,\cdots): W\rightarrow \mathbb{A}^\infty_{\mathbb{Z}}
$$

$$
w_n(T_1,\cdots)=T_1^{p^n}+pT_2^{p^{n-1}}+\cdots+p^n T_n
$$

For Witt vectors, We refer a very good reference at [[1]](https://arxiv.org/pdf/1409.7445.pdf)

Shorten the notation by $\underline{X}:=X_1,\cdots, X_n$We denote the addition law by 
$$
S_n(\underline{X},\underline{Y})
$$

and the multiplication law by 

$$
P_n(\underline{X},\underline{Y})
$$

Those rule are made to ensure

$$
w_n(S_n(\underline{X},\underline{Y}))=w_n(\underline X)+w_n(\underline Y)
$$
and
$$
w_n(P_n(\underline{X},\underline{Y}))=w_n(\underline X)w_n(\underline Y)
$$

We usually denote $W_n$ the sub-ringschme with the first n components.

### CoWitt group

The Witt ring schme is try to reconstruct $\mathbb{Z}_p$ from $\mathbb{F}_p$, we can say the coWitt is try to reconstruct $\mathbb{Q}_p/\mathbb{Z}_p$


**Definition** The coWitt ring scheme is a ring scheme with underlying space

$$
CW = \mathrm{Spec}\left(\mathbb{Z}[T_0,T_{-1},T_{-2},\cdots,T_{-n},\cdots]\right)\cong \mathbb{A}^\infty_{\mathbb{Z}}
$$

the ring scheme structure ($\widehat +$)is defined by following

$$
(X_0,X_{-1},\cdots)\widehat{+}(Y_0,Y_{-1},\cdots) = (Z_0,Z_{-1},Z_{-2},\cdots)
$$
with
$$
Z_{-n}=S_n(X_{-n},X_{1-n},\cdots,X_{0},Y_{-n},Y_{1-n},\cdots,Y_0)
$$

One realize the Witt ring scheme is obtained by 
$$
W=\lim_{\rightarrow} W_n
$$
with natural embedding $W_n\rightarrow W_{n+1}$

The coWitt ring scheme can be realized as 

$$
CW=\lim_{\rightarrow} W_n
$$

with respect to the Verchibang map $V: W_n\rightarrow W_{n+1}$.

The witt ring scheme have an action on Witt coring scheme. Just like $\mathbb{Z}_p$ can act on elements in $\mathbb{Q}_p/\mathbb{Z}_p$.


# References

[[1] The Theory of Witt Vectors](https://arxiv.org/pdf/1409.7445.pdf)