$\Lambda (A)$

DEFINITION 05.1   For any commutative ring $A$,

1. we define
   $$\Lambda(A)=(1+T A[[T]]) \qquad ($$as a set$$)$$
   For any $f \in (1+T A[[T]])$, we denote by $(f)_W$ the corresponding element in $\Lambda (A)$.
2. For any $(f)_W$ , $(g)_W \in \Lambda(A)$, we define their sum by
   $$(f)_W+(g)_W=(fg)_W$$

It is easy to see that $\Lambda (A)$ is an additive group. It also carries the “$T$-addic topology” so that $\Lambda (A)$ is a topological additive group.

The next task is to define multiplicative structure on $\Lambda (A)$. To that end, we do something somewhat different to others.

DEFINITION 05.2   For any commutative ring $A$, we define $E(A)=\operatorname{End}_{{\operatorname{additive}}}(\Lambda(A))$. It has the usual structure of a ring. For any $a\in A$, we define its “Teichmüler” lift $[a]$ as

$$(f(T))_W\mapsto (f(a T))_W.$$

The basic idea is to define $E_0(A)$ as the subalgebra of $E(A)$ topologically generated by all the Teichm"uller lifts $\{[a]; a \in A\}$ and identify $E_0(A)$ with $\Lambda (A)$. To avoid some difficulties doing so, we first do this when $A$ is a very good one:

PROPOSITION 05.3   Assume $A=\Omega$, an algebraically closed field. Then:

1. $\Lambda (A)$ is generated by $\{(1-aT)_W\vert a \in A\}$ as a topological additive group.
2. The subring $E_0(A)$ of $E(A)$ generated by $\{[a]\vert a \in A\}$ as a topological ring is equal to $\{x\in E(A); x$ commutes with all Teichmüller lifts$\}$.
3. $(1-T)_W$ is a generating separating vector of $\Lambda (A)$ over $E_0(A)$. Thus we have a module isomorphism
   $$E_0(A) \ni \varphi \to \varphi((1-T)_W) \in \Lambda(A).$$
   (Note that This isomorphism sends $[a]$ to $(1-aT)_W$.

We may thus identify $E_0(A)$ and $\Lambda (A)$ via this isomorphism and employ a ring structure on $\Lambda (A)$.

Here after, for any algebraically closed field $A$, we employ the ring structure of $\Lambda (A)$ defined as the above proposition. In this language we have:

$$(1-aT)_W \cdot (1-bT)_W= (1-ab T)_W \qquad (a,b \in A)$$

More generally, for any $f(T)\in 1+TA[[T]]$ , we have a formula for multiplication by degree-1-object $(1-aT)_W$:

$$(1-aT)_W \cdot (f(T))_W= (f(aT)_W) \qquad (a \in A)$$

We may extend this formula to any polynomial $g(T)\in 1+TA[T]$ with constant term=1. Indeed, we factorize $g$ as $g(T)=\prod_{j=1}^k (1-\alpha_j T)$ and

$$(g(T))_W\cdot (f(T))_W=\prod_j f(\alpha_j T)$$

EXERCISE 05.1   Compute $(1+ a T + bT^2)_W (1+ p T +q T^2)_W$. Notice that the result of the computation only needs polynomials with coefficents in $\mathbb{Z}[a,b,p,q]$ rather than some extension of the ring.