In Proposition 10.15, let $A$ be Noetherian ring and $\hat{A}$ be $\frak{a}$– adic completion , and we can also complete $\frak{a}$ as $A$ – module to $\hat{\frak{a}}$ with the isomorphism that $\hat{\frak{a}} \cong \hat{A}\otimes_A \frak{a}$
then the book says that since $\hat{A}$ is complete under $\hat{\frak{a}}$– topology(I know why it's complete) then for any $x \in \hat{\frak{a}}$ the $$(1-x)^{-1} = 1+x+x^2+….$$
converge so $(1-x)$ is a unit.
there are two question
- why $1+x+x^2…$ converge if $\hat{A}$ is complete in $\hat{\frak{a}}$ topology
- why the infinite sequence $1+x+x^2+…$ is the inverse of $(1-x)$?
I can't make it clear.
Best Answer
Maybe I can answer my question now:
for the first question by definition $G$ is complete if canonical map $G\to \hat{G}$ is isomorphism, I adopt the definition for $\hat{G}$ to be set of equivalent Cauchy sequence, then we see since the map is surjective we have a point $y\in G$ such that the constant sequence $(y) $ is equivalent to the sequence $(\sum^n_{i=0}x^i)$ ,and easy to check that $y$ is the convergent point of the Cauchy sequence.
Therefore we see the completeness in the commutative algebraic definition immediately implies the Cauchy sequence will converge.
For the second question see answer here