Let $A$ a commutative ring with an arbitrary ideal $I$. Consider the completion $\hat{A}_I := \varprojlim A / I^n$.
I have a few questions about some $length$ relations between $A$ and $\hat{A}_I $:
Why following relations hold:
$length_A (\hat{A} / \hat{I}^n) = length_A A /I^n$ (here $\hat{I}$ induced ideal in $\hat{A}$ by $I$)
$length_{\hat{A}} (\hat{A} / \hat{I}^n)=0$
The term $length$ I'm working with is described here: https://stacks.math.columbia.edu/tag/00IU