Let $ X$ be a proper $ n$ -dimensional $ k$ -scheme and $ x \in X$ a closed point. Consider the stalk $ \mathcal{O}_{X,x}$ . We consider now it’s completion $ O_{X,x}^{\wedge}$ wrt it’s maximal ideal $ m_x$ .

My QUESTIONS concerns the analysis of the structure of $ \mathcal{O}_{X,x}$ :

When $ O_{X,x}^{\wedge}$ has the shape $ k[[x_1,x_2,…, x_m]]/I$ ?

When – more specifically – it has the shape $ k[[x_1,x_2,…, x_{n+1}]]/(f)$ for $ f \in k[[x_1,x_2,…, x_{n+1}]]$ non zero divisor?

Are there any reconmendable reference with thread the structure of such local complete rings rigoriously?

My CONSIDERATIONS /starting point:

The main tool to treat this problem is of course the Cohen strucure theorem. It provides especially that $ $ O_{X,x}^{\wedge} \cong \Lambda [[x_1, \ldots , x_ n]]/I$ $ where $ \Lambda$ is the “mysterious” Cohen ring.

Which criterions are neccessary & sufficient to settle that here $ \Lambda=k$ (references?)?

Another point is under which conditions we obtain a more “concretely” form $ $ O_{X,x}^{\wedge}= k[[x_1,x_2,…, x_{n+1}]]/(f)$ $

?

This question arises from following former thread of mine: Intuition behind RDP (Rational Double Points)