Let $R$ be a local ring (commutative, Noetherian, over an algebraically closed field; if needed Henselian). Suppose one wants to prove some statement.

Suppose $R$ happens to be the ring of "functions", each of which being defined in a small enough neighborhood of the origin, then one can consider some representative of $Spec(R)$, with genuine geometric points off the origin. And to prove the statement pointwise, by going over all the points. Sometimes, this is simpler than to do the proof in the general case of an "abstract" local ring.

e.g. Suppose we are given two ideals $I,J\subset R$ (defined in a complicated way). And we want to check that the ideal they generate together contains a power of the maximal ideal. Geometrically (if $R$ is an analytic ring) this means: the two subschemes $V(I)$, $V(J)\subset Spec(R)$ intersect at the origin only. Then one can just go over the points in the punctured neighborhood of the origin and to check pointwise. This might greatly simplify the proof. (At least the idea of the proof. At least for some people.)

But, when working with complete local rings, one cannot speak of the "points near the origin", etc.

${\bf Question:}$ Is there some analogue of Lefschetz principle, when working with local rings? Something like: if a statement is formulated over an arbitrary local ring, and can be proven for analytic rings (i.e. $k \{ x_1,..,x_n \} /I$), then it is true for an arbitrary local ring (at least henselian, over $k=\bar{k}$)?