# working with local rings: “abstract” vs “geometric” proofs

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 $k=\bar{k}$)?

${\bf upd:}$ In view of the comments, an additional example might be helpful. Consider a matrix $A$ over a local ring. Suppose one wants to re-derive the standard upper bound on the height of the fitting ideal $I_j(A)$. The bound is well known in alg.com. But if this matrix happens to be a matrix of "functions", that are computable in some open neighborhood of the origin, then one can derive this bound also geometrically, as the codimension of the corresponding degeneracy locus. The question is: after I prove geometrically some bound of this type, which invocations should be pronounced to ensure the validity over an arbitrary local ring?

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This may not be exactly what you're looking for, but if the statement you're trying to prove can be reduced to the completion of your local ring, you can use the Cohen structure theorem (en.wikipedia.org/wiki/Cohen_structure_theorem). –  Eric Wofsey Feb 17 at 16:52
@Eric Wofsey: I speak about a statement formulated over an arbitrary local ring. Maybe complete, maybe not. Can't see how Cohen's structure theorem can be helpful here. –  Dmitry Kerner Feb 17 at 17:29
@Dmitry: Many properties hold for a local ring if and only if they hold for its completion. –  Mahdi Majidi-Zolbanin Feb 17 at 17:58
@Mahdi: precisely. That's what I'm asking. For which statements about the local rings it is enough to check the statement just for e.g. localization/henselization of an affine ring? –  Dmitry Kerner Feb 17 at 19:15
There is Artin's Approximation Theorem, which I think is similar to the question you are asking. But then I see that you already asked a question about Artin's Approximation Theorem before, so that tells me this is not what you are looking for? –  Mahdi Majidi-Zolbanin Feb 17 at 21:18
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