Where can I find a clear exposé of the so called "standard reduction to the local artinian (with algebraically closed residue field", a sentence I read everywhere but that is never completely unfold?

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Thank you for your answear dear BCnrd. Here are some precisions since I'm still not able to do it on my own.

EDIT: properness assumption for $f$ has been removed. Questions are still uncorrect.

I say that $(f:X\longrightarrow S,Z,E)$ is a 'context' if:

(1) $f:X\longrightarrow S$ is a morphism of preschemes which is surjective, separated, smooth of pure relative dimension $1$ and finitely presented.  
(2) $Z$ is a closed sub-prescheme that is supposed finite etale and surjective over $S$ (finitely presented over $S$ If needed)  
(3) $E$ is a $O_X$-Module which is locally free of finite rank.

There is some extra data $D$ defined over any 'context' $(X/S,Z,E)$ as before.
I can base-change $D$ along any $S'\longrightarrow S$ and localize it $D$ at any point $x\in X$. (Yes, the base-change and the localization of a context are still a context)

The claim I want to prove is something like:
If some hypothesis $H(X/S,Z,E,D)$ is true then $D$ is zero. 


The hypothesis is stable under any base change $S'\longrightarrow S$ and is stable by localizing over $X$ at any point.



I call a context $(X/S,Z,E)$ 'limited' if

$S=spec(A)$,  
$X=spec(B=A[t])$ (or $X=spec(B=A[[t]])$),  
$Z$ is defined by $spec(tB)$,  
$E=spec(B^n)$ for ANY ring $A$.


First question:
Is it possible to reduce the proof of the claim on any context to the proof of the limited case? If it is, how?

Second question:
If I had some pull-back of the data along any etale map $X'\longrightarrow X$ and I forget the assumption that $f$ is fintely presented would I be able to make the reduction to the limited case also? (maybe with a different argument). If it is, how?