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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?

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.
EDIT: The context has been simplified.

I say that $(f:X\longrightarrow S,Z)$ 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)

There is some extra data $D$ defined over any 'context' $(X/S,Z)$ 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 $H(X/S,Z,D)$ 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 $(X/S,Z)$ 'limited' if

$S=spec(A)$,
$X=spec(B=A[t])$ (or $X=spec(B=A[[t]])$),
$Z$ is defined by $spec(tB)$,
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?

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?

EDIT: Here, was a badly posed question.

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?

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?

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?

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.
EDIT: The context has been simplified.

I say that $(f:X\longrightarrow S,Z)$ 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)

There is some extra data $D$ defined over any 'context' $(X/S,Z)$ 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 $H(X/S,Z,D)$ 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 $(X/S,Z)$ 'limited' if

$S=spec(A)$,
$X=spec(B=A[t])$ (or $X=spec(B=A[[t]])$),
$Z$ is defined by $spec(tB)$,
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?

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?

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

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, proper, 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?

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?

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?