TITLE: Standard reduction to the artinian local case?
QUESTION [20 upvotes]: 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.

REPLY [7 votes]: Since I was somewhat critical of the question, I would offer here some reasons why the phrase: "reduction to the artinian local case" may mean quite differently to an algebraist (I was also vaguely aware of the geometric use BCnrd explained, that's  why I asked the OP to clarify). 
To prove some statements about a commutative ring $R$, a very often used technique is to pass to the quotient of $R$ by a non-zerodivisor. This is a local analogue of "taking the hyperplane section" in geometry. For examples of  this technique, see the proof of the Auslander-Buchsbaum formula in Bruns-Herzog, or this answer. 
Now, if the ring you started with is local and Cohen-Macaulay, then after killing enough regular elements you ended up in a local Artinian ring $(A,m,k)$. There are a lot of nice things about such rings: all f.g. modules have finite lengths, the Grothendieck group is $\mathbb Z$, local duality is simple as $A$ is already complete, the representation theory is well-studied, blah blah...
Sometimes you also wish that the residue field $k$ is algebraically closed. This can be obtained by making a faithfully flat extension by the EGA result BCnrd mentioned. Here are a few examples where such extension is useful:
1) The Auslander-Reiten quiver is easier to write down (all arrows has trivial valuations).
2) If $A$ is a complete intersection, then one can attach to each pair of $A$-modules $M,N$ a $k$-variety, whose dimension equals to the rate of (polynomial) growth of the lengths of $\text{Ext}^i(M,N)$. This has a non-trivial consequence that $\text{Ext}^i(M,N)$  and $\text{Ext}^i(N,M)$ grow at the same rate! (see this paper).
3) To prove that  there are only finitely many semi-dualizing modules over Cohen-Macaulay algebras, the only known proof is to pass to the Artinian case with alg. closed field and then use relevant results from representation theory. 
My hope is although this post did not answer what the OP wanted to ask, perhaps it can be of help to someone who googles "reduction to artinian case" with different motivation and find this MO question.