TITLE: When do we genuinely need Noetherian conditions?
QUESTION [26 upvotes]: When are Noetherian conditions on a scheme genuinely essential in algebraic geometry?  
I am under the impression that most of the time these conditions are imposed for expositional clarity and to simplify the commutative algebra involved (for example throughout Hartshorne and FGA), and that they can be removed by paying more attention to finiteness conditions on the morphisms involved.
So what are the things that genuinely require Noetherian conditions?  Even better, is there a "slogan" that tells you when they will be needed?

REPLY [26 votes]: Here are some examples illustrating the genuine necessity of noetherian assumptions:  
1) Every scheme with just one point is the spectrum of a local artinian ring?
This is true   for every noetherian one point scheme and false for every non-noetherian one point scheme.
A non-noetherian (and thus non-Artinian) example :  $\operatorname {Spec}(\mathbb Q[T_1,T_2,T_3,\cdots]/\langle T_1^2,T_2^2,T_3^2 \rangle )$    
2) A scheme has only finitely many irreducible components ?
A noetherian scheme has only finitely many irreducible components, but a non-noetherian scheme may have infinitely many irreducible components: this is the case for any disjoint union of infinitely many non-empty schemes.  
3) Every scheme has a closed point?
This is true for every noetherian scheme (actually for any quasi-compact scheme), but there exist schemes without any closed point: Qing Liu, Chapter 3, Exercise 3.27, page 114.     
4) Injective modules give injective sheaves?
If $I$ is an injective module over the  ring $A$, then the associated quasi-coherent sheaf $\tilde I$ on  $X=\operatorname {Spec}(A)$ is an injective sheaf of $\mathcal O_X$- Modules if $A$ is noetherian but is not necessarily injective for $A$ non-noetherian: SGA6, Exposé II, Appendice I Un contre-exemple de Verdier, page 195.   
5) A finitely presented sheaf  is coherent?
Given  on a scheme $X$ a sheaf of $\mathcal F$ of $\mathcal O_X$-Modules, does the existence of an open covering $(U_i)$ of $X$ for which one has exact sequences $\mathcal O_{U_i}^{n_i}\to \mathcal O_{U_i}^{m_i}\to \mathcal F\vert _{U_i}\to 0$ imply that $\mathcal F$ is coherent?
The answer is yes if $X$ is noetherian (or even locally noetherian) but no in general: there exist non-noetherian rings $A$ such that the structural sheaf $\mathcal O_X$ on $X=\operatorname {Spec}(A)$ is not coherent!
6) A scheme is affine if all its quasi-coherent sheaves are acyclic ?
Serre's criterion is that a noetherian scheme $X$ is affine if and only if  $H^p(X, \mathcal F)=0$ for all quasi-coherent sheaves $\mathcal F$ on $X$ and all  $p\gt 0$.
This no longer holds if $X$ is not assumed noetherian: given a field $k$, any infinite disjoint sum $X=\coprod \operatorname {Spec}k$ satisfies the cohomology condition but is not affine since it is not quasi-compact.