## Why does finitely presented imply quasi-separated ?

By the EGA definition, a morphism of schemes of finite presentation is required to be quasi-separated. As far as I can see, removing this requirement does not prevent from proving the basic properties such as stability of the notion under composition, products, etc. So my question is :

where exactly in proving important theorems involving morphisms of finite presentation is the quasi-separated assumption crucial ?

Note that a morphism of finite type is not required to be quasi-separated.

All kinds of examples and counter-examples will be appreciated.

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One of the main interests in finitely presented morphisms comes from the various theorems in EGA IV,8. They show that for many questions about morphisms of schemes and sheaves on them, the condition of finite presentation allows one to reduce to a noetherian situation. For these theorems the assumption of quasi-separatedness is crucial.

Let me quickly try to explain why. The heart of the reduction to the noetherian case are theorems like the following: Let X over Spec A be a finitely presented scheme. Then there is a subring $A_0$ of $A$ which is a finitely generated $\mathbb{Z}$-algebra (and in particular noetherian) and an $A_0$-scheme $X_0$ of finite presentation such that $X$ arises from $X_0$ via the base-change $A_0\to A$. If $X$ is affine, this is pretty clear, as $X$ is definied by finitely many equations in an affine space over $A$. In order to pass from the affine case to the general case, it does NOT suffice to know that we can cover $X$ by finitely many affine pieces (which would be the assumption of quasi-compactness), but we also need that the glueing data for the affine pieces are somehow described by a finite number of equations. This is ensured by the assumption that the intersection of two affine pieces is quasi-compact which corresponds precisely to the assumption that $X$ is quasi-separated over A.

I guess that these theorems were the reason for Grothendieck to include this condition in the definition of finitely presented.

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Great ! That's exactly the kind of answer I was expecting. Thanks. – Matthieu Romagny Aug 26 2010 at 12:01
Note also the trivial "converse" that since every f. type map $f_0:X_0 \rightarrow S_0$ to a noetherian $S_0$ is q−septd, any $f:X \rightarrow S$ arising from such $f_0$ by base change must be q-septd. So q-sep'tdness is not only sufficient for a f. type map to admit descent to a "noetherian" situation, but also necessary. For example, if $A$ is a ring and $U$ is a non-qc open in Spec($A$) then the f.type $A$-scheme gluing $X$ of Spec($A$) to itself along $U$ cannot descend to f.type scheme over noetherian ring; stronger than saying the EGA method of proof doesn't apply. – BCnrd Aug 26 2010 at 12:27
small clarification: when I wrote about "sufficient for a f.type map.." I should have said "sufficient for a quasi-compact locally finitely presented map". – BCnrd Aug 31 2010 at 5:25