$\newcommand{\Spec}{\mathrm{Spec}\ }$
Let $(P)$ be a property of rings. I call $(P)$ local when $(P)$ satisfy these two
conditions:

 - if $A$ is a ring satisfying $(P)$, then the distinguished rings $A_f$ also
  satisfy $(P)$;
 - If $\Spec A$ is covered by distinguished open $\Spec A_i$ with the $A_i$ having
  $(P)$, then $A$ satisfy $(P)$.

Now if $(P)$ is local, then it is natural to extend the property $(P)$ to
schemes by saying that a scheme $X$ has property $(P)$ iff for all open affines
$\Spec A$ of $X$, $A$ has property $(P)$.

By definition of locality, then

- a ring $A$ satisfy $(P)$ iff $\Spec A$ satisfy $(P)$;
- a scheme $X$ satisfy $(P)$ iff $X$ can be covered by affines open $\Spec A_i$ with
  $A_i$ satisfying $(P)$.

Likewise in the relative setting, local properties of morphisms of rings
allow to define a corresponding notion for morphisms of schemes.

[I have to point out that sometimes extending a local property $(P)$ of rings
to schemes this way is called $(\mathrm{locally}\ P)$, and a scheme $X$ is said to have
property $(P)$ when $X$ is locally P and satisfy some finiteness condition.
For instance $X$ is noetherian when it is locally noetherian and
quasi-compact; a morphism is of finite presentation when it is locally of
finite presentation and quasi-compact + quasi-separated.]

Now while this is a standard construction explained in all textbooks, it is
harder to find references for what happen to the global sections of non
affine open subschemes.

Indeed, if $X$ has a local property $(P)$, then an open scheme $U$ has also
property $(P)$, but $\Spec O_X(U)$ may not have $(P)$ when $U$ is not affine.

For instance:

- $X$ is noetherian does not mean that each ring of sections $O_X(U)$ is
  noetherian
  (An example is given by the union of two disjoints plane in projective
  space $P^3_k$,
  see http://sma.epfl.ch/~ojangure/nichtnoethersch.pdf);
- $X$ is finitely generated (say over a field $k$) does not mean that each
  ring of sections O_X(U) is finitely generated
  (Same example as above, see also http://math.stanford.edu/~vakil/files/nonfg.pdf);
- $X \to \Spec A$ is flat does not mean that $O_X(X)$ is flat over $A$.
  (see http://mathoverflow.net/questions/65267/global-sections-of-flat-scheme-also-flat).

However there are properties that hold for sections over any open subschemes:

- If $X$ is reduced, then for every open subscheme $U$, $O_X(U)$ is reduced;
- If $X$ is integral, then for every open subscheme $U$, $O_X(U)$ is integral.

I am interested to what happens with other properties $(P)$. I am also
interested to what happens in the relative case: if a morphism $X \to Y$ has
property $(P)$, then does $\Spec_Y(X) \to Y$ also has $(P)$?