I have asked the same question on math.stackexhange here, but thought that is was a good idea to post it here also.
I am learning schemes theory at school and I have for now only lectures notes that I am taking during the course. The professor is quite often using following expressions, without having defined them : if $S$ is a scheme and if $f : X \rightarrow Y$ is a morphism of $S$-schemes he often says that some property P of $f$ is
(1) local on $X$
(2) local on $Y$
(3) local on $X$ and $Y$
(4) local on $X, Y$ and $S$
and I have several question on this, and related to this.
I went to the library to look at all Grothendieck's EGA's and found the same frequent use of the expression la question est locale sur in the same four cases - at least, maybe there are other than these four ? But I found in them no definition of the meaning of la question est locale sur, which means the question is local on in english.
I guess that (1) means that $f$ has the property P if and only if for every open cover $(U_i)_{i\in I}$ of $X$, all restrictions $f_{|U_i} : U_i \rightarrow Y$ have the property P, and that (2) means that $f$ has the property P if and only if for every open cover $(V_i)_{i\in I}$ of $Y$, all corestrictions $f^{-1} (V_i) \rightarrow V_i$ have the property P. For (3) obviously it is equivalent to check (1) and (2), but I cannot state a more synthetic formulation.
Does (3) means that $f$ has property P if and only if for every open cover $(U_i)_{i\in I}$ of $X$ and for every open cover $(V_j)_{j\in J}$ of $Y$ all morphisms $U_i \cap f^{-1} (V_j) \rightarrow V_j$ have the property P ? For (4) it is sufficient to define what means P is local on S. Does this means that $f$ has the property P if and only if for every open cover $(W_i)_{i\in I}$ of $S$, every morphism $p^{-1}(W_i) \cap f^{-1}(q^{-1} (W_i)) \rightarrow q^{-1} (W_i)$ has the property P ? Or is there something more clever than this. Combining this with (3) is equivalent to (4), but here again, is there a more synthetic formulation for this ?
In (1), (2), (3), (4) I used "local morphisms" (restrictions in (1), corestrictions in (2) etc). Is it possible to express these morphisms thanks to fiber products, and if so, how ?
In (1), (2), (3), (4) I used the word every cover but sometimes in the course it suffice to have the fact for only one cover to have it for all. How does this work ? (I call Q this question.)
Sometimes its not only open covers that are used, but open affine covers. For $i\in\{1,2,3,4\}$, is (i) equivalent to the assertion (i) with "open" replaced by "open affine", or even by "affine" ?
Is answering to question Q easier with "open" replaced by "open affine", or only "affine" ? Of this note, what does the following sentence mean : the property P of the morphism of $A$-algebras $\varphi : B' \rightarrow B$ is local on $\textrm{Spec}(A)$ (resp. on $\textrm{Spec}(B')$, resp. on $\textrm{Spec}(B)$, resp. on "combinations of the previous") ?
I have a last question : all the previous questions are local for the topology of the schemes involved, which is a topology in the "classic" sense. Is all of this translatable to Grothendieck topologies ? If so, how ? And then, intuitively, in the case of the Zariski site, is the same ?
I know that this was a lot of question, but all are intimately related to this "local on" stuff, so I preferred to ask all of them in one shot.