"This property is local on" : properties of morphisms of $S$-schemes

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.

• Why don't you ask all of this to your professor in person? Especially for this kind of long but ultimately standard question (which everyone grapples with when first learning algebraic geometry, and generally figures out from the experience of seeing how the ideas are actually used), discussing in person with an expert at a blackboard or with pen and paper is going to be much more efficient than trying to do so via a computer. Jan 15 '15 at 14:44
• @user74230 "discussing in person with an expert at a blackboard or with pen and paper is going to be much more efficient than trying to do so via a computer" This is almost true for all non reference-related question here. ;-) Nevertheless, even experts ask questions here, so don't blame him so quickly ;-) Jan 16 '15 at 1:39
• @AntonFonarev Modulo the definition of reasonable book ;-) Hartshorne is not reasonable for instance. Nor in Eisenbud and Harris. Only local on the base is defining in Liu, afair. In EGA/SGA, no mention to this, afair. Jan 16 '15 at 1:44
• I think one of the reasons that some references don't give a precise definition is that this phrase is often used to have slightly different meanings. For example, all of the definitions you've suggested are reasonable. But they don't cover this slightly different usage: "Theorem: There is a unique isomorphism between the S-schemes X and Y with such and such properties. Proof: The assertion is local on S, ..." Here the phrase "local on S" should be taken to mean that if every point of S has arbitrarily small neighborhoods over which the theorem is true, then the theorem is true for S. (cont)
– Jacob Lurie
Jan 16 '15 at 16:04
• In this example, you wouldn't want to assert that the existence of an open covering of S for which the theorem is true implies that the theorem is true for S, because that wouldn't let you glue the local isomorphisms together to make a global one
– Jacob Lurie
Jan 16 '15 at 16:10

Ok, this will be a long one.

There is one fuzzy question underlying all your questions : for a given topology (the "classic one" as you say for, but also for Grothendieck's ones, like étale, smooth, fppf, fpqc, synthomic, Niesnievitch etc) what are reasonable properties that one can reasonably expect from a class/type of morphisms of schemes ?

For the classic topology, essentially all classes of morphisms have the following properties : closed under composition, base change and they are local on the base, or target - or on $Y$ with your (2)'s notation. For (1) one says local on the source.

The latter means that to ensure that a morphism $f : X \rightarrow Y$ is in the class, you need only to check that corestrictions on an open cover of $Y$ are in the class. You wrote all covers, but usually one suffices, as practically the properties of the class are often nice enough to imply that if you have this for one cover you have it for all. Same remark for (1), (3) and (4), with all replaced by one. Remark : the construction of product $X\times_S Y$ in the category of schemes over a fixed scheme $S$ is local on $X$, $Y$, $S$. See EGA I 3. 3.2 I think. There won't be a definition of (4), but you will feel what it means.

By the way, the properties of the considered class of scheme are often nice enough to ensure that you can replace in what I previously wrote "open covers" by "open affine covers". This is essentialy dued to the fact that reasonable classes of morphism are defined by properties Q of morphisms themselves defined via properties P of schemes defined in the following way : the morphism $f : X\rightarrow Y$ has the property Q if (and only if but it's a definition) for every open affine $V\in Y$ the scheme $f^{-1} (U)$ as the property Q.

On note of affine covers and checking something on one or all over, one question could be asked : if you check some property of morphism via an affine cover and if I check it on another affine cover, what can we say ? That is, what does what I did on my cover implies on yours, etc... For this matter, there are two key facts :

(1) if you take $U,V$ open affine subschemes of a scheme $X$, then $U\cap V$ is union of open subsets of $U$ and $V$ simultanously (obviously) that are distinguished in $U$ and $V$. (Remember that an open $V \subseteq \textrm{Spec}(A)$ is distinguished if $V = D(f)$ for some $f\in A$.

(2) Let P be a property that affine open subset of a scheme $X$ may have such that :

(a) If an open affine subset $U \simeq \textrm{Spec}(A)$ of $X$ has the property P, then for any $f\in A$ the distinguished affine subset $V \simeq \textrm{Spec}(A_f)$ of $X$ has also the property P

(b) If $f_1,\ldots,f_d \in A$ generate $A$ and if for each $i\in\{1,\ldots,d\}$ the open affine subset $V_i \simeq \textrm{Spec}(A_{f_i})$ of $X$ has the property P then the open affine subset $U \simeq \textrm{Spec}(A)$ of $X$ has the property P

Then if $X = \cup_{i\in I} U_i$ where each open affine subset $U_i \simeq \textrm{Spec}(A_i)$ of $X$ has the property (P), then every open affine subset of $X$ has the property.

Example of properties local on the target : quasi-compact, finite type, open immersion, closed immersion, immersion, finite, quasi-finite, etc

Example of properties local on the base and on the target : locally of finite type, locally of finite presentation, flat, étale, unramified, smooth, etc

For your question on properties of morphism of rings that are local on various spectra, you may guess that these properties often "translate" to the associated morphisms of affine scheme, and that various localness can be checked thanks to what we have discussed previously. There are plenty of nice of fun exemples of this in *Anneaux locaux henséliens" from Michel Raynaud, LNM, 1st chapter for instance. For sure in other books also. (SGA 1 also...)

Finally, how what preceeds could be translated to the étale topology for instance, or others Grothendieck topologies ? This is intimately linked to descent, but I will give only a simple definition : let $\mathscr{T}$ be a Grothendieck topology from the following list : Zariski, fpqc, fppf, syntomic, smooth, étale. Let P be a property of schemes over some base $S$. P is said to be local on the base for the topology $\mathscr{T}$ if the following is true : for any $S$-morphism $f : X \rightarrow Y$ and any $\mathscr{T}$-covering $\{Y_i\rightarrow Y\}$ of $Y$, the morphism $f$ has the property P if and only if for each $i$ the morphism $Y_i \times_Y X \rightarrow Y_i$ has the property P.

Note that as isomrphisms are always $\mathscr{T}$-coverings, any property local on the base for $\mathscr{T}$ is preserved by base change. You may understand why base is important, and maybe why I required it earlier in reasonable properties of a class of morphisms of schemes. The reason is the following : if $V$ is (classic) open in $Y$, the open $f^{-1} (V)$ is but isomorphic to $V \times_Y X$ and $f : f^{-1} (V) \rightarrow V$ is none but $f : V \times_Y X \rightarrow V$, base change of $f$ by the open immersion $V\rightarrow Y$. You see that in fact one mimics classic topology (rejoining earlier definitions) in the setting of Grothendieck topologies, thanks to categoric constructions. Finally, how to relate definitions in the classic topology case to the Zariski topology. They are identical, by definition of a Zariski covering (a family of open immersions with covering images) and by the previous categoric remark.

I am almost off-topic so I will stop here, saying that SGA IV 1. first exposés or Milne's Étale Cohomology are good places to learn about all of this.

• Comments appreciated ;-) Jan 16 '15 at 1:45
• Thx a lot for all your answers and details, that was great ! I will try to prove myself the results that you have stated on "On note of affine covers and checking something on one or all cover" and come back to you if I am stuck. Thx again ! Jan 16 '15 at 10:28