I'm currently studying the basic theory of schemes from Q.Liu's book "AG & Arithmetic Curves".
I have ran across several times in a type of argument where the author has to prove some fact about a general scheme $X$, and in the proofs he begins with "we may assume that $X$ is affine", or some variants of this like situations when he need to prove some fact about a general open set $U$, and he just says "by covering $U$ with affine opens, we may assume $U$ is affine".
There are occasions when I understand what he is doing, but not always. I believe he is always using the same reasoning for each and everyone of these situations, so I am disappointed by the fact that I can understand what he is doing in some cases but not in all cases.
I would like to understand if there is some general pattern of reasoning that the author is following in each one of the cases, which I fail to see.
I will give two examples:
First example
Fact: Any open subscheme of a noetherian scheme is noetherian; and for any generic open of a noetherian scheme, its ring of sections is a noetherian ring.
Sketch of proof by reduction to the affine case:
Step1: A generic open of an affine noetherian scheme is noetherian. Step2: Given a finite covering of affines, the intersections with the generic open $U$ is a finite cover of $U$ by opens which are noetherian by step1, so we conclude. For the second part of the statement, since we saw that $U$ must be noetherian we cover it with a finite number of noetherian affines $U_j$, $j=1,...,n$. Given an ideal $I \subseteq \mathcal{O}_X(U)$, its extension $I_j=I\mathcal{O}_X(U_j)$ is finitely generated for any $j$. Pulling back the generators of $I_j$ for each $j$, we get a finite subset in $\mathcal{O}_X(U)$ and we consider the ideal $J$ generated by it. We have then $I_j=I\mathcal{O}_X(U_j)=J\mathcal{O}_X(U_j)$ for every $j$. Next, passing to the stalks we get that $I\mathcal{O}_{U_j,x}=J\mathcal{O}_{U_j,x}$ for every point $x \in U$, and we conclude $I=J$ by a Nakayama argument. Done.
Second example:
Fact: Let $X$ be an integral scheme with generic point $\xi$, $U$ a generic open subscheme, and recall that we can view both $\mathcal{O}_X(U)$ and $\mathcal{O}_{X,x}$ as subrings of $\mathcal{O}_{X,\xi}$. We have that $\cap_{x\in U}\mathcal{O}_{X,x}=\mathcal{O}_X(U)$.
In the proof, Q.Liu's just says "by covering $U$ with affine opens, we may assume $U$ is affine".
I see that here there is a (partial) explanation of this second example. But the answer there is actually not very illuminating for the purpose of my main question above.
It does seem that the reduction to the affine case is a general strategy of proof that is realized with some random trick which is different for every situation. Is it so?
This is slightly different, but close enough to warrant an answer, I think. Vakil, in The Rising Sea, unifies many such properties as "affine-local." A property $P$ of affine open subschemes in a scheme $X$ is affine-local if it satisfies the following two properties.
i. If an affine open subscheme $\operatorname{Spec}A$ in $X$ satisfies $P$, then so does the smaller open set $\operatorname{Spec}A_f \subset \operatorname{Spec A}$. ii. If $\operatorname{Spec}A$ is an affine open in $X$, with $(f_1,\ldots, f_n)$ a generating set for $A$, and all the smaller opens $\operatorname{Spec}A_{f_i}$ satisfy $P$ (notice they cover $\operatorname{Spec}A$), then $\operatorname{Spec} A$ also satisfies $P$.
Why should we care about such properties? Well, if $X$ is covered by affine opens that satisfy an affine-local property $P$, then any affine open in $X$ satisfies $P$. This result is called the Affine Communication Lemma (Section 5.3 in Vakil's notes). In particular, proving an affine-local property holds on some affine cover implies it holds on any affine cover.
Being Noetherian, reduced, etc are affine-local. As a result the definitions can be stated in terms of the existence of an affine cover with certain properties, but actually mean that any affine has certain properties.