Serre's Structure Sheaf on Affine Space - Clarification

Question

I am working out of this English translation of Serre's FAC. In Section 31, p.38 we begin to put the structure sheaf on the affine space equipped with Zariski topology. I have a few clarification questions about things he says. Here is the selection from the text containing the relevant information:

[Serre] For $x \in X$ we denote by $\mathscr{O}_x$ the local ring of $x$; recall that this is the subring of the field $K(X_1, \dots, X_r)$ consisting of those fractions which can be put in the form $R = P/Q$, where $Q(x) \neq 0.$

Such a fraction is said to be regular in x; for all points $x \in X$ for which $Q(x) \neq 0$, the function $x \mapsto P(x)/Q(x)$ is a continuous function with values in $K$ ($K$ being given the Zariski topology) which can be identified with $R$, the field $K$ being infinite. The $\mathscr{O}_x$, $x \in X$ thus form a subsheaf $\mathscr{O}$ of the sheaf $\mathscr{F}(X)$ of germs of functions on $X$ with values in $K$; the sheaf $\mathscr{O}$ is a sheaf of rings.

The precise definition of a subsheaf is given on page 12 and the sheaf of germs is defined on page 10 to be the sheaffification of a system of functions, however it is mentioned that the system was a sheaf to begin with.

Questions

1. So he is saying $\mathscr{O}_x = \{\text{fractions that are regular in } x \}$ = $\{R = P/Q \, | \, Q(x) \neq 0 \}$ and this ring is in fact local because it is the localization $(K[x_1, \dots, x_r])_{\mathfrak{m}}$ for the maximal ideal determined by the point $x$, such a localization is always a local ring.

2. Is there a particular reason he wants to develop the structure sheaf as a subsheaf of the sheaf of germs? It seems we could construct the desired sheaf in several other ways, will there be advantages to doing it this way?

3. By using the definition of subsheaf given on page 12 - to verify subsheaf we will first want to see that $\mathscr{O}_x$ is a subring of $\mathscr{F}(X)_x$. Now by the construction of $\mathscr{F}(X)$ on page 10, the stalk over $x$ will be a direct limit of functions, so the elements of $\mathscr{F}(X)_x$ are equivalence classes of functions that agree on a neighborhood of $x$. So to see the subring, lets just identify the fraction $R = P/Q$ with the function $P(x)/Q(x)$ and send it to its equivalence class in $\mathscr{F}(X)_x$? However we need this to be injective, so we need to verify that if two functions are distinct as elements of the localization then they will be distinct on arbitrarily small neighborhood of $x$? How do I show this. I know that when K is infinite we have a correspondence between formal polynomials and functions $K^n \to K$, can I extend that here?

4. The next part of showing the subsheaf is condition (b) or its equivalent on page 12. How to do this?

5. It seems easier to me to say what the rings $\mathscr{O}_U$ are for an open set $U$, which would clearly be a subsystem of the system of germs so sheaffifying both should yield subsheaf? Is there a reason Serre's approach is going to be more fruitful?

6. The next paragraph on page 38 might actually explain why he is involving the sheaf of germs - he induces the structure sheaf from $X$ onto a locally closed subset $Y$ by defining a restriction morphism $\epsilon_x \colon \mathscr{F}(X)_x \to \mathscr{F}(Y)_x$ for all $x \in Y$. This is a stalk map from the sheaf of germs on $X$ to sheaf of germs on $Y$ given by restriction. Let me make sure I understand. An element of $\mathscr{F}(X)_x$ is an equivalence class of functions with domain $X$ who agree on some neighborhood of $x$, then we send them over by just changing them to domain $Y$? Serre then uses this to put the structure sheaf on $Y$ by saying $\mathscr{O}_x,Y$ is the image of $\mathscr{O}_x$ under the map $\epsilon_x$, and then he says these stalks form a subsheaf of the sheaf of germs on $Y$. Again, why did he do it this why? Is the resulting sheaf any different from if we just restricted the sheaf $\mathscr{O}_X$ to the subspace $\mathscr{Y}$ in the usual way he outlines on page 10?

6.5. - I think part of my issue in question 7 is that I realized just now I do not really understand the map $\epsilon_x$. Pick some arbitrary element of $\mathscr{F}(X)_x$. The element looks like this: $[f]$, where $f: U \to K$ is a function for some open $U \subset X$ with $x \in U$, and $[f] = \{g: V \to K \, | \, V \text{ is open in } X \text{ and there is } W \text{ open in } X \text{ such that } f|_{W} = g|_{W}\}$. How do we take this equivalence class into $\mathscr{F}(Y)_x$? Is it like this: $[f] \mapsto [f|_{Y \cap U}] = \{g: V \to K \, | \, V \text{ is open in } Y \text{ and there is } W \text{ open in } Y \text{ such that } f|_{W} = g|_{W}\}$. This should be well defined since if you agree on an open subset in $X$, you will agree on an open subset in $Y$, but what else? Is it injective? Is it surjective? Is there geometric intuition?

7. He follows up by saying that "a section of $\mathscr{O}_Y$ over an open subset $V$ of $Y$ is thus by definition a function $f: V \to K$ which is equal, in the neighborhood of any point of $x \in V$, to a restriction to $V$ of a rational function which is regular at $x$. Such a function is said to be regular on $V$. Ok, I need to make sure I see what happened here. We know from page 10 that for the sheaf $\mathscr{F}(Y)$, we have $\Gamma(V, \mathscr{F}(Y)) \cong \mathscr{F}(Y)_V$ which is to say that we can identify the sections of the sheaf with the functions $f: V \to K$ for every $V$ open in $Y$. Now lets start with $s \in \Gamma(V, \mathscr{O}_Y)$ and identify it with the section $s \in \Gamma(V, \mathscr{F}(Y))$. Now we know that for each $y \in V$, $s(y) \in \mathscr{O}_{y,Y} = \epsilon_y(\mathscr{O}_y)$, thus $s(y)$ is a restriction of a rational function $P/Q$ where $Q(y)$ is not zero and the rational function is regular on a neighborhood of $y$. Thus we can identify the section $s$ itself with a function $s \colon V \to K$ such that for each point of $x \in V$ we have a neighborhood where $s$ is given by the restriction of a function regular at $x$.

Answer 1

Hm. It's typically good to ask one question in a question.

However, while you have many questions, they are (sort of) narrow in scope, pertaining as they do to this one section of FAC.

That said, in order to address them, I will have to go through a bit more of FAC than is frankly ideal.

1. Yes.
2. I can't properly answer that without reading significantly more of FAC than I'm willing to do for a MSE answer. However, as a first approximation, the answer is likely, no of course not. Any equivalent definition will do for defining an object. Since FAC was apparently more or less foundational to introducing sheaves into AG (see this MO answer, I'm no historian), it should be recognized that the article is old, and likely there has been innovation in the definitions and their presentation. It's not really a good question to ask why a particular definition was chosen in an old paper (unless you're a math historian), when there is a pretty reasonable answer: because that definition was the best available at the time in the opinion of the author.
3. If two rational functions agree on any neighborhood of a point, then they are equal. (Simple corollary of the fact that any polynomial which is zero everywhere in a Zariski open set is zero).

4. I don't like the espace etale definition of sheaf. However, let me give it a go here.

   Note that the topology on $\mathscr{F}$ has as a base $[f,U]=\{f_x:x\in U\}$ (where $f_x=\phi_x^U(f)$ in the notation in the pdf) with $f:U\to K$ and $U$ open in $X$. Given $g\in \mathcal{O}_x$, $g$ is regular on an open set $U$ (the complement of the pole set), and the neighborhood $[g,U]$ of $g$ in $\mathscr{F}$ is a subset of $\mathcal{O}$. Hence since every element of $\mathcal{O}$ has a neighborhood contained inside $\mathcal{O}$, $\mathcal{O}$ is open in $\mathscr{F}$ as desired.

5. They are equivalent, and I personally think Serre's approach was fairly brief, and he didn't have to then prove that $\mathcal{O}_U$ was a sheaf already, also the sheafification is necessary for more complicated varieties (such as ones with coordinate rings that aren't UFDs), so we might as well define the sheaf by its stalks now, since that will generalize better, though later material suggests that this won't be necessary anyway, so perhaps that's not why. Again, I don't know that this is a useful question to ask.
6. It seems likely that this is why he wanted to prove that the structure sheaf is a subsheaf of the sheaf of germs, yes. Also yes, it is different from the restriction defined on page 10. If we just did the restriction defined on page 10, the stalks on $Y$ would be germs on $X$ rather than germs on $Y$, by which I mean, they would differ even if they agree in an open nhood of the point in $Y$ as long as they differ in an open nhood of the point in $X$. For example, given the origin in the affine line, the restriction of the structure sheaf as done on page 10 would have stalk $k[x]_{(x)}$, where as the restriction defined here gives a structure sheaf of the point which is just $k$. In the first case, we're still retaining information about points in $X$ outside of $Y$.

6.5: Your impression of the map $\epsilon_x$ is correct, as in, yes, that is how it is defined. As for whether or not the map is injective or surjective, the answers are that $\epsilon_x$ is always surjective, and it is injective if and only if there exists a neighborhood $U$ of $x$ in $X$ with $U \subseteq Y$, or $U\cap X = U\cap Y$, i.e. locally at $x$, $X=Y$. As for geometric intuition, I personally find the modern definition of sheaf more clear intuitively, however, the idea is that if we have $Y\subseteq X$, and $x\in Y$, then if we have a function defined on some neighborhood $U$ of $x$ in $X$, then we can restrict that function to the neighborhood of $x$ in $Y$, $U\cap Y$. We forget about all of its values off of $Y$. This is sort of analogous to how smooth functions on a subset of a smooth manifold are sometimes defined. If $Y\subset M$, then $f:Y\to \Bbb{R}$ is smooth if for all $y\in Y$, there exists a neighborhood $U$ of $y$ in $M$ such that there is a smooth function $g:U\to \Bbb{R}$ such that $f|_U=g|_{U\cap Y}$. I.e., a function on a subset is smooth if locally it can be extended to a smooth function on an open neighborhood of any point in the domain.

7. Your interpretation of what's going on here looks correct to me.