Question

Background I have searched a bit for the definition/constructions on how to "semi-localize" a scheme, but have been unsuccessful in finding a good reference; I apologize in advance if this topic has been covered in detail elsewhere (e.g. in a book or article) and would be happy for a reference!

This question arose from a problem I had been working on in finding an étale morphism into affine space.

Much of the terminology here will be from EGA I.

(Aside: The constructions below take place in the Zariski site but I think some of them go through in the étale/Nisnevich site)

Definitions A local scheme is the spectrum of a local ring and a semi-local scheme the spectrum of a semilocal ring. In the constructions below a ring $O_{X,C}$ is given, so then the candidate semi-local scheme is $Spec \; O_{X,C}$.

NB: for some reason I was having trouble with "\varinjlim" here, so I'm using "lim" below to mean direct limit i.e. colimit.

Question

Given a scheme $X$ we can localize $X$ at a point $x\in X$ by taking $$O_{X,x} := \lim_{U\ni x} O_X(U). $$

Suppose now that we are given a finite set of closed points $x_1,\ldots, x_n \in X$. Let $C :=$ {$x_1,\ldots, x_n $}. How can we 'localize' $X$ around $C$?

There are at least three ways I know how to do this procedure and would be happy to hear about other methods as well as comments (especially geometric ones) regarding the following constructions:

$1.$ Define $$O_{X,C} : = \lim_{U\supset C} O_X(U).$$

This construction is similar to the localization construction above in that we take opens $U$ of $X$ containing $C$ and then take the direct limit; the case $n=1, C = {x_1}$ is then a special case. NB: we can 'see' this direct limit in the sense that for each $x_i$ we find an open $U_i\ni x_i$, then taking the (finite!) union of the $U_i$ we obtain an open $U$ containing $C$. Just as in the local case above, this direct limit is filtered by inclusion.

$2.$ Further assume now that $X$ is locally noetherian and regular. Let $A_i: = O_{X,x_i}$ and then define $$O_{X,C}:= \prod_i A_i .$$

Using the hypothesis that $X$ is regular, we can argue that the maximal ideals here correpsond to the $x_i$: the maximal ideals in $\prod_i A_i$ are generated by elements of the form $(1,1,\ldots, b_{ij},1,\ldots, 1)$ where the $b_{ij}$ generate $x_i$ (here is where we are using the two added hypothesis), i.e. that $(b_{ij})_{1\leq j\leq n_i} = m_i$ where $m_i$ is the max ideal corresponding to $x_i$ and $n_i = dim O_{X,x_i}$.

This construction is more ad-hoc (I think) vs. 1. Moreover, the geometry here is slightly more explicit in that this $Spec \; O_{X,C}$ is a finite disjoint union of local schemes, whereas in case 1, the topology is less disjoint when looking at neighborhoods of the $x_i$.

$3.$ With $X$ any scheme (no additional hypothesis as in 2), let $F_i : = O_{X,x_i}/m_i$ where $m_i$ is the maximal ideal corresponding to the closed point $x_i$. Define: $$O_{X,C}: = \prod_i F_i .$$ This construction is the most disjoint of the three in that the spectrum is now we have a finite coproduct of ''points''.

Closing remarks Presently, for me the most useful of the three is 1 and I would appreciate feedback on where the process of semi-localization has been defined. A professor that I admire very much once said (during a lecture) "from now on and for the rest of your life, every time you see something in commutative algebra, try to relate it to geometry, and vice versa" (I'm paraphrasing).

Answer 1

Your construction number (1) seems completely natural and correct to me, for the following reason:

If $A$ is a (Noetherian, commutative, unital) ring and $I$ is an ideal of $A$, then the localaization of $A$ away from $I$ is $A_I=S^{-1}A$, where

$$S=\{a\in A: a\mod{I}\mbox{ is not a zero divisor is }A/I\}$$

If $I$ is prime, then this is what you expect.

Next, suppose that $I$ is radical and that $\mathfrak{p}_i$ are the finitely many minimal prime ideals containing it (so $I$ is the intersection of the $\mathfrak{p}_i$). Then the zero divisors in the reduced ring $A/I$ are exactly the union of the minimal prime ideals of the ring, implying that $S=\bigcup_i\mathfrak{p}_i$. From this it follows that the localization $A_I$ is a semi-local ring whose maximal ideals correspond to the $\mathfrak{p}_j$. Moreover, $A_I$ will exactly by the direct limit you describe in (1), if you take $X=\mbox{Spec}A$ and $C=\{\mathfrak{p}_i\}$.