On semi-local schemes

2010-05-31T11:03:08Z

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 by Matthew Morrow for On semi-local schemes

2010-05-31T11:23:50Z

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 $C=\{\mathfrak{p}_i\}$ .

On semi-local schemes - MathOverflow

On semi-local schemes

Answer by Matthew Morrow for On semi-local schemes