# Meaning of shut factors of a system

This is an inquiry in Liu's publication.

Allow $X$ be a quasi-compact system. Show that $X$ has a shut factor.

Well I'm incapable to do this inquiry, so any kind of aid would certainly be valued. This inquiry additionally makes me interested to find out about the meaning/use of shut factors of a system as a whole - by that I suggest a system which is not an algebraic variety/local system over an area, which has a geometric definition. Many thanks!

Sorry for uploading once more on this old string yet I would certainly such as to recognize what is incorrect, if something, with the adhering to, due to the fact that I assume it needs to jobs.

We have an open affine covering $X = \bigcup_{i} U_{i}$ with $U_{i} = \mbox{Spec}(A_{i})$. Allow $\xi_{1}$ be a shut factor in $U_{1}$ (simply take a topmost perfect). If $\xi_{1}$ is still enclosed $X$ we are done. Or else, take $\xi_{2} \neq \xi_{1}$ with $\xi_{2} \in \overline{\left\{\xi_{1}\right\}}$. Therefore $\xi_{2} \in U_{i}$ for some $i \neq 1$, claim $i = 2$. After that duplicate the thinking over once more. The procedure has to stop given that there are just finitely several $U_{i}$ therefore $X$ has a shut factor.

Kevin Lin's solution pertaining to the definition of shut factors is fairly practical, especialy in case when the system concerned underlies a timeless selection. I intend to add some added statements and also instances for thinking of even more basic systems.

Below are some tautological statements : remember that a factor x in a system X is called a field of expertise of y if x hinges on the Zariski closure of y (and also y is called a generalization of x). So tautologically, a shut factors is one that can not be specialized any kind of more (equally as a common factor can not be generalised any kind of more). What does field of expertise actually suggest : ring in theory, it suggests taking the photo under a homomorphism ; so if p and also q are prime perfects of a ring A, after that q is a field of expertise of p in Spec A if and also just if q has p, i.e. if A/p surjects onto A/q. It is probably best to consider an instance : claim A is C [x, y ],. p is the prime excellent gen would certainly by (x - 1) and also q is the prime (in fact topmost perfect) gen would certainly by (x - 1, y). After that in A/p, we have actually "specialized" the value of x to equivalent 1 (due to the fact that we have actually proclaimed x - 1 = 0) yet y is still a free variable. When we pass to the more ratio A/q, we have actually specialized both x and also y : x is specialized to 1 and also y is specialized to 0. Now, we can not specialize anymore ; practically, this is due to the fact that q is a topmost perfect of A,. so a shut factor of Spec A ; with ease, it is due to the fact that both x and also y have actually currently both been "specialized" to real numbers, therefore we can not specialize any kind of better.

Yet intend currently we set B = Z [x, y ], and also take p and also q to be the very same, i.e. gen would certainly by x - 1 and also by (x - 1, y) specifically. After that q is *not * topmost ; there is even more ability for field of expertise.
Just how is this? Well, x and also y are currently taking values in Z (as opposed to the area C) therefore we can additionally lower both x and also y modulo some prime, claim 5 ; this offers a prime excellent r = (x - 1, y,5) in B having q. Currently r *is * topmost, therefore we are done specializing.

So if you have a system that is limited type over Z, the shut factors will certainly represent "real factors", in Kevin's terminology, yet specified over limited areas. The factors of the system whose works with are integers, claim, will certainly *not * be shut. One has the selection of assuming them of them as "real factors" which however can be specialized better by lowering modulo tops, or as subvarieties as opposed to "real factors", by recognizing them.
with their Zariski closures (for an image of this, see the illustration of Mumford that Kevin web links to).

Zorn's lemma indicates there is a marginal nonempty shut set $F \subset X$ without correct shut parts (due to the fact that the shut collections have the limited junction building because seemingly - density). It suffices to locate a shut factor in $F$. Currently $F$ remains in itself a system, and also it has an open part $U=\mathrm{Spec}\, A$ for $A$ a ring. This have to be every one of $F$ by minimality. A topmost perfect in $A$ offers a shut factor in $F$, therefore in $X$.

There are individuals below that can offer a better response to your various other inquiry, so I'll leave it.

Closed factors need to be taken being "real factors", whereas non - shut factors can represent all type of various points : subvarieties, "fat" or "blurry" factors, common factors, etc You could be curious about reviewing this blog post concerning Mumford's illustration of $\operatorname{Spec} \mathbb{Z}[x]$.

One feasible means to warrant the case that shut factors are the "real factors" is the reality that if we have, as an example, a smooth selection over $\mathbb{C}$, after that its analytification will certainly be an intricate manifold. The shut factors of the previous will certainly after that match specifically to the factors of the last.

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