Encouraging Example for Algebraic Geometry/Scheme Theory

I remain in the process of attempting to find out algebraic geometry using systems and also am asking yourself if there are straightforward encouraging instances of why you would certainly intend to take into consideration these frameworks.

I assume my largest concern is the adhering to: I recognize (and also actually like) the suggestion of passing from a room to features on a room. In passing from $k^n$ to $R:=k[x_1,\ldots,x_n]$, we might recoup the factors by considering the topmost suggestions of $R$. Yet why take into consideration $\operatorname{Spec} R$ as opposed to $\operatorname{MaxSpec} R$? Why is it handy to have non-closed factors that do not have an analog to factors in $k^n$? On a wikipedia write-up, it stated that the Italian college made use of a (obscure) idea of a common indicate confirm points. Exists a (reasonably) straightforward instance where we can see the energy of non-closed factors?

2019-05-06 01:35:11
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Answers: 4

I assume I can at the very least encourage you that it's an excellent suggestion to collaborate with non - lowered rings, which aren't actually recorded by their topmost perfects. The suggestion is that you get even more functors. Equally as $\text{Hom}(-, \mathbb{C})$ is the functor which sends out a finitely - created domain name over $\mathbb{C}$ to its set of $\mathbb{C}$ - factors, it ends up that $\text{Hom}(-, \mathbb{C}[x]/x^2)$ sends out a finitely - created domain name over $\mathbb{C}$ to its set of $\mathbb{C}$ - factors along with a selection of tangent vector. Actually this is one means to specify the Zariski tangent room. So, on the system side, it's an excellent suggestion to research morphisms from $\text{Spec } \mathbb{C}[x]/x^2$ to your system, and also you can not do this if you recognize $\mathbb{C}[x]/x^2$ with its set of $\mathbb{C}$ - factors. (You can not also do this if you recognize $\mathbb{C}[x]/x^2$ with its prime perfects ; you actually require the whole framework sheaf.)

(An actually wonderful application : an algebraic definition of the Lie algebra of an algebraic team.)

2019-05-08 12:48:05

See my solution here for a quick conversation of just how factors that are enclosed one optic (sensible remedies to a Diophantine formula, which are shut factors on the selection over $\mathbb Q$ affixed to the Diophantine formula) come to be non - enclosed an additional optic (when we clear and also consider the Diophantine formula as specifying a system over $\mathbb Z$).

In regards to rings (and also attaching to Qiaochu's solution), under the all-natural map $\mathbb Z[x_1,...,x_n] \to \mathbb Q[x_1,...,x_n]$, the preimage of topmost perfects are prime, yet not topmost.

These instances might offer impact that non - shut factors are crucial in math scenarios, yet in fact that is not the instance. The ring $\mathbb C[t]$ acts just like $\mathbb Z$, therefore one can have the very same conversation with $\mathbb Z$ and also $\mathbb Q$ changed by $\mathbb C[t]$ and also $\mathbb C(t)$. Why would certainly one do this?

Well, intend you have a formula (like $y^2 = x^3 + t$) which you intend to research, where you consider $t$ as a parameter. To research the common practices of this formula, you can consider it as a selection over $\mathbb C(t)$. Yet intend you intend to research the geometry for one certain value of $t_0$ of $t$. After that you require to pass from $\mathbb C(t)$ to $\mathbb C[t]$, to make sure that you can use the homomorphism $\mathbb C[t] \to \mathbb C$ offered by $t \mapsto t_0$ (field of expertise at $t_0$). This is entirely similar to the scenario taken into consideration in my connected solution, of taking indispensable remedies to a a Diophantine formula and afterwards lowering them mod $p$.

What is the result? Primarily, any kind of significant research of selections in family members (whether math family members, i.e. systems over $\mathbb Z$, or geometric family members, i.e. parameterized family members of selections) calls for system - logical strategies and also the factor to consider of non - shut factors.

(Of training course, significant such researches were made by the Italian geometers, by Lefschetz, by Igusa, by Shimura, and also by several others prior to Grothendieck's development of systems, yet the entire factor of systems is to clarify what came in the past and also to offer a specific and also practical concept that incorporates every one of the contexts taken into consideration in the "old days", and also is additionally extra organized and also extra effective than the older strategies.)

2019-05-08 10:58:31

To start, the discussion at Sbseminar has remarks from great deals of individuals that in fact recognize algebraic geometry, and also if anything I claim negates something they claim, please trust fund them and also not me.

One factor is that you shed the functoriality of $Spec$ if you adhere to $MaxSpec$ : the inverted photo of a topmost perfect is not always topmost. However, if you adhere to systems of limited type over an area, this holds true (it's primarily a variation of the Nullstellensatz). Specifically, in Serre's FAC paper he specifies a "selection" by gluing with each other normal affine algebraic embed in the feeling of timeless algebraic geometry. Yet this is much less basic. One all-natural instance of a system which is not of limited type over an area is merely $Spec \mathbb{Z}$. After that offered a system $X$ over this (well, unquestionably every system $X$ is a system over $Spec \mathbb{Z}$ in an approved means), the fibers at the non - shut factor of $Spec \mathbb{Z}$ is still intriguing and also primarily total up to researching polynomial formulas over $\mathbb{Q}$ (when $X \to \mathbb{Z}$ is of limited type).

As a (straightforward) instance of just how common factors can be made use of, one can confirm that a systematic sheaf on a noetherian indispensable system is free on a thick open part. Why? Due to the fact that it has to be free at the common factor (given that the neighborhood ring there is an area), and also it is a basic reality that 2 systematic sheaves whose stalks are isomorphic are isomorphic in an area. (This holds true in fact for sheaves of limited discussion over a ringed room.)

2019-05-08 10:52:28

Here's a straightforward junction logical instance.

Take the junction of the line $y=0$ and also the parabola $y=x^2$. Typically, the junction is a factor. Yet note that there is even more to the junction than simply the factor ; there is the reality that both contours are tangent then. System - in theory, the junction is $\operatorname{Spec} k[x,y]/(y,y-x^2) \cong \operatorname{Spec} k[x]/(x^2)$. This mirrors the tangency. If the junction were transverse, after that the system - logical junction would certainly have been simply $\operatorname{Spec} k$.

Greater order tangencies can be seen in the system - logical junction too ; as an example, repeat this workout with $y=x^3$ instead of $y=x^2$.

2019-05-08 10:08:05