A function that is a one-to-one mapping from CN to CN is called a universal function. This theorem is known as the Abroton-Leak theorem. The claim itself is elementary, but the proof is not so straightforward. A famous proof of this theorem uses model theory. For example, refer to Tanaka Ikko's Mathematical Foundations and the Snow Removal Theorem 6.4.8. I will omit the details, but in that proof, the compactness theorem is used to reduce the statement about algebraic groups of characteristic zero to a statement about algebraic groups of positive characteristic. Then, it is connected to the proposition that if a finite set is injective, it is universal. The lower Deglon-Leak significantly generalizes this proposition in Volume 4 of EGA, proving the following proposition using large-scale scientific methods: Let S be a scheme and X be a finite S-scheme. In this case, the rational points of X over S are universal. The term rational will be explained later, and I would like to prove the Abroton-Leak theorem following the proof of this proposition.

First, I will introduce the Jacobson scheme, which is a class of schemes. The general theory of Jacobson schemes can be found in Volume 4 of EGA (Éléments de Géométrie Algébrique), so if you are interested, please refer to that. To define a Jacobson scheme, we need the concept of a Jacobson space, so I will introduce that. A topological space X is called a Jacobson space if for any irreducible closed subset Z of X, the condition Z = Z^qcl(X) holds. Here, cl(X) represents the set of all closed subsets of X. In a figurative sense, this means that the closed sets are densely packed in X. For example, the spectrum of a ring or the low-dimensional spaces in algebraic geometry are Jacobson spaces. 

Now, consider a local ring R that is not maximal, and let X = Spec(R). In this case, X is not a Jacobson space because cl(X) consists of the closed sets that come from the maximal ideals. Jacobson spaces have the following properties: Let X be a topological space, and when we give cl(X) the relative topology, the following conditions are equivalent: 1. X is a Jacobson space. 2. For any irreducible closed subset Z of X, the closure of Z in cl(X) is a constructible subset of X.

I will confirm the key terms related to the front-end of the clx technical feasible subset. The technical feasible subset refers to the collection of finite sets U1 to UN and V1 to VN after the EA surgery, which can be expressed as the union of the intersection of the complements of UI and VI. Typically, the available subsets are often referred to simply as feasible subsets, but here I will adhere strictly to the terminology. The definition of feasible subsets in EGA is a subset that can similarly be expressed using finite compact sets from U1 to UN and V1 to V. There is likely no established term for retro-compact and post-operative, but for now, I will translate it as simple compact. A subset Z of a topological space X is considered simple compact if, for every compact cover U of 2, the intersection Z ∩ U is compact. In data spaces, all subsets are simple compact, so technical feasible subsets and feasible subsets become equivalent concepts. In the following discussion, since I will primarily deal with nearly all Neater spaces, I will refer to all technical feasible subsets as feasible subsets.

There is no problem even if we receive it. Now, let's return to the main topic. The above proposition states that the topological space X is a Jacobson space if and only if the set of all constructible subsets of X and CL X correspond to each other. From this proposition, we can easily understand the following: If the constructible subset Z of the Jacobson space X includes all closed sets of X, then X = Z. In fact, since X_cl X = Z_cl X = cl X, we can conclude that X = Z from the injectivity of the correspondence Z ↔ Z_cl X. We should remember this result as we will use it later. By the way, in the above proposition, I only mentioned the correspondence of constructible subsets, but actually, a similar statement can be made regarding open sets and closed sets. However, I will omit those as they are not necessary for this discussion. Now that we have introduced the concept of a Jacobson space, let's define a Jacobson scheme using this concept. Additionally, I will introduce the concept of a Jacobson ring. A scheme X is said to be a Jacobson scheme if the underlying topological space X is a Jacobson space.

ジャコブソン間であるということは、ステックイがジャコブソンスキームであるということを明確に示しています。例は先ほど挙げたもので、差し当たり十分でしょう。ジャコブソン間に関して、次の非常に重要な命題が成立します。Bがジャコブソン艦であり、Aが有限性整B台数であるとき、Aもジャコブソン感であるということです。この性質は非常に嬉しいですね。証明は省略しますが、特にZ上あるいは体上の有限生生大数はジャコブソン感であることが分かります。

以上でジャコブソンスキームの導入は終わりにしますが、ここでジャコブソンスキームがどのようなものなのかという概念的な話をします。E4の3観10章の初めに次のように書かれています。閉店の集合が至るところ中立であるという事実によって、多くの証明において閉店以外の点を考えなくて済むようになります。そして再び台数多様体、すなわち我々の言葉で言うところの体上の大数的善スキームの

There was a connection between the words of schemes and the words of manifolds or the spaces of sheaves, as mentioned earlier. In Jacobson schemes, closed points carry a lot of information. Therefore, it often occurs that in Jacobson schemes, one can focus on closed points. This is very beneficial because, for example, even when considering the affine space An in the context of a large number of algebraic varieties, the existence of non-closed points complicates the discussion. However, if one can avoid non-closed points, it becomes akin to handling coordinate spaces TN, making the discussion much simpler. 

Finally, let's look at an example where this can be felt more intuitively. This example will be used later, so I will define the concept of a Radicel scheme to describe it. Regarding schemes, the following four conditions are equivalent, and when these are satisfied, F is said to be a Radicel scheme. This is in relation to the sheaf Z from Y, which provides the defined transformation of F.

Prime is a set of vehicles that is injective to the large number battalion K. From Homset KX to Homset KY, it is injective. The set of vehicles is also injective, and KX is a quasi-proportional extension of KFX with respect to X. Here, KX represents the upper body at x. The typhoon that emerges is universal for all vehicles. I will omit the proof of this homogeneity. For example, please refer to Proposition 1.7.1 of a certain homology theory. Here, it can be shown that a certain formal function, namely the one we have been considering, is defined by F, which is a desil. In fact, we can demonstrate this. It suffices to show any one of the four conditions, but here we will demonstrate the fourth, namely that the typhoon defined by F is universal for all vehicles. First, let's confirm that this proposition seems plausible based on a non-rigorous argument. Let’s temporarily forget about schemes and consider a simple set CN. In this case, the proposition is evident. Here, the typhoon defined by the set...

This is determined by the diagram of the set of swords as shown above. The fiber space determined by the diagram in the lower right becomes a set like the one above. However, due to the injectivity of F, this can be identified with the diagonal set of CN under CN. Therefore, the diagonal image of the fiber is the entire fiber. Now, let's consider the case of schemes. In this case, it is not as simple as in the previous case. This is because the set anc under ENC includes points that are not closed, so we must also show that these points are included in the image of the fiber, which is surprisingly difficult. However, using the property of the Jacobson space simplifies the discussion considerably. If we recall the theorem introduced earlier, the finite type over the field is between Jacobson spaces, so now anc and anc under anc are Jacobson schemes. Since delta F is a finite morphism between Neta schemes, by the theorem of Shibale, delta F anc is a constructible subset of anc under anc. However, in order to show that de F ENC = anc under ENC, we must demonstrate that all closed points of an anc under anc are included in the image of the fiber.

By demonstrating this, we have eliminated the need to consider points other than the closure. Of course, even though we refer to it as a closure, it does not correspond to points in the usual sense, so we cannot say it is exactly the same as in the case of sets. However, if we consider points more generally, almost the same argument holds. Here, the fact that C is a large number of soldiers comes into play. In this way, we can demonstrate the full property of delta F, that is, that F is a Radichellian object. From here, we will think about reducing the scheme anc to a more manageable scheme. Specifically, we will utilize the limits of schemes to reduce our discussion to finite-type schemes over Z. The arguments using limits that will be introduced below are commonly used in the theory of large number sciences. First, let me present a fundamental proposition concerning limits of schemes. Let S0 be a scheme, and let A and lambda be pure connections. Let A = Comm A lambda. At this point, Spec A is the limit of the inverse system Spec A lambda in the context of schemes. This proposition applies to C.

Using this, we can particularly say the following: First, we fix one subalgebra A0 over Z that contains all coefficients of F without finite generation. To construct this, we can set A0 = ZA1, ..., an, where A1 to an are the coefficients of F. We denote the collection of all subalgebras Aλ that contain A0 as those generated over Z without finite generation. Then, C is the colimit of the Aλ's, which means it is a co-limit of the Aλ's. Similarly, the polynomial rings over C are the colimits of the polynomial rings over Aλ's, so these spectra will be the limits of the spectra of the Aλ's and the polynomial rings over Aλ's. Note that each polynomial ring over Aλ is a finitely generated Z-algebra, so they are Jacobson rings. In this situation, for all λ, the formal function fλ corresponding to F is determined over Aλ. Moreover, these limit functions Fλ converge to F. Naturally, Fλ is finitely presented. Additionally, both F and the second Fλ can be obtained as change of basis from F0. Now, with the discussion so far, let's rewrite the scheme's vehicle in terms of limits.

ここで重要になるのが次の命題です。すなわち、極限の車が何かしらの性質を持つことと、十分大きいラムダで同様の性質を持つことは同値です。Eには全車やル社以外にも、極限に関して同様に振る舞う多くの性質が列挙されていますが、今回は使わないので割愛します。この命題によって次のことが分かります。一つに、アルラムダが存在して、Fラムダはラディシェルであるということが分かります。これは先ほど示した通り、ケフがラディシェルであったからです。もう一つに、エフが全車であることを示すためには、エフラムダが全車であることを示せば十分であるということが分かります。さらに、anラムダはZ上有限型なので、ジャコブソンスキームであり、ラムダが全車であることを示すためには、ラムダ ANAラムダがANAラムダの全ての閉点を含むことを示せば十分です。以上のことから問題は、エフラムダ、すなわちZ上有限型なスキームの閉点に関する話に帰

We were able to reduce it to a finite-type scheme. Here, let’s review some general properties regarding finite-type schemes over Z. For a finite-type Z scheme Y, the following holds: 1. Y being closed is equivalent to its residue field Y being a finite field. For a prime number P and a generator D, the scheme KY is a finite extension of the field FP, and the degree of this extension is a divisor of D. The set KPD consists of those Y such that the aforementioned conditions are satisfied, and it is a finite set. From the main assertion, the source of KPD is the closed points of Y, and we should also note that the closed points of Y are included in KPD with respect to RKD. In other words, the collection of all closed points of Y forms the union of all keys corresponding to D in TTD. Now, let’s prove the proposition. First, if Y is closed, then PiY is also closed, as can be seen from the following general lemma: Let X be a Noetherian Jacobson scheme, Y be a Noetherian scheme, and f be a finite morphism from Y to X. Then, F maps closed points to closed points. Now, since Spec Z is a Noetherian Jacobson scheme, Y is a Noetherian scheme, and F is a finite morphism, we have that...

The lemma satisfies the conditions, and if Y is closed, then it can be concluded that Y is also closed. This lemma will be proven later. Next, when K is a finite field, the neighborhood will also be closed. We will show this by contradiction. Let's assume that the neighborhood is not closed, meaning it is zero. In this case, since there exists KY from the field map K0=9, K cannot be a finite field. Therefore, PiY must be closed. From this discussion, we can see that to demonstrate the claim in 1, it is permissible to add the condition that PiY is closed. At this point, considering the fibers above Y, the desired property can be derived from the following proposition. Let K be a pair and Y be a finite-type K-scheme. In this case, the fact that Y is closed is equivalent to KY being a finite K-algebra. Finally, we will prove the lemma we used earlier. Let Y be the closure of Y. In this case, the set consisting of points from Y is a quasi-compact subset of Y, thus a constructible subset. According to the theorem, the set consisting of FY=FY will be a quasi-constructible subset of x. If we assume that FY is not closed, then clxFY is empty.

However, since X is a Jacobson scheme, ZMAX to ZCclx gives the front of the possible subsets of x and the possible subsets of clx, so FY becomes an empty set. This is clearly a contradiction. Therefore, we can conclude that FY is closed. By the way, in this proof, we assumed Noetherian and finite-type conditions in order to use Shilov's theorem, but in fact, a similar fact can be demonstrated more broadly when X is a Jacobson scheme and F is a locally finite-type object. For the proof, please refer to Eeg Volume 4, Section 10.4.7. Now, let's move on to the second claim. First, since Y is an affine scheme of finite type, we can assume from the beginning that Y is an affine scheme. Therefore, we can express Y as a finite type Z-algebra C, say y = c. Let T1 to TN be the generators of C over Z. Considering Y as the limit of TPD, KY is a finite type extension of FP, and since the degree of the extension is a divisor of D, KY is isomorphic to a subfield of FPD. Thus, there exists a morphism from KY to FPD. For each YintPD, it is naturally induced from C to KY and from the previous KY.

The image file corresponding to fpdf is determined from the vehicle C synthesized with fpd. Here, there are multiple ways to take fpd from KY, but we will choose one and fix it with respect to nucleus Y. At this time, the mapping corresponding to the fine inverse ideal with respect to the time is a retrograde of the mapping, so it is one-to-one. The fpd from observer C is determined by the destination from the generation limit T1 to TN of C, but since fpd is a finite set, there can only be a finite number of candidates for the destination. Therefore, the finiteness of Hom(fpd) is evident, and since Hom(C, fpd) exists from the gratitude of gpd, it can be concluded that gpd is also a finite set. Thus, the proof is complete. Furthermore, in the proof of ega, it is written that Hom(C, fpd) is a front-end entity from the gradual mapping TPD, but this seems to be incorrect. For example, if C = F4, Q = 2, and b = 2, then T2 2 contains the only point of the stack F4, which is a singleton, but Hom(F4, F4) includes two entities, namely the higher entity and OM2 squared. However, as shown here, there exists a one-to-one mapping to Hom(C, fpd).

There are no particular problems, just one more breath. I will reconsider the original idea of the Shifram. Therefore, DPD will be a finite subset of cln lambda. The following proposition holds regarding DPD and F lambda: For a prime number P and an integer D, Ftkd is included in ttd. This proposition is almost obvious. For Zinttd, the upper residue field K lambda Z induces KZ. Thus, we can see that K lambda Z is isomorphic to a subfield of KZ. Since Z is the source of TPD, the degree of the extension of KZ over FP is a divisor of D. Furthermore, from the fact that K lambda Z is isomorphic to a subfield of KZ, we can conclude that the degree of the extension of KF lambda Z is also a divisor of D. It can be shown that the lower field F lambda Z is included in TPD. With this, we have completed the proof. The proof of the Abtani-Leak theorem is almost finished. From the previous proposition, it can be seen that the restriction of lambda to TPD is a mapping from TPD to itself. Since lambda was originally assumed to be a Radichelle field, the restriction of lambda to TPD is also injective.

TPD is a finite set, so this applies to all vehicles. The intersection of all KPDs is the entire set of closed subsets of ANA lambda. Therefore, it can be understood that the image of lambda includes all closed subsets of ANA lambda. As seen in the Jacobson scheme, this shows that F lambda is a finite set. Thus, by the proposition concerning the limit of vehicles, we have shown that F is a finite set. This concludes the proof, but I would like to add a few remarks. First, the previous discussion can be applied not only to ANC but also to finite-type schemes over two-dimensional algebraic spaces. In particular, a similar theorem holds for algebraic varieties over algebraically closed fields. Additionally, the proposition introduced at the beginning, which is written in EGA, is more general than the proposition shown here. However, by considering each fiber of S, it can be reduced to the case where S is finite, allowing for a similar argument using limits. Finally, I would like to refer to some literature; the cohomology theory discussed can be found in Chapter 1.

The PDF has been published, and the extreme discussions introduced in the video are summarized in section 1.10 of the same document, so please refer to it if you are interested. This video is basically created based on volume 3 of EGA4, which can be viewed online. Additionally, these slides are available on the author's website. That’s all. Thank you for watching.

Ax–Grothendieck の定理

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