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Chapter 2: Problem 16

Noetherian Induction. Let \(X\) be a noetherian topological space, and let \(\mathscr{P}\) be a property of closed subsets of \(X .\) Assume that for any closed subset \(Y\) of \(X,\) if \(\mathscr{P}\) holds for every proper closed subset of \(Y\), then \(\mathscr{P}\) holds for \(Y\). (In particular, \(\mathscr{P}\) must hold for the empty set.) Then \(\mathscr{P}\) holds for \(X\)

Short Answer

Expert verified
The property \(\mathscr{P}\) holds for all of \(X\), confirmed by Noetherian induction. The key is understanding the property \(\mathscr{P}\) in the context of Noetherian topological spaces and applying it to all proper closed subsets of \(X\), then \(X\) itself.

Step by step solution

01

Understand the property \(\mathscr{P}\)

The property \(\mathscr{P}\) is some predicate that holds for closed subsets of \(X\). We are assuming that if \(\mathscr{P}\) holds for all proper closed subsets of \(Y\), then it holds for \(Y\) itself.
02

Apply the assumption for a closed subset

For a given closed subset \(Y\) of \(X\), assume that \(\mathscr{P}\) holds for every proper closed subset of \(Y\). This implies that \(\mathscr{P}\) should hold for \(Y\) per our assumption.
03

Prove for the space \(X\)

We can consider \(X\) itself as a closed subset of \(X\). By step 2, since \(\mathscr{P}\) holds for every proper closed subset of \(X\), \(\mathscr{P}\) must hold for \(X\) itself.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Noetherian space
Noetherian spaces are a special type of topological space characterized by their unique property with respect to descending chains of closed subsets. The defining feature of a Noetherian space is that it does not allow infinitely descending chains of closed subsets. Instead, any descending sequence of closed sets eventually stabilizes, meaning it stops changing after a finite number of steps.
This concept originates from algebra, specifically from Noetherian rings, which have a similar property for ideals. In a topological context, a space being Noetherian provides a useful framework for using induction arguments, particularly in complex settings like algebraic geometry.
  • A Noetherian space ensures that every set is compact. This simplifies many theoretical discussions about the structure and behavior of spaces.
  • It eases the use of Noetherian induction, a powerful tool for proving properties about such spaces.
Understanding Noetherian spaces lays the groundwork for many further explorations in topology and related fields.
Closed subsets
Closed subsets are a central concept in the study of topological spaces. In any given topological space, a subset is called closed if it contains all its limit points. Equivalently, a subset is closed if its complement within the space is open. This makes closed sets essential for defining continuity and other topological properties.
In the context of Noetherian spaces, closed subsets play a critical role due to their behavior under descending chains. Since a Noetherian space does not allow infinitely descending chains of closed subsets, every sequence of closed subsets should eventually become constant.
  • Closed subsets in Noetherian spaces often exhibit properties that mimic finite structures, simplifying many proofs.
  • They make inductive proofs feasible by ensuring termination points beyond which properties must hold.
Exploring closed subsets reveals many hidden facets of topological spaces and their inherent properties.
Inductive reasoning
Inductive reasoning is a powerful logical tool used to prove statements for all members of a certain class. It works by proving a base case and then showing that if the statement holds for one case, it must hold for the next. This step-by-step approach is immensely helpful, especially in mathematical contexts.
In the case of Noetherian induction, this reasoning is applied to prove properties about closed subsets in a Noetherian space. Here, you would assume a property holds for smaller subsets and use this assumption to demonstrate the property holds for progressively larger subsets or eventually for the entire space.
  • Inductive reasoning in this context ensures a step-by-step build-up, preventing logical gaps in the proof.
  • It is particularly effective in Noetherian spaces due to the invariability in chain lengths of closed subsets.
By mastering inductive reasoning, particularly its Noetherian application, you gain an efficient pathway to proving complex topological propositions.
Topological space
A topological space provides a framework for discussing continuity, compactness, connectedness, and other foundational concepts in topology. Formally, a topological space is a set endowed with a topology, which is a collection of open sets satisfying certain axioms.
Understanding topological spaces is essential for any study in topology. They are the stage where geometrical and analytical phenomena play out. They allow mathematicians to abstractly define and examine the shape, structure, and properties of various kinds of mathematical objects without needing their precise form.
A topological space must satisfy the following conditions:
  • The union of any collection of open sets is open.
  • The intersection of any finite number of open sets is open.
  • The space itself and the empty set must be open.
These axioms provide the skeleton upon which more complex topological constructs are built, allowing the exploration of advanced mathematical theories.

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Most popular questions from this chapter

Let \(S\) be a graded ring, generated by \(S_{1}\) as an \(S_{0}\) -algebra, let \(M\) be a graded \(S\) module, and let \(X=\) Proj S. (a) Show that there is a natural homomorphism \(\alpha: M \rightarrow \Gamma_{*}(\tilde{M})\) (b) Assume now that \(S_{0}=A\) is a finitely generated \(k\) -algebra for some field \(k\) that \(S_{1}\) is a finitely generated \(A\) -module, and that \(M\) is a finitely generated S-module. Show that the map \(\alpha\) is an isomorphism in all large enough degrees, i.e., there is a \(d_{0} \in \mathbf{Z}\) such that for all \(d \geqslant d_{0}, \alpha_{d}: M_{d} \rightarrow \Gamma(X, \tilde{M}(d))\) is an isomorphism. [Hint: Use the methods of the proof of \((5.19) .]\) (c) With the same hypotheses, we define an equivalence relation \(\approx\) on graded \(S\) -modules by saying \(M \approx M^{\prime}\) if there is an integer \(d\) such that \(M_{\geqslant d} \cong M_{\geqslant d^{*}}^{\prime}\) Here \(M_{\geqslant d}=\bigoplus_{n \geqslant d} M_{n} .\) We will say that a graded \(S\) -module \(M\) is quasifinitely generated if it is equivalent to a finitely generated module. Now show that the functors \(^{\sim}\) and \(\Gamma_{*}\) induce an equivalence of categories between the category of quasi-finitely generated graded \(S\) -modules modulo the equivalence relation \(\approx,\) and the category of coherent \(\mathscr{O}_{X}\) -modules.

Affine Morphisms. A morphism \(f: X \rightarrow Y\) of schemes is affine if there is an open affine cover \(\left\\{V_{i}\right\\}\) of \(Y\) such that \(f^{-1}\left(V_{i}\right)\) is affine for each \(i\) (a) Show that \(f: X \rightarrow Y\) is an affine morphism if and only if for every open affine \(V \subseteq Y, f^{-1}(V)\) is affine. [Hint: Reduce to the case \(Y\) affine, and use (Ex. 2.17).] (b) An affine morphism is quasi-compact and separated. Any finite morphism is affine (c) Let \(Y\) be a scheme, and let \(\mathscr{A}\) be a quasi-coherent sheaf of \(\emptyset_{Y}\) -algebras (i.e., a sheaf of rings which is at the same time a quasi-coherent sheaf of \(\mathscr{O}_{Y}\) -modules. Show that there is a unique scheme \(X\), and a morphism \(f: X \rightarrow Y\), such that for every open affine \(V \subseteq Y, f^{-1}(V) \cong \operatorname{Spec} \mathscr{A}(V),\) and for every inclusion \(U\) s \(V\) of open affines of \(Y\), the morphism \(f^{-1}(U)\) s \(f^{-1}(V)\) corresponds to the restriction homomorphism \(\mathscr{A}(V) \rightarrow \mathscr{A}(U)\). The scheme \(X\) is called Spec \(\mathscr{A}\). [Hint: Construct \(X\) by glueing together the schemes Spec \(\mathscr{A}(V)\) for \(V \text { open affine in } Y .]\) (d) If \(\mathscr{A}\) is a quasi-coherent \(\mathscr{V}_{\boldsymbol{r}}\) -algebra, then \(f: X=\operatorname{Spec} \mathscr{A} \rightarrow Y\) is an affine morphism, and \(\mathscr{A} \cong f_{*} \mathscr{O}_{X} .\) Conversely, if \(f: X \rightarrow Y\) is an affine morphism, then \(\mathscr{A}=f_{*} \mathscr{O}_{X}\) is a quasi-coherent sheaf of \(\mathscr{O}_{Y^{-}}\) algebras, and \(X \cong\) Spec \(\mathscr{A}\) (e) Let \(f: X \rightarrow Y\) be an affine morphism, and let \(\mathscr{A}=f_{*} O_{X} .\) Show that \(f_{*}\) induces an equivalence of categories from the category of quasi-coherent \(\mathscr{O}_{X}\) -modules to the category of quasi-coherent \(\mathscr{A}\) -modules (i.e., quasi-coherent \(\mathcal{O}_{Y}\) -modules having a structure of \(\mathscr{A}\) -module). [Hint: For any quasi- coherent \(\mathscr{A}\) -module \(\mathscr{M},\) construct a quasi-coherent \(\mathscr{O}_{X}\) -module \(\mathscr{M},\) and show that the functors \(f_{*}\) and \(^{\sim}\) are inverse to each other.

Adjoint Property of \(f^{-1}\). Let \(f: X \rightarrow Y\) be a continuous map of topological spaces. Show that for any sheaf \(\mathscr{F}\) on \(X\) there is a natural \(\operatorname{map} f^{-1} f_{*} \mathscr{F} \rightarrow \mathscr{F},\) and for any sheaf \(\mathscr{G}\) on \(Y\) there is a natural \(\operatorname{map} \mathscr{G} \rightarrow f_{*} f^{-1} \mathscr{G}\). Use these maps to show that there is a natural bijection of sets, for any sheaves \(\overline{\mathscr{F}}\) on \(X\) and \(\mathscr{S}\) on \(Y\) $$\operatorname{Hom}_{x}\left(f^{-1} \mathscr{G}, \mathscr{F}\right)=\operatorname{Hom}_{Y}\left(\mathscr{G}, f_{*} \mathscr{F}\right)$$ Hence we say that \(f^{-1}\) is a left adjoint of \(f_{*},\) and that \(f_{*}\) is a right adjoint of \(f^{-1}\)

Let \(\varphi: \mathbf{P}_{h}^{n} \rightarrow \mathbf{P}_{h}^{m}\) be a morphism. Then: (a) either \(\varphi\left(\mathbf{P}^{n}\right)=p t\) or \(m \geqslant n\) ana \(\operatorname{dim} \varphi\left(\mathbf{P}^{n}\right)=n\) (b) in the second case. \(\varphi\) can be obtained as the composition of (1) a \(d\) -uple embedding \(\mathbf{P}^{n} \rightarrow \mathbf{P}^{\prime}\) for a uniquely determined \(d \geqslant 1,(2)\) a linear projection \(\mathbf{P}^{\backslash}-\mathrm{L} \rightarrow \mathbf{P}^{m},\) and (3) an automorphism of \(\mathbf{P}^{m} .\) Also, \(\varphi\) has finite fibres.

If \(X\) is a scheme of finite type over a field, show that the closed points of \(X\) are dense. Give an example to show that this is not true for arbitrary schemes.
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