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Chapter 2: Problem 8

Let \(\mathscr{P}\) be a property of morphisms of schemes such that: (a) a closed immersion has \(\mathscr{P}\) (b) a composition of two morphisms having \(\mathscr{P}\) has \(\mathscr{P}\) (c) \(\mathscr{P}\) is stable under base extension. Then show that: (d) a product of morphisms having \(\mathscr{P}\) has \(\mathscr{P}\) (e) if \(f: X \rightarrow Y\) and \(g: Y \rightarrow Z\) are two morphisms, and if \(g\) fhas \(\mathscr{P}\) and \(g\) is separated, then \(f\) has \(\mathscr{P}\) (f) If \(f: X \rightarrow Y\) has \(\mathscr{P},\) then \(f_{\text {icd }}: X_{\text {idd }} \rightarrow Y_{\text {tad }}\) has \(\mathscr{P}\) \([\text {Hint}:\) For (e) consider the graph morphism \(\Gamma_{f}: X \rightarrow X \times_{z} Y\) and note that it is obtained by base extension from the diagonal morphism \(\left.\Delta: Y \rightarrow Y \times_{Z} Y .\right]\)

Short Answer

Expert verified
To summarize, the property \(\mathscr{P}\) has been verified for a product of morphisms, for a morphism \(f\) under given conditions, and for an induced function \(f_{\text{icd}}\) derived from a morphism \(f\) which satisfies property \(\mathscr{P}\) by using the given properties of morphisms and algebraic geometry concepts of closed immersions, compositions, and base extensions.

Step by step solution

01

Understanding Properties

It should be first clarified that \(\mathscr{P}\) is a property of morphisms that gets preserved under certain transformations. We need to use facts that closed immersions, compositions and base extensions all preserve property \(\mathscr{P}\).
02

Demonstrate Property for Product

To show that a 'product' of morphisms has the property \(\mathscr{P}\), consider two morphisms \(f: A \rightarrow B\) and \(g: C \rightarrow D\) both have property \(\mathscr{P}\). The product morphism \(f \times g: A \times C \rightarrow B \times D\) can be written as the composition \( (g \circ pr_2) \circ (f \circ pr_1)\) where \(pr_1: A \times C \rightarrow A\) and \(pr_2: A \times C \rightarrow C\) are the natural projections and therefore closed immersions. From the given property, \(\mathscr{P}\) is preserved under closed immersions and compositions, hence \(f \times g\) should also have property \(\mathscr{P}\). This validates (d).
03

Verification of Property for f in given condition

To demonstrate (e), given \(f: X \rightarrow Y, g: Y \rightarrow Z\), and \(g\) has \(\mathscr{P}\) and is separated, consider the graph morphism \(\Gamma_f : X \rightarrow X \times_{Z} Y\), which is defined by the diagram \(X \stackrel{\Gamma_f}{\rightarrow} X \times_{Z} Y \stackrel{pr_Y}{\rightarrow} Y\) where \(pr_Y\) is a base extension of the diagonal morphism \(\Delta: Y \rightarrow Y \times_{Z} Y\). From the hint, the graph morphism \(\Gamma_f\) is retrieved by base extension from \(\Delta\), which is a closed immersion as \(\Delta\) is separated. As \(\mathscr{P}\) is preserved under base extensions and closed immersions, we conclude that \(\Gamma_f\) has \(\mathscr{P}\). This means \(\mathscr{P}\) is also preserved by \(f\), which completes (e).
04

Induced Property for Function

To verify property (f), consider a morphism \(f\) that satisfies property \(\mathscr{P}\). The morphism \(f_{\text{icd}}\) is then obtained as a base extension of \(f\) which, due to property (c), preserves \(\mathscr{P}\). Hence, \(f_{\text{icd}}\) has property \(\mathscr{P}\), completing the proof for (f).

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Key Concepts

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

Scheme Theory
Scheme theory is a fundamental part of algebraic geometry, expanding on the concepts established by classical algebraic geometry. A scheme is a space that provides a unified approach to study geometric and algebraic structures. It's built upon the idea of 'spec' of a ring, which represents all of its prime ideals in a topological space.

Within this framework, morphisms are functions that map one scheme to another. They are analogous to functions in calculus but operate within the complex landscape of algebraic structures. The property \(\mathscr{P}\) in our exercise is a characteristic that these morphisms possess, and certain operations such as closed immersions, composition, and base extensions can preserve this property.

Understanding how these morphisms work and how they interact through these operations is key to grasping the broader implications of scheme theory, and specifically, in verifying properties like \(\mathscr{P}\) under various transformations.
Closed Immersion
Closed immersion is a specific type of morphism in scheme theory. Informally, it can be thought of as an 'inclusion' of one subscheme into another, similar to how one might embed a line into a plane in classical geometry. Formally, for schemes \( X \) and \( Y \) a morphism \( i: X \rightarrow Y \) is a closed immersion if \( i \) induces a homeomorphism onto its image and the induced map on the structure sheaves is surjective.

A vital attribute of a closed immersion, as stated in the given exercise, is that it preserves certain morphic properties, such as \(\mathscr{P}\). This concept is critical to proving that \(\mathscr{P}\) is an inherent characteristic of certain scheme morphisms, particularly in the context of product morphisms and the graph morphism in our exercise.
Base Extension
Base extension is another important procedure in algebraic geometry, particularly within the realm of scheme theory. It allows us to 'pull back' schemes and morphisms over a new base scheme. In simpler terms, it's a way of adapting a given structure to a new context or 'base.' For a morphism \( f: X \rightarrow Y \) and a morphism \( g: Y' \rightarrow Y \) the base extension of \( X \) by \( g \) is \( X \times_Y Y' \) which results in a new scheme over \( Y' \).

The significance of base extension in our exercise is its ability to preserve the property \(\mathscr{P}\). When a property is stable under base extension, it means that the property is retained even when the scheme or the morphism is transformed through this process. This conservation enables mathematicians to project properties across different schemes, aiding in the proof of complex results.
Graph Morphism
Graph morphism plays a crucial role in understanding the behavior of functions between schemes. It's a geometric representation of a morphism \( f: X \rightarrow Y \) as a sub-scheme of \( X \times_Z Y \) where \( Z \) is a common base scheme. In many ways, it's similar to the plot of a function in calculus, except that in this setting, it conveys information about a morphism of schemes.

In the context of our exercise, the graph morphism \( \Gamma_f: X \rightarrow X \times_Z Y \) constructed via the graph of \( f \) is a powerful tool. By using the graph morphism and its properties, we can infer characteristics about the original morphism \( f \) based on the traits of \( \Gamma_f \), such as its nature as a closed immersion and its preservation of the property \(\mathscr{P}\) under base extensions, giving insight into both the composition of morphisms and the verifying process for property \(\mathscr{P}\).

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Most popular questions from this chapter

The Grothendieck Group \(K(X) .\) Let \(X\) be a noetherian scheme. We define \(K(X)\) to be the quotient of the free abelian group generated by all the coherent sheaves on \(X,\) by the subgroup generated by all expressions \(\mathscr{F}-\mathscr{F}^{\prime}-\mathscr{F}^{\prime \prime},\) whenever there is an exact sequence \(0 \rightarrow \mathscr{F}^{\prime} \rightarrow \mathscr{F} \rightarrow \mathscr{F}^{\prime \prime} \rightarrow 0\) of coherent sheaves on \(X\) If \(\mathscr{F}\) is a coherent sheaf, we denote by \(\gamma(\mathscr{F})\) its image in \(K(X)\) (a) If \(X=\mathbf{A}_{k}^{1},\) then \(K(X) \cong \mathbf{Z}\) (b) If \(X\) is any integral scheme, and \(\mathscr{F}\) a coherent sheaf, we define the rank of \(\mathscr{F}\) to be \(\operatorname{dim}_{\kappa} \mathscr{F}_{\xi},\) where \(\xi\) is the generic point of \(X,\) and \(K=\mathscr{C}_{\xi}\) is the function field of \(X .\) Show that the rank function defines a surjective homomorphism \(\operatorname{rank}: K(X) \rightarrow \mathbf{Z}\) (c) If \(Y\) is a closed subscheme of \(X\), there is an exact sequence \\[ K(Y) \rightarrow K(X) \rightarrow K(X-Y) \rightarrow 0 \\] where the first map is extension by zero, and the second map is restriction. \([\text {Hint}: \text { For exactness in the middle, show that if } \mathscr{H} \text { is a coherent sheaf on } X\) whose support is contained in \(Y\), then there is a finite filtration \(\overline{\mathscr{H}}=\overline{\mathscr{H}}_{0} \supseteq\) \(\mathscr{H}_{1} \supseteq \ldots \supseteq \cdot \overline{\mathscr{H}}_{n}=0,\) such that each \(\mathscr{H}_{i} / \mathscr{H}_{i+1}\) is an \((\mathrm{r}-\) module. To show surjectivity on the right, use (Ex. \(5.15 \text { ). }]\) For further information about \(K(X)\), and its applications to the generalized Riemann-Roch theorem, see Borel Serre [1], Manin [1]. and Appendix A.

Let \(A\) be a ring. Show that the following conditions are equivalent: (i) Spec \(A\) is disconnected : (ii) there exist nonzero elements \(e_{1}, e_{2} \in A\) such that \(e_{1} e_{2}=0, e_{1}^{2}=e_{1}, e_{2}^{2}=e_{2}\) \(e_{1}+e_{2}=1\) (these elements are called orthogonal idempotents): (iii) \(A\) is isomorphic to a direct product \(A_{1} \times A_{2}\) of two nonzero rings.

Describe the spectrum of the zero ring, and show that it is an initial object for the category of schemes. (According to our conventions, all ring homomorphisms must take 1 to \(1 .\) since \(0=1\) in the zero ring, we see that each ring \(R\) admits a unique homomorphism to the zero ring, but that there is no homomorphism from the zero ring to \(R\) unless \(0=1\) in \(R\).)

A topological space is quasi-compact if every open cover has a finite subcover. (a) Show that a topological space is noetherian (I, \(\$ 1)\) if and only if every open subset is quasi-compact. (b) If \(X\) is an affine scheme. show that \(\operatorname{sp}(X)\) is quasi- compact. but not in general noetherian. We say a scheme \(X\) is quati-ciompact if \(\operatorname{sp}(X)\) is. (c) If \(A\) is a noetherian ring. show that spiSpec 1 ) is a nocthcrian topological space. (d) Give an example to show that sp(Spec \(A\) ) can be noetherian even when \(A\) is not.

(a) Let \(\varphi: \overline{\mathscr{H}} \rightarrow \mathscr{G}\) be a morphism of sheaves on \(X\). Show that \(\varphi\) is surjective if and only if the following condition holds: for every open set \(U \subseteq X,\) and for everys \(\in \mathscr{G}(L)\), there is a covering \(\left\\{U_{i} \text { ; of } U \text { , and there are elements } t_{i} \in \mathscr{F}\left(U_{i}\right)\right.\) such that \(\varphi\left(t_{1}\right)=>\left.\right|_{l},\) for all \(i\) (b) Give an example of a surjective morphism of sheaves \(\varphi: \mathscr{F} \rightarrow \mathscr{S},\) and an open set \(U\) such that \(\varphi(U): \mathscr{F}(U) \rightarrow \mathscr{G}(U)\) is not surjective.
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