/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 7 Let \(X\) be a scheme. For any \... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 7

Let \(X\) be a scheme. For any \(x \in X\). let \(C_{\lambda}\) be the local ring at \(x\), and \(m_{\lambda}\) its maximal ideal. We define the residue field of \(x\) on \(X\) to be the field \(k(x)=C_{x} m_{x} .\) Now let \(K\) be any field. Show that to give a morphism of Spec \(K\) to \(X\) it is equivalent to give a point \(x \in X\) and an inclusion \(\operatorname{map} k(x) \rightarrow K\).

Short Answer

Expert verified
To make a morphism from Spec \(K\) to \(X\), it's necessary to give a point \(x\) in \(X\) and an inclusion map \(k(x) 鈫� K\). Conversely, if given a point \(x\) and the map \(k(x) 鈫� K\), a morphism from Spec \(K\) to \(X\) can be constructed. Thereby, the two situations are equivalent.

Step by step solution

01

Understanding The Statement

The first step is to understand the problem statement itself. In algebraic geometry, a scheme is a type of space which is defined by gluing together the spectrum of commutative rings. For a given point \(x\) on the scheme \(X\), we have the local ring \(C_x\) and its maximal ideal \(m_x\). The field defined by \(k(x) = C_x / m_x\) is called the residue field of \(x\) on \(X\). Now, we need to show that giving a morphism from the spectrum of a field \(K\) to \(X\) is equivalent to giving a point \(x\) on \(X\) along with an inclusion map from \(k(x)\) to \(K\).
02

Morphism From Spec K to X

Let us define a morphism from Spec \(K\) to \(X\). Since \(K\) is a field, Spec \(K\) is a one-point space (we call this point as \(u\)). Suppose that \(f\) is the morphism from Spec \(K\) to \(X\). Since the image of \(f\) contains a single point, we can denote the image of \(f\) as \(f(u) = x\). Then, we consider the local ring homomorphism \(C_x 鈫� f_u^*C_u\) where \(f_u^*C_u\) indicates the pullback of the structure sheaf at \(u\) under \(f\). We know that if \(f\) is a morphism of schemes, then \(f_u^*C_u\) is a local \(C_x\)-algebra and it maps \(m_x\) to the maximal ideal of the corresponding local ring.
03

Morphism From Residue Field To K

Now, we will define an inclusion \(g: k(x) 鈫� K\). We consider the exact sequence \(0 鈫� m_x 鈫� C_x 鈫� k(x) 鈫� 0\) and the local ring homomorphism \(C_x 鈫� K\). With this map and the naturality of the exact sequence, the sequence \(0 鈫� m_x 鈫� C_x 鈫� K 鈫� 0\) is also an exact sequence. This indicates that there exists a unique map \(k(x) 鈫� K\) making the second sequence commute, and since \(K\) is a field, this map is injective. Hence, we have the desired inclusion map.
04

Demonstrating The Equivalence

Note that in step 2, we gave a morphism from Spec\(K\) to \(X\) and derived a point \(x\) in \(X\) and a local ring homomorphism \(C_x 鈫� f_u^*C_u\) which induces an inclusion \(k(x) 鈫� K\). Conversely, in step 3, we began by giving a point \(x\) in \(X\) and an inclusion \(k(x) 鈫� K\) and were able to construct a morphism \(f: Spec\(K\) 鈫� \(X\). Therefore, we are able to demonstrate the statement: 'giving a morphism from Spec\(K\) to X is equivalent to giving a point \(x\) on \(X\) and an inclusion map \(k(x) 鈫� K\)'.

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Key Concepts

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

Residue Field
In algebraic geometry, understanding the residue field is fundamental to grasping how algebraic structures localize at specific points. Consider a scheme, which is an algebraic space built from the spectrums of rings. At any point x in this scheme, we can focus on the local behaviors by examining the local ring at x, denoted C_x, and its maximal ideal m_x. The residue field, symbolized as k(x) = C_x / m_x, is essentially the field obtained by taking the quotient of the local ring at x by its maximal ideal.

This concept is not just theoretical; the residue field encapsulates the 'local algebraic properties' of the scheme at point x and plays a pivotal role when exploring morphisms of schemes, particularly when dealing with morphisms involving a field. The property and structure of the residue field serve as a bridge connecting the abstract point x on the scheme to the concrete algebraic inclusions within a field K.
Local Ring at a Point
A local ring at a point x on a scheme X, denoted by C_x, is a critical tool for zeroing in on the algebraic structure surrounding x. This ring differentiates itself from other rings by having a unique maximal ideal, m_x, which contains all the non-units of C_x. Locally, when dealing with a scheme, behavior at x can often be entirely described by the local ring.

  • It's this uniqueness that simplifies considerations down to a 'neighborhood' around our point of interest.
  • The local ring also provides a pathway to define morphisms, which are functions between schemes that preserve the algebraic structure.
  • Understanding the local ring is crucial for defining the residue field and for describing the scheme's behavior at x.
When we discuss the morphism from the spectrum of a field K, the local ring is where we anchor our map, giving us the handle we need to grasp the interaction between the point x and the field K.
Morphism of Schemes
Morphisms in algebraic geometry are akin to functions between schemes, preserving the rich algebraic structures inherent within. In the context of our problem, we are particularly interested in a special kind of morphism that starts from the spectrum of a field K, or Spec(K), and ends at a scheme X. This spectrum represents a very simple scheme; with K being a field, its spectrum consists of just a single point.

  • A morphism of schemes in this scenario is defined by a map that assigns to this point from Spec(K) a specific point x in our target scheme X.
  • This assignment must align with the algebraic structures, involving the local ring at x and the residue field k(x).
  • The particularity lies in the induced maps between local rings and their residue fields, capturing the essence of the morphism's action.
Our goal is to draw an equivalence between such a morphism and providing simply a point and an injective field map, a concept that illuminates the deep interplay between geometry and algebra in the scheme framework.
Spectrum of a Field
The spectrum of a field K, denoted Spec(K), is a foundational concept that links algebraic fields with geometric objects in algebraic geometry. It is a scheme in its own right, albeit a very simple one, comprising just a single point. This singleton scheme is incredibly useful because it acts as a 'probe' that we can 'insert' into more complex schemes to understand their structure at a particular point.

  • The spectrum of a field is essential for 'testing' morphisms between schemes.
  • When a map from Spec(K) is used effectively, it reveals information about the local algebraic structure of the scheme at the image of this single point.
  • The concept underscores the elegant duality between algebraic fields and geometric points, ensuring that questions of algebraic geometry can be translated into tangible geometric manifestations.
The capacity for simple objects like fields to encode complex geometries via their spectrum is a cornerstone of algebraic geometry's power and versatility, and is a key concept for students to master.

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Most popular questions from this chapter

Vector Bundles. Let \(Y\) be a scheme. \(A\) (geometric) vector bundle of rank \(n\) over \(Y\) is a scheme \(X\) and a morphism \(f: X \rightarrow Y\), together with additional data consisting of an open covering \(\left\\{U_{i}\right\\}\) of \(Y\), and isomorphisms \(\psi_{i}: f^{-1}\left(U_{i}\right) \rightarrow \mathbf{A}_{U_{i}}^{n}\) such that for any \(i, j,\) and for any open affine subset \(V=\operatorname{Spec} A \subseteq U_{i} \cap U_{j}\) the automorphism \(\psi=\psi_{j} \circ \psi_{i}^{-1}\) of \(\mathbf{A}_{V}^{n}=\operatorname{Spec} A\left[x_{1}, \ldots, x_{n}\right]\) is given by a linear automorphism \(\theta\) of \(A\left[x_{1}, \ldots, x_{n}\right],\) i.e., \(\theta(a)=a\) for any \(a \in A,\) and \(\theta\left(x_{i}\right)=\) \(\sum a_{i j} x_{j}\) for suitable \(a_{i j} \in A\) An isomorphism \(g:\left(X, f,\left\\{U_{i}\right\\},\left\\{\psi_{i}\right\\}\right) \rightarrow\left(X^{\prime}, f^{\prime},\left\\{U_{i}^{\prime}\right\\},\left\\{\psi_{i}^{\prime}\right\\}\right)\) of one vector bundle of rank \(n\) to another one is an isomorphism \(g: X \rightarrow X^{\prime}\) of the underlying schemes, such that \(f=f^{\prime} \circ g,\) and such that \(X, f,\) together with the covering of \(Y\) consisting of all the \(U_{i}\) and \(U_{i}^{\prime},\) and the isomorphisms \(\psi_{i}\) and \(\psi_{i}^{\prime} \circ g,\) is also a vector bundle structure on \(X\) (a) Let \(\mathscr{E}\) be a locally free sheaf of rank \(n\) on a scheme \(Y\). Let \(S(\mathscr{E})\) be the symmetric algebra on \(\mathscr{E},\) and let \(X=\operatorname{Spec} S(\mathscr{E}),\) with projection morphism \(f: X \rightarrow Y\) For each open affine subset \(U \subseteq Y\) for which \(\left.\mathscr{E}\right|_{U}\) is free, choose a basis of \(\mathscr{E}\) and let \(\psi: f^{-1}(U) \rightarrow \mathbf{A}_{U}^{n}\) be the isomorphism resulting from the identification of \(S(\mathscr{E}(U))\) with \(\mathscr{O}(U)\left[x_{1}, \ldots, x_{n}\right] .\) Then \((X, f,\\{U\\},\\{\psi\\})\) is a vector bundle of rank \(n\) over \(Y\), which (up to isomorphism) does not depend on the bases of \(\mathscr{E}_{U}\) chosen. We call it the geometric vector bundle associated to \(\delta,\) and denote it by \(\mathbf{V}(\mathscr{E})\). (b) For any morphism \(f: X \rightarrow Y\), a section of \(f\) over an open set \(U \subseteq Y\) is a morphism \(s: U \rightarrow X\) such that \(f \circ s=\) id \(_{U} .\) It is clear how to restrict sections to smaller open sets, or how to glue them together, so we see that the presheaf \(U \mapsto\\{\text { set of sections of } f \text { over } U\\}\) is a sheaf of sets on \(Y\), which we denote by \(\mathscr{S}(X / Y) .\) Show that if \(f: X \rightarrow Y\) is a vector bundle of \(\operatorname{rank} n,\) then the sheaf of sections \(\mathscr{S}(X / Y)\) has a natural structure of \(\mathscr{O}_{Y}\) -module, which makes it a locally free \(\mathscr{O}_{Y}\) -module of rank \(n\). [Hint: It is enough to define the module structure locally, so we can assume \(Y=\operatorname{Spec} A\) is affine, and \(X=\mathbf{A}_{Y}^{n} .\) Then a section \(s: Y \rightarrow X\) comes from an \(A\) -algebra homomorphism \(\theta: A\left[x_{1}, \ldots, x_{n}\right] \rightarrow\) \(A,\) which in turn determines an ordered \(n\) -tuple \(\left\langle\theta\left(x_{1}\right), \ldots, \theta\left(x_{n}\right)\right\rangle\) of elements of \(A .\) Use this correspondence between sections \(s\) and ordered \(n\) -tuples of elements of \(A \text { to define the module structure. }]\) (c) Again let \(\delta\) be a locally free sheaf of rank \(n\) on \(Y\), let \(X=\mathbf{V}(\delta)\), and let \(\mathscr{S}=\) \(\mathscr{S}(X / Y)\) be the sheaf of sections of \(X\) over \(Y\). Show that \(\mathscr{S} \cong \mathscr{E}^{\curlyvee},\) as follows. Given a section \(s \in \Gamma\left(V, \delta^{\curlyvee}\right)\) over any open set \(V\), we think of \(s\) as an element of \(\operatorname{Hom}\left(\left.\mathscr{E}\right|_{V}, \mathcal{O}_{V}\right) .\) So \(s\) determines an \(\mathscr{O}_{V^{-} \text {algebra homomorphism }} S\left(\left.\mathscr{E}\right|_{V}\right) \rightarrow \mathcal{O}_{V}\) This determines a morphism of spectra \(V=\operatorname{Spec} O_{V} \rightarrow \operatorname{Spec} S\left(\left.\mathscr{E}\right|_{V}\right)=\) \(f^{-1}(V),\) which is a section of \(X / Y .\) Show that this construction gives an isomorphism of \(\mathscr{E}^{\curlyvee}\) to \(\mathscr{S}\) (d) Summing up, show that we have established a one-to-one correspondence between isomorphism classes of locally free sheaves of rank \(n\) on \(Y\), and isomorphism classes of vector bundles of rank \(n\) over \(Y\). Because of this, we sometimes use the words "locally free sheaf" and "vector bundle" interchangeably, if no confusion seems likely to result.

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 \(\left.\Delta: Y \rightarrow Y \times_{Z} Y .\right]\)

Shyscraper Sheares. Let \(X\) be a topological space. let \(P\) be a point, and let \(A\) be an abelian group. Define a sheaf \(i_{P}(A)\) on \(X\) as follows: \(i_{p}(A)(\mathcal{L})=A\) if \(P \in L, 0\) otherwise. Verify that the stalk of \(i_{P}(A)\) is \(A\) at every point \(Q \in\left\\{P \text { ; }^{-} \text {, and } 0\right.\) elsewhere, where \(\\{P \text { ; - denotes the closure of the set consisting of the point } P\) Hence the name "skyscraper sheaf." Show that this sheaf could also be described \(\operatorname{as} i_{*}(A),\) where \(A\) denotes the constant sheaf \(A\) on the closed subspace \(\\{P\\}^{-},\) and \(i:\\{P\\}^{-} \rightarrow X\) is the inclusion.

Let \(X\) be a regular nocthcrian scheme, and \(\delta\) a locally free coherent sheaf of rank \(\geqslant 2\) on \(X\) (a) Show that Pic \(\mathbf{P}(\delta) \cong \operatorname{Pic} X \times \mathbf{Z}\) (b) If \(f^{\prime}\) is another locally free coherent sheafon \(X\). show that \(\mathrm{P}(\mathcal{E)} \cong \mathbf{P}(\mathcal{E} \text { ' ) lover } X\) ' if and only if there is an invertible sheaf \(\mathscr{Y}\) on \(X\) such that \(\mathscr{E}^{\prime} \cong \delta \otimes \mathscr{Y}\)

Prove the analogue of (5.6) for formal schemes, which says, if \(\mathcal{K}\) is an affine formal scheme. and if \\[ 0 \rightarrow \widetilde{\psi} \rightarrow \widetilde{\psi} \rightarrow \widetilde{\psi}^{\prime \prime} \rightarrow 0 \\] is an exact sequence of \(C_{\mathrm{r}}\) -modules. and if \(\tilde{\psi}\) is coherent, then the sequence of global sections \\[ 0 \rightarrow \Gamma\left(\boldsymbol{x}, \widetilde{\boldsymbol{x}}^{\prime}\right) \rightarrow \Gamma(\boldsymbol{x}, \widetilde{\boldsymbol{\varphi}}) \rightarrow \Gamma\left(\boldsymbol{x}, \widetilde{x}^{\prime \prime}\right) \rightarrow 0 \\] is exact. For the proof. proceed in the following steps. (a) Let 3 be an ideal of definition for \(\mathfrak{X},\) and for each \(n>0\) consider the exact sequence \\[ 0 \rightarrow \mathfrak{F}^{\prime} / \mathfrak{I}^{n} \mathfrak{F}^{\prime} \rightarrow \mathfrak{F} / \mathfrak{I}^{n} \mathfrak{F}^{\prime} \rightarrow \mathfrak{F}^{\prime \prime} \rightarrow 0 \\] Use \((5.6),\) slightly modified. to show that for every open affine subset \(21 \subseteq\) the sequence is exact. (b) Now pass to the limit, using \((9.1),(9.2),\) and \((9.6) .\) Conclude that \(\widetilde{x} \cong \lim \tilde{x} \sqrt{x}_{x}\) and that the sequence of global sections above is exact.
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