/*! 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 1 Let \(\left(X, C_{X}\right)\) be... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 1

Let \(\left(X, C_{X}\right)\) be a ringed space, and let \(\delta\) be a locally free \(C_{X}\) -module of finite rank. We define the dual of \(\mathscr{E}\), denoted \(\tilde{\delta}\), to be the sheaf \(\Varangle\) om \(_{e x}\left(\mathscr{E}, \mathcal{O}_{x}\right)\) (a) Show that \((\tilde{\delta})^{\sim} \cong \mathcal{E}\). (b) For any \(C_{x}\) -module \(\mathscr{F}, \mathscr{H}\) om \(_{e_{X}}(\mathscr{E}, \mathscr{F}) \cong \check{\mathscr{E}} \otimes_{\text {ox }} \mathscr{F}\). (c) For any \(C_{x}\) -modules \(\mathscr{F} . \mathscr{S}, \operatorname{Hom}_{\epsilon_{x}}(\mathscr{E} \otimes \mathscr{F}, \mathscr{S}) \cong \operatorname{Hom}_{\mathscr{C}_{X}}\left(\mathscr{F}, \mathscr{H}_{O} m_{e_{x}}(\mathscr{E}, \mathscr{S})\right)\). (d) (Projection Formula). If \(f:\left(X, O_{X}\right) \rightarrow\left(Y, O_{Y}\right)\) is a morphism of ringed spaces, if \(\mathscr{F}\) is an \(\mathscr{O}_{X}\) -module, and if \(\mathscr{E}\) is a locally free \(\mathscr{O}_{Y}\) -module of finite rank, then there is a natural isomorphism \(f_{*}\left(\mathscr{F} \otimes_{\mathcal{O}_{\mathbf{X}}} f^{*} \mathscr{E}\right) \cong f_{*}(\mathscr{F}) \otimes_{\mathcal{O}_{\mathbf{Y}}} \mathcal{E}\).

Short Answer

Expert verified
The process can be said to follow the principles of homomorphisms and isomorphisms within ringed spaces leading to the conclusion that: (a) \((\tilde{\delta})^{\sim} \cong \mathcal{E}\), (b) \(\mathscr{H}\) om \(_{e_{X}}(\mathscr{E}, \mathscr{F}) \cong \check{\mathscr{E}} \otimes_{\text{ox}} \mathscr{F}\), (c) \(\operatorname{Hom}_{\epsilon_{x}}(\mathscr{E} \otimes \mathscr{F}, \mathscr{S}) \cong \operatorname{Hom}_{\mathscr{C}_{X}}\left(\mathscr{F}, \mathscr{H}_{O} m_{e_{x}}(\mathscr{E}, \mathscr{S})\right)\) and (d) \(f_{*}\left(\mathscr{F} \otimes_{\mathcal{O}_{\mathbf{X}}} f^{*} \mathscr{E}\right) \cong f_{*}(\mathscr{F}) \otimes_{\mathcal{O}_{\mathbf{Y}}} \mathcal{E}\).

Step by step solution

01

Show that \((\tilde{\delta})^{\sim} \cong \mathcal{E}\)

Noting the definition of \(\tilde{\delta}\), the dual of \(\mathscr{E}\), the first step is to establish an isomorphism between the dual of the dual of \(\mathscr{E}\) and \(\mathscr{E}\) itself. This involves showing that \((\tilde{\delta})^{\sim}\) and \(\mathscr{E}\) have the exact same structure. This can be accomplished by considering any element \(x\) in \(C_X\) and demonstrating that the mapping \(x \mapsto \tilde{\delta}(x)\), and its inverse, are well-defined and bijective.
02

Show that \(\mathscr{H}\) om \(_{e_{X}}(\mathscr{E}, \mathscr{F}) \cong \check{\mathscr{E}} \otimes_{\text{ox}} \mathscr{F}\)

In this step, we have to show the isomorphism between two complex terms: \(\mathscr{H}\) om \(_{e_{X}}(\mathscr{E}, \mathscr{F})\) and \(\check{\mathscr{E}} \otimes_{\text{ox}} \mathscr{F}\). The goal is to state that \(\mathscr{H}\) om \(_{e_{X}}(\mathscr{E}, \mathscr{F})\) has the same structure as \(\check{\mathscr{E}} \otimes_{\text{ox}} \mathscr{F}\). We do this by choosing arbitrary \(x,y\) in \(C_{X}\) and showing that the mapping \(x \mapsto \mathscr{H}\) om \(_{e_{X}}(\mathscr{E}, \mathscr{F})(x,y)\), and its inverse, are well-defined and bijective.
03

Show that \(\operatorname{Hom}_{\epsilon_{x}}(\mathscr{E} \otimes \mathscr{F}, \mathscr{S}) \cong \operatorname{Hom}_{\mathscr{C}_{X}}\left(\mathscr{F}, \mathscr{H}_{O} m_{e_{x}}(\mathscr{E}, \mathscr{S})\right)\)

This step involves showing an isomorphism between the homomorphisms of tensor product of modules \(\mathscr{E}\) and \(\mathscr{F}\) into \(\mathscr{S}\) and the homomorphisms of \(\mathscr{F}\) into the tensor product of \(\mathscr{E}\) and \(\mathscr{S}\). The functions chosen in this step should not only be bijective, but also preserve the structure of the original modules. Verify that if \(x \in \mathscr{E} \otimes \mathscr{F}\) and \(y \in \mathscr{S}\), then \((x, y) \mapsto \operatorname{Hom}_{\epsilon_{x}}(x,y)\) and its inverse are well-defined and bijective.
04

Show that \(f_{*}\left(\mathscr{F} \otimes_{\mathcal{O}_{\mathbf{X}}} f^{*} \mathscr{E}\right) \cong f_{*}(\mathscr{F}) \otimes_{\mathcal{O}_{\mathbf{Y}}} \mathcal{E}\)

The Projection formula requires the establishment of an isomorphism between \(f_{*}\left(\mathscr{F} \otimes_{\mathcal{O}_{\mathbf{X}}} f^{*} \mathscr{E}\right)\) and \(f_{*}(\mathscr{F}) \otimes_{\mathcal{O}_{\mathbf{Y}}} \mathcal{E}\). When we show this, we can say that the pushing forward of the tensor product of \(\mathscr{F}\) and \(f^{*} \mathscr{E}\) is the same as the tensor product of the pushing forward of \(\mathscr{F}\) and \(\mathcal{E}\). To establish this isomorphism, we can assume \(x \in f_{*}(\mathscr{F})\) and \(y \in f^{*} \mathscr{E}\) and show that the mapping \(x \mapsto f_{*}(x,y)\), and its inverse, are well-defined and bijective.

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Key Concepts

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

Sheaf Theory
Sheaf theory is a mathematical tool used to systematically track local data attached to the open sets of a topological space and then "glue" this local data together. This concept emerges from topology and is widely used in various fields of mathematics such as algebraic geometry, differential geometry, and complex analysis. A sheaf on a topological space provides a way to move between local and global properties in a flexible and coherent manner.

A sheaf consists of a space, a collection of open sets over the space, and a rule that assigns to each open set a particular type of data (like functions, sections of a bundle, etc.). Importantly, this assignment should respect inclusions of open sets and allow us to effectively glue together the local pieces of data. In the context of ringed spaces, sheaves can represent modules or functions that live comfortably within the structure.
  • Presheaf: An assignment of data to open sets fulfilling the basic rules (no gluing requirement).
  • Sheaf: A presheaf that satisfies additional axioms for gluing local data consistently.
Sheaves offer the advantage of translating geometrical problems into algebraic terms, making complex calculations more manageable.
Locally Free Modules
Locally free modules are a special class of modules in which each point of a given space has a neighborhood where the module looks like free modules of finite rank. Imagine having a module tied to each point of the space such that, locally around any point, it resembles a familiar algebraic structure.

These are incredibly useful in algebraic geometry and topology because they allow for the intuitive use of vector fields, while maintaining an algebraic framework. Locally free modules can be thought of as bundles that look (locally) like direct sums of the structure sheaf. The concept is central to understanding vector bundles and coherent sheaves.
  • Locally, they are expressed as free modules, simplifying numerous calculations.
  • Global properties are studied by examining the transition functions between these local free modules.
This local-to-global perspective is powerful in solving many problems where local information needs to be pieced together.
Homomorphisms
In mathematics, homomorphisms are foundational concepts that involve structure-preserving maps between two algebraic structures, such as groups, rings, or modules. A homomorphism between modules maintains the operations of addition and scalar multiplication.

Learning about homomorphisms in the context of ringed spaces and sheaves is essential because it allows us to study and relate different algebraic structures in a way that respects the intrinsic properties of these structures. Homomorphisms help us understand equivalency and transformations between different modules or sheaves.
  • Represent the concept of functions between structures that respect given operations.
  • Central to algebraic structures because they preserve algebraic operations.
In sheaf theory, homomorphisms relate sheaves just as functions relate sets, enabling complex analysis and mappings to be handled effectively.
Projection Formula
The Projection Formula plays a vital role in algebraic geometry, enabling the manipulation of sheaves under a continuous map between two spaces. This formula establishes a relationship for transferring information from one space to another.

The formula can be stated as follows: for a morphism of ringed spaces, a locally free module of finite rank, and another module, there exists a natural isomorphism relating the pushforward of the tensor product with the tensor product of the pushforward. This essentially allows calculations to be "projected" in a meaningful way:

\[ f_{*}( ext{Module}_1 imes f^{*}( ext{Module}_2)) \cong f_{*}( ext{Module}_1) imes ext{Module}_2\]
  • This relates the various modules across different spaces in an algebraically coherent manner.
  • The isomorphism highlights how geometric transformations reflect in associated structures.
Understanding how and why the Projection Formula works is key to many areas, allowing for efficient handling of geometrical problems using algebraic techniques.

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Most popular questions from this chapter

Some Examples of Sheares on Varieties. Let \(X\) be a variety over an algebraically closed field \(k,\) as in \(\mathrm{Ch}\). I. Let \(C_{\lambda}\) be the sheaf of regular functions on \(X(1.0 .1)\) (a) Let \(Y\) be a closed subset of \(X\). For each open set \(U \subseteq X,\) let \(\mathscr{H},(U)\) be the ideal in the ring \(\left(\begin{array}{ll}i & U & \text { ) consisting of those regular functions which vanish }\end{array}\right.\) at all points of \(Y \cap U\). Show that the presheaf \(U \mapsto \mathscr{H}_{1}(U)\) is a sheaf. It is called the sheaf of ideal, \(\mathscr{Y}\), of \(Y\), and it is a subsheaf of the sheaf of rings \((\) " (b) If \(Y\) is a subvariety, then the quotient sheaf \(C, \quad y,\) is isomorphic to \(i_{3}(C, 1)\) where \(i: Y \rightarrow X\) is the inelusion, and \((, \text { is the sheaf of regular functions on } Y\). (c) Now let \(X=\mathbf{P}^{1}\), and let \(Y\) be the union of two distinct points \(P . Q \in X\). Then there is an exact sequence of sheaves on \(X\). where \(\mathscr{F}=i_{*} C_{P} \oplus i_{*} C_{0}\) $$0 \rightarrow \vartheta_{1} \rightarrow c_{1} \rightarrow \bar{y} \rightarrow 0$$ Show however that the induced map on global sections \(I(X, C, 1 \rightarrow I(X, F)\) is not surjective. This shows that the global section functor \(\Gamma(X: \cdot)\) is not exact fef. (Ex. 1.8 ) which shows that it is left exact. (d) Again let \(X=\mathbf{P}^{\prime},\) and let \(C\) be the sheaf of regular functions. Let \(x\) be the constant sheaf on \(X\) associated to the function field \(K\) of \(X .\) Show that there is a natural injection \(C \rightarrow \mathscr{K}\). Show that the quotient sheaf. \(\%\) ( is isomorphic to the direct sum of sheaves \(\sum_{P \in \lambda} i_{P}\left(I_{P}\right),\) where \(I_{P}\) is the group \(K C_{P},\) and \(i_{P}\left(I_{P}\right)\) denotes the sk yscraper sheaf (Ex. 1.17 ) given by \(I_{P}\) at the point \(P\) (e) Finally show that in the case of (d) the sequence $$0 \rightarrow \Gamma(X, C) \rightarrow \Gamma(X, X) \rightarrow \Gamma(X, \mathscr{K} \quad C) \rightarrow 0$$ is exact. (This is an analogue of what is called the "first Cousin problem" in several complex variables. See Gunning and Rossi \([1, \mathrm{p} .248] .\)

Let \(X\) be a noetherian scheme, and let \(\mathscr{F}\) be a coherent sheaf. (a) If the stalk \(\mathscr{F}_{x}\) is a free \(\mathscr{C}_{x}\) -module for some point \(x \in X,\) then there is a neighborhood \(U\) of \(x\) such that \(\left.\mathscr{F}\right|_{v}\) is free. (b) \(\mathscr{F}\) is locally free if and only if its stalks \(\mathscr{F}_{x}\) are free \(\mathscr{O}_{x}\) -modules for all \(x \in X\) (c) \(\mathscr{F}\) is invertible (i.e., locally free of rank 1 ) if and only if there is a coherent sheaf \(\mathscr{G}\) such that \(\mathscr{F} \otimes \mathscr{G} \cong \mathscr{O}_{X} .\) (This justifies the terminology invertible: it means that \(\mathscr{F}\) is an invertible element of the monoid of coherent sheaves under the operation \(\otimes .\)

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.

singular Curves. Here we give another method of calculating the Picard group of a singular curve. Let \(X\) be a projective curve over \(k\), let \(\tilde{X}\) be its normalization, and let \(\pi: \tilde{X} \rightarrow X\) be the projection \(\operatorname{map}(\mathrm{Ex} .3 .8) .\) For each point \(P \in X,\) let \(C_{P}\) be its local ring, and let \(\tilde{C}_{P}\) be the integral closure of \(C_{P} .\) We use a \(*\) to denote the group of units in a ring. (a) Show there is an exact sequence \\[ 0 \rightarrow \bigoplus_{P \in X} \tilde{\mathscr{C}}_{P}^{*} / \mathcal{O}_{P}^{*} \rightarrow \operatorname{Pic} X \stackrel{\pi^{*}}{\rightarrow} \operatorname{Pic} \tilde{X} \rightarrow 0 \\] \([\text {Hint}: \text { Represent Pic } X \text { and } \operatorname{Pic} \tilde{X}\) as the groups of Cartier divisors modulo principal divisors, and use the exact sequence of sheaves on \(X\) \\[ 0 \rightarrow \pi_{*} \mathscr{O}_{\dot{X}}^{*} / \mathcal{O}_{X}^{*} \rightarrow \mathscr{K}^{*} / \mathcal{O}_{\dot{X}}^{*} \rightarrow \mathscr{K}^{*} / \pi_{*} \mathcal{O}_{\bar{X}}^{*} \rightarrow 0 \\] (b) Use (a) to give another proof of the fact that if \(X\) is a plane cuspidal cubic curve, then there is an exact sequence \\[ 0 \rightarrow \mathbf{G}_{a} \rightarrow \operatorname{Pic} X \rightarrow \mathbf{Z} \rightarrow 0 \\] and if \(X\) is a plane nodal cubic curve, there is an exact sequence \\[ 0 \rightarrow \mathbf{G}_{m} \rightarrow \operatorname{Pic} X \rightarrow \mathbf{Z} \rightarrow 0 \\]

Complete Intersections in \(\mathbf{P}^{n}\). A closed subscheme \(Y\) of \(\mathbf{P}_{k}^{n}\) is called a (strict, global) complete intersection if the homogeneous ideal \(I\) of \(Y\) in \(S=k\left[x_{0}, \ldots, x_{n}\right]\) can be generated by \(r=\operatorname{codim}\left(Y, \mathbf{P}^{n}\right)\) elements (I, Ex. 2.17). (a) Let \(Y\) be a closed subscheme of codimension \(r\) in \(\mathbf{P}^{n}\). Then \(Y\) is a complete intersection if and only if there are hypersurfaces (i.e., locally principal subschemes of codimension 1) \(H_{1}, \ldots, H_{r},\) such that \(Y=H_{1} \cap \ldots \cap H_{r}\) as schemes, i.e., \(\mathscr{I}_{Y}=\mathscr{I}_{H_{1}}+\ldots+\mathscr{I}_{H_{r}} .[\text { Hint }:\) Use the fact that the unmixedness theorem \(\text { holds in }S \text { (Matsumura }[2, \mathrm{p} .107]) .]\) (b) If \(Y\) is a complete intersection of dimension \(\geqslant 1\) in \(P^{n},\) and if \(Y\) is normal, then \(Y\) is projectively normal (Ex. 5.14). \([\text {Hint}: \text { Apply }(8.23)\) to the affine cone over \(Y .]\) (c) With the same hypotheses as (b), conclude that for all \(l \geqslant 0\), the natural map \(\Gamma\left(\mathbf{P}^{n}, \mathcal{O}_{\mathbf{p}^{n}}(l)\right) \rightarrow \Gamma\left(Y, \mathcal{O}_{\mathbf{Y}}(l)\right)\) is surjective. In particular, taking \(l=0,\) show that \(Y\) is connected. (d) Now suppose given integers \(d_{1}, \ldots, d_{r} \geqslant 1,\) with \(r
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