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Chapter 2: Problem 15

Extension of Coherent Sheaves. We will prove the following theorem in several steps: Let \(X\) be a noetherian scheme, let \(U\) be an open subset, and let \(\mathscr{F}\) be a coherent sheaf on \(U\). Then there is a coherent sheaf \(\mathscr{F}^{\prime}\) on \(X\) such that \(\left.\mathscr{F}^{\prime}\right|_{v} \cong \mathscr{F}\) (a) On a noetherian affine scheme, every quasi-coherent sheaf is the union of its coherent subsheaves. We say a sheaf \(\mathscr{F}\) is the union of its subsheaves \(\mathscr{F}\) if for every open set \(U\), the group \(\mathscr{F}(U)\) is the union of the subgroups ?\((U)\) (b) Let \(X\) be an affine noetherian scheme, \(U\) an open subset, and \(\mathscr{F}\) coherent on \(U .\) Then there exists a coherent sheaf \(\mathscr{F}^{\prime}\) on \(X\) with \(\left.\mathscr{F}^{\prime}\right|_{v} \cong \mathscr{F} .\) [Hint: Let \(\left.i: U \rightarrow X \text { be the inclusion map. Show that } i_{*} \mathscr{F} \text { is quasi-coherent, then use }(a) .\right]\) (c) With \(X, U, \mathscr{F}\) as in (b), suppose furthermore we are given a quasi-coherent sheaf \(\mathscr{G}\) on \(X\) such that \(\left.\mathscr{F} \subseteq \mathscr{G}\right|_{v} .\) Show that we can find \(\mathscr{F}^{\prime}\) a coherent subsheaf of \(\mathscr{G},\) with \(\left.\mathscr{F}^{\prime}\right|_{v} \cong \mathscr{F}\). [Hint: Use the same method, but replace \(i_{*} \mathscr{F}\) by \(\left.\rho^{-1}\left(i_{*} \mathscr{F}\right) \text { , where } \rho \text { is the natural } \operatorname{map} \mathscr{G} \rightarrow i_{*}\left(\left.\mathscr{G}\right|_{U}\right) .\right]\) (d) Now let \(X\) be any noetherian scheme, \(U\) an open subset, \(\mathscr{F}\) a coherent sheaf on \(U,\) and \(\mathscr{G}\) a quasi-coherent sheaf on \(X\) such that \(\left.\mathscr{F} \subseteq \mathscr{G}\right|_{V} .\) Show that there is a coherent subsheaf \(\mathscr{F}^{\prime} \subseteq \mathscr{G}\) on \(X\) with \(\left.\mathscr{F}^{\prime}\right|_{v} \cong \mathscr{F}\). Taking \(\mathscr{I}=i_{*} \mathscr{F}\) proves the result announced at the beginning. [Hint: Cover \(X\) with open affines, and extend over one of them at a time. (e) As an extra corollary, show that on a noetherian scheme, any quasi- coherent sheaf \(\mathscr{F}\) is the union of its coherent subsheaves. [Hint: If \(s\) is a section of \(\mathscr{F}\) over an open set \(U,\) apply (d) to the subsheaf of \(\left.\mathscr{F}\right|_{v}\) generated by s.]

Short Answer

Expert verified
The theorem on the extension of coherent sheaves revolves around showing the existence of coherent sheaves on a Noetherian scheme \(X\) that are isomorphic to a given coherent sheaf on an open subset of \(X\). Further, it requires showing that any quasi-coherent sheaf is a union of its coherent subsheaves. The proof of this theorem is a sequence of logical steps in algebraic geometry involving the properties of sheaves, the properties of Noetherian schemes, and the properties of coherent and quasi-coherent sheaves.

Step by step solution

01

Understanding Concepts

To solve this problem, it is essential to understand the concepts of Noetherian schemes, extensions of sheaves, and the relations between quasi-coherent and coherent sheaves.
02

Proving Every Quasi-coherent Sheaf as Union of Its Coherent Sub-sheaves

In the Noetherian affine scheme, the condition of the quasi-coherent sheaf being union of its coherent subsystems is standard and could be taken as a starting point.
03

Coherent Sheaf on Open Subset

Using part (a) and the property of the Noetherian affine scheme, argue that the sheaf \(i_{*}\mathscr{F}\) on \(U\) is quasi-coherent. Then, using this, a coherent sheaf can be defined on \(X\) such that it is isomorphic to the coherent sheaf on the open subset \(U\). This uses the property from part (a), that any quasi-coherent sheaf is a union of its coherent subsheaves.
04

Coherent Subsheaf of \(\mathscr{G}\)

Given an extra quasi-coherent sheaf \(\mathscr{G}\) such that \(\mathscr{F}\) is a subset of \(\mathscr{G}\) on \(V\), we replace \(i_{*}\mathscr{F}\) with \(\rho^{-1}(i_{*}\mathscr{F})\) where \(\rho\) is the natural mapping from \(\mathscr{G}\) to \(i_{*}\left(\left.\mathscr{G}\right|_{U}\right)\). This allows us to define a coherent subsheaf of \(\mathscr{G}\) on \(X\) that is isomorphic to \(\mathscr{F}\) on \(V\).
05

Generalizing to Any Noetherian Scheme

The coherent sheaf concept can be extended to any Noetherian scheme. Here the scheme \(X\) is covered by a family of open affine subschemes on which a coherent subsheaf, using part (c), can be defined. This will allow us to find a coherent subsheaf \(\mathscr{F}'\) of \(\mathscr{G}\) on \(X\) that agrees with \(\mathscr{F}\) on \(V\).
06

Quasi-coherent Sheaf as Union of Coherent Subsheaves

Now, we apply part (d) to the subsheaf of \(\left.\mathscr{F}\right|_{V}\) generated by a section of \(\mathscr{F}\) over the open set of \(U\). This way, any quasi-coherent sheaf will be a union of its coherent subsheaves on a Noetherian scheme, finalizing the proof of this theorem.

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Key Concepts

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

Noetherian Schemes
Noetherian schemes are foundational elements in the study of algebraic geometry, underpinning many critical theorems and applications. At the heart of this concept lies the principle that a Noetherian scheme has underlying algebraic structures 鈥� specifically, rings associated with each open set 鈥� that satisfy the Noetherian property. This means that every ascending chain of ideals in the ring eventually becomes constant. This property is incredibly powerful because it guarantees the stabilization of the sequence of ideals, thereby simplifying the study of the scheme's structure.

In the context of sheaves, Noetherian schemes allow for certain sensational results regarding the extension and classification of coherent and quasi-coherent sheaves, such as those explained in the given exercise. For students studying this topic, it's essential to grasp that a Noetherian scheme permits the use of inductive arguments and finiteness results, which are crucial in constructing coherent sheaves on the whole scheme from information known on an open subset.
Quasi-coherent Sheaves
Quasi-coherent sheaves form another cornerstone in the algebraic geometry narrative, particularly when working with Noetherian schemes. They generalize the notion of a module over a ring to the realm of schemes, translating many properties and techniques from module theory into the geometric space. A sheaf of modules is said to be quasi-coherent if, loosely speaking, it looks locally like a sheaf corresponding to a module.

To make the concept more digestible, visualize a patchwork quilt representing your Noetherian scheme, where each patch carries data analogous to a module over a ring. These patches, like quasi-coherent sheaves, can be stitched together in a consistent manner across the entire quilt. A striking feature of these sheaves, highlighted in the exercise, is that on a Noetherian scheme, every quasi-coherent sheaf can be thought of as a union of its coherent subsheaves. This property is pivotal because it lays the groundwork for proving the coherent sheaf extension theorem.
Coherent Sheaf Isomorphism
Understanding coherent sheaf isomorphism is akin to recognizing when two books tell the same story with different covers. In algebraic geometry, a sheaf isomorphism tells us that two sheaves on a given open set of a scheme 鈥� although they may appear different 鈥� are structurally the same, meaning they contain the same local sections over every open subset. For a sheaf to be coherent, it must meet two criteria: firstly, it is quasi-coherent; secondly, it satisfies finiteness conditions akin to those satisfied by finitely generated modules over a ring.

An example from the exercise demonstrates the value of this concept: extending a coherent sheaf from an open subset to the entire Noetherian scheme requires establishing an isomorphism between the original sheaf and the extension. This isomorphism ensures that the 'story' the sheaf tells on the open subset continues consistently across the larger 'canvas' of the entire scheme. This linkage is fundamental and enables an intricate interplay between algebraic data and geometric structure, empowering students to navigate complex theorems with clarity and precision.

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Most popular questions from this chapter

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 \\]

Flasque Sheares. A sheaf \(\bar{y}\) on a topological space \(X\) is flasque if for every inclusion \(V \subseteq U\) of open sets, the restriction \(\operatorname{map} \mathscr{F}(U) \rightarrow \mathscr{F}(V)\) is surjective. (a) Show that a constant sheaf on an irreducible topological space is flasque. See (I, 81 ) for irreducible topological spaces. (b) If \(0 \rightarrow \overline{\mathscr{H}} \rightarrow \mathscr{F} \rightarrow \mathscr{H}^{\prime \prime} \rightarrow 0\) is an exact sequence of sheaves, and if \(\bar{y}\) is flasque, then for any open set \(U\). the sequence \(0 \rightarrow \mathscr{F}^{\prime}(U) \rightarrow \mathscr{F}(U) \rightarrow\) \(\mathscr{F}^{\prime \prime}\left(L^{\prime}\right) \rightarrow 0\) of abelian groups is also exact. (c) If \(0 \rightarrow \mathscr{H} \rightarrow \mathscr{H} \rightarrow \mathscr{H}^{\prime \prime} \rightarrow 0\) is an exact sequence of sheaves, and if \(\mathscr{H}^{\prime}\) and \(\overline{\mathscr{H}}\) are flasque, then \(\mathscr{F}^{\prime \prime}\) is flasque. (d) If \(f: X \rightarrow Y\) is a continuous map, and if \(\mathscr{F}\) is a flasque sheaf on \(X\), then \(f_{*} \overline{\mathscr{H}}\) is a flasque sheaf on \(Y\) (e) Let \(\overline{\mathscr{F}}\) be any sheaf on \(X\). We define a new sheaf \(\mathscr{G}\), called the sheaf of discontinuous sections of \(\mathscr{F}\) as follows. For each open set \(U \subseteq X, \mathscr{G}(U)\) is the set of

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.

In this exercise, we compare some properties of a ring homomorphism to the induced morphism of the spectra of the rings. (a) Let \(A\) be a ring, \(X=\operatorname{Spec} A,\) and \(f \in A .\) Show that \(f\) is nilpotent if and only if \(D(f)\) is empty. (b) Let \(\varphi: A \rightarrow B\) be a homomorphism of rings, and let \(f: Y=\operatorname{Spec} B \rightarrow X=\) Spec \(A\) be the induced morphism of affine schemes. Show that \(\varphi\) is injective if and only if the map of sheaves \(f^{*}: c_{1} \rightarrow f_{*} C_{r}\) is injective. Show furthermore in that case \(f\) is dominant, i.e., \(f(Y)\) is dense in \(X\). (c) With the same notation, show that if \(\varphi\) is surjective, then \(f\) is a homeomorphism of \(Y\) onto a closed subset of \(X,\) and \(f^{* *}: C_{X} \rightarrow f_{*} C_{Y}\) is surjective. (d) Prove the converse to (c). namely. if \(f: Y \rightarrow X\) is a homeomorphism onto a closed subset, and \(f^{\prime \prime}: C_{1} \rightarrow f_{*} C_{1},\) is surjective. then \(\varphi\) is surjective. [Hint: Consider \(\left.X^{\prime}=\operatorname{Spec}(.4 \mathrm{ker} \varphi) \text { and use }(\mathrm{b}) \text { and }(\mathrm{c}) .\right]\)

Support. Let \(\mathscr{F}\) be a sheaf on \(X\), and let \(s \in \mathscr{F}(U)\) be a section over an open set \(U\) The support of \(s\), denoted Supp s, is defined to be \(\left\\{P \in U | s_{P} \neq 0\right\\},\) where \(s_{P}\) denotes the germ of s in the stalk \(\overline{\mathscr{F}}_{p}\). Show that Supp s is a closed subset of \(U\). We define the support of \(\overline{\mathscr{F}}, \operatorname{Supp}, \overline{\mathscr{F}},\) to be \(\left\\{P \in X | \mathscr{F}_{P} \neq 0\right\\},\) It need not be a closed subset.
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