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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

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.

Dimension. Let \(X\) be an integral scheme of finite type over a field \(k\) (not necessarily algebraically closed). Use appropriate results from (I, 81 ) to prove the following. (a) For any closed point \(P \in X, \operatorname{dim} X=\operatorname{dim} \mathscr{O}_{P},\) where for rings, we always mean the Krull dimension. (b) Let \(K(X)\) be the function field of \(X\) (Ex. 3.6). Then \(\operatorname{dim} X=\operatorname{tr.d.} K(X) / k\) (c) If \(Y\) is a closed subset of \(X,\) then \(\operatorname{codim}(Y, X)=\inf \left\\{\operatorname{dim} \mathscr{O}_{P, X} | \mathrm{P} \in Y\right\\}\) (d) If \(Y\) is a closed subset of \(X\), then \(\operatorname{dim} Y+\operatorname{codim}(Y, X)=\operatorname{dim} X\) (e) If \(U\) is a nonempty open subset of \(X,\) then \(\operatorname{dim} U=\operatorname{dim} X\) (f) If \(k \subseteq k^{\prime}\) is a field extension, then every irreducible component of \(X^{\prime}=X \times_{k} k^{\prime}\) has dimension \(=\operatorname{dim} X\)

Describe Spec \(\mathbf{Z}\), and show that it is a final object for the category of schemes. i.e., each scheme \(X\) admits a unique morphism to Spec \(\mathbf{Z}\).

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.

The Grothendieck Group of a Nonsingular Curce. Let \(X\) be a nonsingular curve over an algebraically closed field \(k .\) We will show that \(K(X) \cong \operatorname{Pic} X \oplus \mathbf{Z},\) in several steps. (a) For any divisor \(D=\sum n_{i} P_{i}\) on \(X\), let \(\psi(D)=\sum n_{i \hat{i}}\) ' \(\left(k\left(P_{i}\right)\right) \in K(X),\) where \(k\left(P_{i}\right)\) is the skyscraper sheaf \(k\) at \(P_{t}\) and 0 elsewhere. If \(D\) is an effective divisor, let \(\mathrm{C}_{D}\) be the structure sheaf of the associated subscheme of codimension \(1,\) and show that \(\psi(D)=\dot{\gamma}\left(C_{D}\right) .\) Then use (6.18) to show that for any \(D, \psi(D)\) depends only on the linear equivalence class of \(D,\) so \(\psi\) defines a homomorphism \(\psi: \mathrm{Cl} X \rightarrow K(X)\) (b) For any coherent sheaf \(\mathscr{F}\) on \(X\), show that there exist locally free sheaves \(\delta_{0}\) and \(\mathscr{E}_{1}\) and an exact sequence \(0 \rightarrow \mathscr{E}_{1} \rightarrow \mathscr{B}_{0} \rightarrow \mathscr{H} \rightarrow 0 .\) Let \(r_{0}=\operatorname{rank} \delta_{0}\) \(r_{1}=\operatorname{rank} \delta_{1},\) and define det \(\tilde{\mathscr{H}}=\left(\bigwedge^{r_{0}} \mathscr{E}_{0}\right) \otimes\left(\bigwedge^{r_{1}} \delta_{1}\right)^{-1} \in \operatorname{Pic} X .\) Here \(\wedge \mathrm{de}\) notes the exterior power (Ex. 5.16). Show that det \(\mathscr{H}\) is independent of the resolution chosen, and that it gives a homomorphism det: \(K(X) \rightarrow\) Pic \(X\) Finally show that if \(D\) is a divisor, then \(\operatorname{det}(\psi(D))=\mathscr{L}(D)\) (c) If \(\mathscr{F}\) is any coherent sheaf of rank \(r,\) show that there is a divisor \(D\) on \(X\) and an exact sequence \(0 \rightarrow \mathscr{P}(D)^{\oplus r} \rightarrow \mathscr{H} \rightarrow \mathscr{I} \rightarrow 0\), where \(\mathscr{J}\) is a torsion sheaf. Con- clude that if \(\mathscr{F}\) is a sheaf of rank \(r,\) then \(\gamma(\mathscr{F})-r \gamma\left(\mathcal{O}_{X}\right) \in \operatorname{Im} \psi\) (d) Using the maps \(\psi,\) det, rank,and \(1 \mapsto \gamma\left(C_{X}\right)\) from \(\mathbf{Z} \rightarrow K(X),\) show that \(K(X) \cong\) Pic \(X \oplus \mathbf{Z}\)
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