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Chapter 2: Problem 7

Let \(\varphi: \overline{\mathscr{H}} \rightarrow \mathscr{G}\) be a morphism of sheaves. (a) Show that im \(\varphi \cong \mathscr{F} /\) ker \(\varphi\) (b) Show that coker \(\varphi \cong \mathscr{G} / \mathrm{im} \varphi\)

Short Answer

Expert verified
The image of the morphism \(\varphi\) is isomorphic to the quotient sheaf of \(\mathscr{F}\) by the kernel of \(\varphi\). The cokernel of the morphism \(\varphi\) is isomorphic to the quotient sheaf of \(\mathscr{G}\) by the image of \(\varphi\).

Step by step solution

01

Understand the Quotient Sheaf

The quotient sheaf, denoted \(\mathscr{F} / \) ker \(\varphi\), is a sheaf whose sections over an open set U are the equivalence classes of sections of \(\mathscr{F}\) over U. These equivalence classes are such that two sections are equivalent if their difference lies in the kernel of \(\varphi\). Hence the quotient sheaf is a way of 'modding out' the kernel of the morphism.
02

Show that im \(\varphi \cong \mathscr{F} /\) ker \(\varphi\)

Consider a section f in \(\mathscr{F}\) over an open set U. Under the morphism \(\varphi\), this section gets mapped to \(\varphi(f)\) in \(\mathscr{G}\). A section is in the image of \(\varphi\) if and only if it can be written in this form. Now, consider the equivalence class of f in the quotient sheaf \(\mathscr{F} / \) ker \(\varphi\). This equivalence class is [f], and any g in [f] gets mapped to the same thing under \(\varphi\) because their difference is in the kernel. Thus, each [f] gets mapped to a unique thing in the image under \(\varphi\), showing an isomorphism.
03

Understand the Cokernel

In category theory, the cokernel of a morphism \(\varphi\) is a way to measure how far \(\varphi\) is from being surjective, analogous to how the kernel measures how far \(\varphi\) is from being injective. It is defined as the quotient of \(\mathscr{G}\) by the image of \(\varphi\) .
04

Show that coker \(\varphi \cong \mathscr{G} / \mathrm{im} \varphi\)

The cokernel of a morphism \(\varphi\) in this context is the sheaf associated to the presheaf U -> \(\mathscr{G}(U)/\varphi(\mathscr{F}(U))\). The sheafification process is necessary because the presheaf is not always a sheaf. However, when dealing with sheaves of abelian groups (as we are), it turns out that the presheaf is already a sheaf. Thus, we conclude that the cokernel of \(\varphi\) (a sheaf) is isomorphic to the quotient of \(\mathscr{G}\) by the image of \(\varphi\).

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Key Concepts

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

Morphisms of Sheaves
A morphism of sheaves is a fundamental concept used to understand how different sheaves relate to each other. Specifically, if we have two sheaves, say \(\mathscr{F}\) and \(\mathscr{G}\), a morphism \(\varphi: \mathscr{F} \rightarrow \mathscr{G}\) is essentially a rule that assigns each section of \(\mathscr{F}\) to a section of \(\mathscr{G}\).
This morphism respects the operations and structures on those sections, meaning it behaves in a consistent manner with respect to restrictions and stalks.
  • Preservation of Structure: Morphisms must preserve the algebraic structure found within the stalks, ensuring operations like addition or restriction are consistent.
  • Application in Geometry: Morphisms are crucial for gluing together local data into global geometric constructs.
They also provide a way to compare sheaves, discerning whether they have similar properties or differentials between them.
Quotient Sheaf
The concept of a quotient sheaf is similar to quotient groups or spaces in algebra and topology. Here, the quotient sheaf \(\mathscr{F} / \text{ker } \varphi\) is used to describe how sections of a sheaf can be grouped into equivalence classes modulo the kernel of a morphism.
The resulting sheaf contains sections that differ by something in the kernel, effectively simplifying the original sheaf.
  • Construction: The sections over an open set \(U\) in \(\mathscr{F} / \text{ker } \varphi\) are formed by taking equivalence classes of sections from \(\mathscr{F}(U)\).
  • Purpose: By focusing only on significant elements (those not in the kernel), the quotient sheaf helps in reducing complexity, providing a clearer picture of the sheaf's behavior.
Quotient sheaves are essential when analyzing how much of a sheaf is "used up" by a morphism.
Cokernel in Algebraic Geometry
In algebraic geometry, the cokernel of a morphism is used to measure non-surjectivity. If \(\varphi: \mathscr{F} \rightarrow \mathscr{G}\) is a morphism, the cokernel helps identify what is missing for \(\varphi\) to be onto.
It provides a view into the gaps left after applying the morphism.
  • Definition: Formally, the cokernel is defined as \(\text{coker } \varphi = \mathscr{G} / \text{im } \varphi\).
  • Role: It tells us how much \(\mathscr{G}\) extends beyond the reach of \(\varphi\).
  • Sheafification: While normally requiring sheafification, in the case of sheaves of abelian groups, this step is often inherently satisfied.
The cokernel's importance lies in refining our grasp of the entire image and range of a morphism, allowing for deeper insight into the differences between the sheaves.
Kernel in Algebraic Geometry
The kernel is a pivotal component in determining how close a morphism is to being injective. For a morphism \(\varphi: \mathscr{F} \rightarrow \mathscr{G}\), the kernel consists of the sections of \(\mathscr{F}\) that map to zero in \(\mathscr{G}\).
This set acts as a measure of redundancy or non-injectivity within the morphism.
  • Definition: The kernel is \(\text{ker } \varphi = \{ f \in \mathscr{F} \mid \varphi(f) = 0 \text{ in } \mathscr{G} \}\).
  • Injectivity Check: If the kernel is only the zero section, the morphism is injective.
  • Significance: Provides foundational understanding for constructing quotient sheaves and further connects with cokernels in broader diagrammatic representations.
Understanding kernels allows us to filter out and discern truly meaningful sections, emphasizing injective characteristics of morphisms.

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Most popular questions from this chapter

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.]

If \(\widetilde{\psi}\) is a coherent sheaf on a noetherian formal scheme \(\vec{x},\) which can be generated by global sections, show in fact that it can be generated by a finite number of its global sections.

Espace Etale of a Presheuf. (This exercise is included only to establish the connection between our definition of a sheaf and another definition often found in the literature. See for example Godement [1. Ch. II, \$1.2].) Given a presheaf \(\mathscr{F}\) un \(X\), we define a topological space Sp茅( \(\bar{y}\) ), called the espuce 茅ral茅 of \(\mathscr{F},\) as follows. As a set. Sp茅(. \(\overline{\mathscr{F}})=\cup_{P e}, x^{-} \overline{\mathscr{H}}_{P} .\) We define a projection map \(\pi: \operatorname{Sp茅}(\mathscr{F}) \rightarrow X\) by sending \(s \in \overline{\mathscr{H}}_{p}\) to \(P\). For each open set \(U \subseteq X\) and each section \(s \in \overline{\mathscr{F}}(\mathcal{L}),\) we obtain a \(\operatorname{map} \bar{\Im}: L \rightarrow \operatorname{Spei}(\mathscr{F})\) by sending \(P \mapsto s_{P},\) its germ at \(P .\) This map has the property that \(\pi \quad \bar{s}=\) id \(_{l},\) in other words, it is a "section" of \(\pi\) over \(U\). We now make Sp茅(. \(\overline{\mathscr{H}}\) ) into a topological space by giving it the strongest topology such that show that the sheaf \(\bar{y}^{+}\) associated to \(\bar{y}\) can be described as follows: for any open set \(l \subseteq X, \overline{\mathscr{H}}^{+}(U)\) is the set of continuous sections of \(\operatorname{Spei}(\mathscr{F})\) over \(U\). In particular, the original presheaf \(\mathscr{I}\) was a sheaf if and only if for each \(U, \mathscr{F}(U)\) is equal to the set of all continuous sections of Sp茅(. \(\overline{\mathcal{F}}\) ) over \(U\)

Support. Recall the notions of support of a section of a sheaf, support of a sheaf, and subsheaf with supports from (Ex. 1.14 ) and (Ex. 1.20 ). (a) Let \(A\) be a ring, let \(M\) be an \(A\) -module, let \(X=\operatorname{Spec} A,\) and let \(\mathscr{F}=\tilde{M}\) For any \(m \in M=\Gamma(X, \overline{\mathscr{F}}),\) show that Supp \(m=V(\text { Ann } m),\) where Ann \(m\) is the annihilator of \(m=\\{a \in A | a m=0\\}\) (b) Now suppose that \(A\) is noetherian, and \(M\) finitely generated. Show that \(\operatorname{Supp} \mathscr{F}=V(\operatorname{Ann} M)\) (c) The support of a coherent sheaf on a noetherian scheme is closed. (d) For any ideal a \(\subseteq A,\) we define a submodule \(\Gamma_{\mathrm{a}}(M)\) of \(M\) by \(\Gamma_{\mathrm{a}}(M)=\) \(\left\\{m \in M | a^{n} m=0 \text { for some } n>0\right\\} .\) Assume that \(A\) is noetherian, and \(M\) any \(A\) -module. Show that \(\Gamma_{\mathrm{a}}(M)^{\sim} \cong \mathscr{H}_{Z}^{0}(\mathscr{F}),\) where \(Z=V(\mathrm{a})\) and \(\mathscr{F}=\tilde{M}\) \([\text {Hint}: \text { Use (Ex. } 1.20)\) and (5.8) to show a priori that \(\mathscr{H}_{Z}^{0}(\mathscr{F})\) is quasi-coherent. Then show that \(\left.\Gamma_{\mathrm{a}}(M) \cong \Gamma_{\mathrm{z}}(\mathscr{F}) .\right]\) (e) Let \(X\) be a noetherian scheme, and let \(Z\) be a closed subset. If \(\mathscr{F}\) is a quasicoherent (respectively, coherent) \(O_{X}\) -module, then \(\mathscr{H}_{Z}^{0}(\mathscr{F})\) is also quasicoherent (respectively, coherent).

Let \(f: X \rightarrow Y\) be a morphism of separated schemes of finite type over a noetherian scheme \(S\). Let \(Z\) be a closed subscheme of \(X\) which is proper over \(S\). Show that \(f(Z)\) is closed in \(Y,\) and that \(f(Z)\) with its image subscheme structure (Ex. \(3.11 d\) ) is proper over \(S .\) We refer to this result by saying that "the image of a proper scheme is proper." [Hint: Factor \(f\) into the graph morphism \(\Gamma_{f}: X \rightarrow X \times_{s} Y\) followed by the second projection \(\left.p_{2}, \text { and show that } \Gamma_{f} \text { is a closed immersion. }\right]\)
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