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Chapter 2: Problem 6

(a) Let \(\mathscr{F}^{\prime}\) be a subsheaf of a sheaf \(\mathscr{F}\). Show that the natural map of \(\mathscr{F}\) to the quotient sheaf \(\mathscr{F}^{\prime} \mathscr{F}^{\prime}\) is surjective, and has kernel \(\mathscr{F}\) : Thus there is an exact sequence $$0 \rightarrow \mathscr{F}^{\prime} \rightarrow \mathscr{F} \rightarrow \mathscr{H} / \mathscr{F}^{\prime} \rightarrow 0$$ (b) Conversely, if \(0 \rightarrow \overline{\mathscr{F}}^{\prime} \rightarrow \widetilde{\mathscr{F}} \rightarrow \widetilde{\mathscr{F}}^{\prime \prime} \rightarrow 0\) is an exact sequence, show that \(\mathscr{F}^{\prime}\) is isomorphic to a subsheaf of \(\overline{\mathscr{F}},\) and that \(\mathscr{F}^{\prime \prime}\) is isomorphic to the quotient of \(\mathscr{F}\) by this subsheaf.

Short Answer

Expert verified
The natural map of a sheaf to the quotient sheaf is surjective, with kernel being the subsheaf \(\mathscr{F}^{\prime}\), thus producing the exact sequence \(0 \rightarrow \mathscr{F}^{\prime} \rightarrow \mathscr{F} \rightarrow \mathscr{F} / \mathscr{F}^{\prime} \rightarrow 0\). An exact sequence given implies that \(\mathscr{F}^{\prime}\) is isomorphic to a subsheaf of \(\overline{\mathscr{F}}\) and \(\mathscr{F}^{\prime\prime}\) is isomorphic to the quotient of the sheaf by this subsheaf.

Step by step solution

01

Preliminary Definitions

Let's first establish the definitions of terms used in the problem such as sheaf, subsheaf, quotient sheaf, natural map, kernel, exact sequence, and isomorphism. Knowing these definitions will help in understanding the problem and creating a solid foundation.
02

Show Surjectivity of the Natural Map

This requires showing that for each element in the codomain \(\mathscr{F}^{\prime} / \mathscr{F}^{\prime}\), there exists a corresponding element in the domain \(\mathscr{F}\). This can be demonstrated by selecting an arbitrary element in the codomain, and then showing that a corresponding element can be found in the domain.
03

Show Kernel

Next, determine the kernel of the natural map. This involves finding the elements in the domain that are mapped to the 'zero' element in the codomain. Since the 'zero' element of the sheaf is \(\mathscr{F}^{\prime}\), the aim is to find the elements of \(\mathscr{F}\) that are mapped to \(\mathscr{F}^{\prime}\). Beginning with the conditions of the sheaf, find these elements and show that the kernel is \(\mathscr{F}^{\prime}\).
04

Show Exact Sequence

Show that the given sequence is exact. This means confirming that the image of each morphism is the kernel of the following morphism, verifying that this holds for the given sequence.
05

Converse Statement

Given an exact sequence, show that \(\mathscr{F}^{\prime}\) is isomorphic to a subsheaf of \(\overline{\mathscr{F}}\), and that \(\mathscr{F}^{\prime \prime}\) is isomorphic to the quotient of \(\mathscr{F}\) by this subsheaf. Use the properties of the sequence and the isomorphism to relate \(\mathscr{F}^{\prime}\) to the subsheaf of \(\overline{\mathscr{F}}\) and \(\mathscr{F}^{\prime \prime}\) to the quotient of \(\mathscr{F}\).

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Key Concepts

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

Subsheets
In sheaf theory, a **subsheaf** is a foundational concept used to understand the structure within sheaves. A sheaf, in algebraic geometry, is like a tool that helps us keep track of functions on various pieces of a space. When we place a subsheaf inside a sheaf, we're identifying a subset of these functions that behaves nicely relative to the larger sheaf. Every sheaf has associated properties, like sections over open sets, which are key in classifying the sheaf structure. A subsheaf retains these properties but exists as a 'subset' within a larger context.
  • A subsheaf can help better understand or simplify complex sheaf relationships
  • It is crucial for defining operations and transformations within the realm of algebraic geometry
By understanding subsheaves, one gains insight into sheaf morphisms, kernels, and images, which are essential for further explorations in sheaf theory.
Exact Sequence
An **exact sequence** in algebraic geometry is a sequence of sheaf homomorphisms that neatly connects various mathematical objects in a way that preserves essential algebraic properties. This sequence can be written as: \[ 0 \rightarrow \mathscr{F}^{\prime} \rightarrow \mathscr{F} \rightarrow \mathscr{H} / \mathscr{F}^{\prime} \rightarrow 0 \] The term 'exact' means that at each part of the sequence, the image of one map matches precisely with the kernel of the next map.
  • The initial '0' represents a zero object, indicating that the map is injective at the start
  • The matching kernels and images help show the sequence's structure and draw out important relationships
These sequences are key in algebraic geometry because they provide a powerful means of tracking how objects connect and transform across different spaces.
Natural Map
In algebraic geometry, a **natural map** is a homomorphism that respects the structure of the sheaves involved. It's akin to a bridge that 'naturally' or 'obviously' connects two sheaves based on their given properties. Natural maps are often involved when discussing quotient sheaves or embeddings. They allow transformations while retaining inherent sheaf qualities such as locality and gluing of sections.
  • A natural map is often surjective, meaning it covers every element in the quotient sheaf
  • Such maps showcase how elements of one sheaf can be projected or mapped onto another, respecting the algebraic nature of the sheaves
These maps simplify complex relationships by providing straightforward connections between different mathematical constructs.
Kernel in Algebraic Geometry
The concept of a **kernel** in algebraic geometry aligns with its usage in broader algebraic contexts, acting as the 'inverse shadow' cast by a homomorphism. Specifically, the kernel of a sheaf homomorphism is the collection of elements that map to zero in the target sheaf. In the context of sheaf theory, finding the kernel helps identify where a map loses information. This has significant importance in exact sequences, where determining the kernel can be a step to showing the sequence's precise continuation.
  • The kernel is fundamental when examining morphisms between sheaves
  • It helps define key relationships and guide the properties of exact sequences
A strong understanding of kernels is vital for insights into sheaf properties and algebraic structures, guiding deeper exploration into algebraic geometry's complex landscape.

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Most popular questions from this chapter

Let \(X\) be a scheme of finite type over a field \(k\) (not necessarily algebraically closed). (a) Show that the following three conditions are equivalent (in which case we say that \(X\) is geometrically irreducible). (i) \(X \times_{k} \bar{k}\) is irreducible, where \(\bar{k}\) denotes the algebraic closure of \(k .\) abuse of notation, we write \(X \times_{k} \bar{k}\) to denote \(X \times_{\text {spec } k}\) Spec \(\bar{k} .\) (ii) \(X \times_{k} k_{s}\) is irreducible, where \(k_{s}\) denotes the separable closure of \(k\) (iii) \(X \times_{k} K\) is irreducible for every extension field \(K\) of \(k\) (b) Show that the following three conditions are equivalent (in which case we say \(X\) is geometrically reduced) (i) \(X \times_{k} \bar{k}\) is reduced. (ii) \(X \times_{k} k_{p}\) is reduced, where \(k_{p}\) denotes the perfect closure of \(k\) (iii) \(X \times_{k} K\) is reduced for all extension fields \(K\) of \(k\) (c) We say that \(X\) is geometrically integral if \(X \times_{k} \bar{k}\) is integral. Give examples of integral schemes which are neither geometrically irreducible nor geometrically reduced.

Blowing up a Nonsingular Subvariety. As in \((8.24),\) let \(X\) be a nonsingular variety, let \(Y\) be a nonsingular subvariety of codimension \(r \geqslant 2\), let \(\pi: \tilde{X} \rightarrow X\) be the blowing-up of \(X\) along \(Y\), and let \(Y^{\prime}=\pi^{-1}(Y)\) (a) Show that the maps \(\pi^{*}:\) Pic \(X \rightarrow\) Pic \(\tilde{X}\), and \(\mathbf{Z} \rightarrow\) Pic \(X\) defined by \(n \mapsto\) class of \(n Y^{\prime},\) give rise to an isomorphism Pic \(\tilde{X} \cong \operatorname{Pic} X \oplus \mathbf{Z}\) (b) Show that \(\omega_{\tilde{X}} \cong f^{*} \omega_{X} \otimes \mathscr{L}\left((r-1) Y^{\prime}\right) .[\) Hint: By (a) we can write in any \(\operatorname{case} \omega_{\tilde{X}} \cong f^{*} \mathscr{M} \otimes \mathscr{L}\left(q Y^{\prime}\right)\) for some invertible sheaf \(\mathscr{M}\) on \(X,\) and some integer \(q .\) By restricting to \(\tilde{X}-Y^{\prime} \cong X-Y\), show that \(\mathscr{M} \cong \omega_{X} .\) To determine \(q\) proceed as follows. First show that \(\omega_{Y^{\prime}} \cong f^{*} \omega_{X} \otimes O_{Y^{\prime}}(-q-1) .\) Then take a closed point \(y \in Y\) and let \(Z\) be the fibre of \(Y^{\prime}\) over \(y .\) Then show that \(\omega_{z} \cong\) \(\left.\mathcal{O}_{\mathbf{z}}(-q-1) . \text { But since } Z \cong \mathbf{P}^{r-1}, \text { we have } \omega_{\mathbf{Z}} \cong \mathcal{O}_{\mathbf{z}}(-r), \text { so } q=r-1 .\right]\)

Zariski Spaces. A topological space \(X\) is a Zariski space if it is noetherian and every (nonempty) closed irreducible subset has a unique generic point (Ex. 2.9 ). For example, let \(R\) be a discrete valuation ring, and let \(T=\operatorname{sp}(\operatorname{Spec} R)\). Then \(T\) consists of two points \(t_{0}=\) the maximal ideal, \(t_{1}=\) the zero ideal. The open subsets are \(\varnothing,\left\\{t_{1}\right\\},\) and \(T .\) This is an irreducible Zariski space with generic point \(t_{1}\). (a) Show that if \(X\) is a noetherian scheme, then \(\operatorname{sp}(X)\) is a Zariski space. (b) Show that any minimal nonempty closed subset of a Zariski space consists of one point. We call these closed points. (c) Show that a Zariski space \(X\) satisfies the axiom \(T_{0}\) : given any two distinct points of \(X\), there is an open set containing one but not the other (d) If \(X\) is an irreducible Zariski space, then its generic point is contained in every nonempty open subset of \(X\) (e) If \(x_{0}, x_{1}\) are points of a topological space \(X,\) and if \(x_{0} \in\left\\{x_{1}\right\\}^{-},\) then we say that \(x_{1}\) specializes to \(x_{0},\) written \(x_{1} \leadsto \rightarrow x_{0} .\) We also say \(x_{0}\) is a specialization of \(x_{1},\) or that \(x_{1}\) is a generization of \(x_{0} .\) Now let \(X\) be a Zariski space. Show that the minimal points, for the partial ordering determined by \(x_{1}>x_{0}\) if \(x_{1} \leadsto x\) \(x_{0},\) are the closed points, and the maximal points are the generic points of the irreducible components of \(X .\) Show also that a closed subset contains every specialization of any of its points. (We say closed subsets are stable under specialization. . Similarly, open subsets are stable under generization. (f) Let \(t\) be the functor on topological spaces introduced in the proof of (2.6) If \(X\) is a noetherian topological space, show that \(t(X)\) is a Zariski space. Furthermore \(X\) itself is a Zariski space if and only if the \(\operatorname{map} \alpha: X \rightarrow t(X)\) is a homeomorphism.

Subshect with Supports. Let \(Z\) be a closed subset of \(X\), and let \(\mathscr{F}\) be a sheaf on \(X\) We define \(\Gamma_{X}(X, \overline{\mathscr{F}})\) to be the subgroup of \(\Gamma(X, \overline{\mathscr{H}})\) consisting of all sections whose support (Ex. 1.14 ) is contained in \(Z\). (a) Show that the presheaf \(V \mapsto \Gamma_{z \cap v}\left(V,\left.\bar{y}\right|_{V}\right)\) is a sheaf. It is called the subsheaf of \(\overline{\mathscr{F}}\) with supports in \(Z,\) and is denoted by \(\mathscr{H}_{Z}^{0} \cdot \overline{\mathscr{F}}\) ). (b) Let \(U=X-Z,\) and let \(j: U \rightarrow X\) be the inclusion. Show there is an exact sequence of sheaves on \(X\) $$0 \rightarrow \mathscr{H}_{Z}^{0}(\mathscr{F}) \rightarrow \mathscr{H} \rightarrow i_{*}\left(\left.\mathscr{F}\right|_{C}\right)$$ Furthermorc, if \(\mathscr{F}\) is flasque, the \(\operatorname{map} \mathscr{F} \rightarrow i_{*}\left(\left.\mathscr{F}\right|_{l}\right)\) is surjective

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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