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Chapter 2: Problem 3

Let \(X=\) Spec \(A\) be an affine scheme. Show that the functors \(^{\sim}\) and \(\Gamma\) are adjoint, in the following sense: for any \(A\) -module \(M\), and for any sheaf of \(\mathscr{C}_{X}\) -modules \(\mathscr{F}\) there is a natural isomorphism \\[ \operatorname{Hom}_{A}(M, \Gamma(X, \mathscr{F})) \cong \operatorname{Hom}_{\sigma_{X}}(\tilde{M}, \mathscr{F}) \\].

Short Answer

Expert verified
The adjointness of the functors \(^{\sim}\) and \(\Gamma\) can be proven by defining a mapping between the respective hom-sets of \(A\)-module \(M\) and sheaf \(\mathscr{F}\), and demonstrating that the map is a well-defined isomorphism. The naturality of this isomorphism further underscores the adjointness of the functors.

Step by step solution

01

Understanding the adjoint functors

The functors \(^{\sim}\) and \(\Gamma\) are related to an \(A\)-module \(M\) and a sheaf \(\mathscr{F}\) in a specific way. In the category of \(A\)-modules, \(M\) is an object and in the category of sheaves of \(\mathscr{O}_X\) modules \(\mathscr{F}\) is an object. The functor \(^{\sim}\) sends an \(A\)-module to the corresponding sheaf, while the functor \(\Gamma\) sends a sheaf to the corresponding \(A\)-module.
02

Identifying the hom-sets

To prove the claim, we need to demonstrate an isomorphism between two hom-sets, specifically, \(\operatorname{Hom}_{A}(M, \Gamma(X, \mathscr{F}))\) and \(\operatorname{Hom}_{\mathscr{O}_X}(\widetilde{M}, \mathscr{F})\). The first one refers to the set of all \(A\)-module homomorphisms from \(M\) to \(\Gamma(X, \mathscr{F})\), whereas the second one refers to the set of all homomorphisms in the category of sheaves that send \(\widetilde{M}\) to \(\mathscr{F}\).
03

Constructing a natural isomorphism

To explicitly show an isomorphism between these two sets, a mapping should be defined from one set to another and then demonstrated to be a bijection. A map \(\phi: \operatorname{Hom}_{A}(M, \Gamma(X, \mathscr{F})) \rightarrow \operatorname{Hom}_{\mathscr{O}_X}(\widetilde{M}, \mathscr{F})\) can be defined using universal property of \(\Gamma\) and \(^{\sim}\) functors, for each \(f: M \rightarrow \Gamma(X, \mathscr{F})\) \(A\)-module homomorphism, there corresponds a unique \(\phi(f): \widetilde{M} \rightarrow \mathscr{F}\) sheaf homomorphism, making the triangle commute. The inverse mapping can be similarly defined.
04

Confirming the isomorphism

To confirm the map \(\phi\) actually yields an isomorphism, it is important to verify that both \(\phi\) and its inverse are well-defined i.e., they both exist, and they indeed reverse one another, thus certifying the bijection proof.
05

Proving naturality

Finally, the naturality of the isomorphism must be shown, through a commutative diagram. This step includes specific checks to verify that for any \(A\)-module homomorphisms \(g: M \rightarrow M'\) or \(h: \mathscr{F} \rightarrow \mathscr{F}'\), the induced map commutes with the original maps in terms of \(\phi\), thus ensuring that we have a natural transformation.

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Key Concepts

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

Understanding Affine Schemes
Affine schemes are one of the foundational concepts in algebraic geometry, forming the building blocks for the broader study of schemes. An affine scheme, denoted as \(X =\) Spec \(A\), is constructed from the prime spectrum of a commutative ring \(A\), where \(A\) is typically thought of as the ring of functions on the space \(X\). This space is equipped with the Zariski topology, where open subsets are determined by the non-vanishing of functions.

Furthermore, on the affine scheme \(X\), we have a structure sheaf, \(\mathscr{O}_X\), which assigns to each open subset of \(X\) the ring of functions that are defined on that subset. By providing the data of both a topological space and a sheaf of rings on it, affine schemes present a way to study geometric properties using algebraic tools.

In the exercise, the correspondence between \(A\)-modules and sheaves of \(\mathscr{O}_X\)-modules, expressed through the functors \(^{\sim}\) and \(\Gamma\), showcases the deep interplay between algebra and geometry that characterizes much of algebraic geometry.
Sheaves of Modules
A sheaf of modules over an affine scheme is a fundamental structure that captures the varying algebra across the scheme. If \(X =\) Spec \(A\) is an affine scheme, as in our exercise, a sheaf of \(\mathscr{O}_X\)-modules, denoted \(\mathscr{F}\), can be thought of as an \(A\)-module that changes, or 'sheafs' over the various open subsets of \(X\).

Each open subset of \(X\) is assigned an \(\mathscr{O}_X(U)\)-module by the sheaf \(\mathscr{F}\), which can be intuitively understood as analogous to sections of a bundle in topology 鈥� albeit with more algebraic structure, reflecting the module operations. Sheaves of modules provide a powerful language for speaking about 'locally defined' algebraic objects, which are essential when we want to glue local data to obtain global properties.

In the context of the exercise, we're looking at a sheaf of modules over an affine scheme and exploring the relationship between the global sections functor \(\Gamma\), which 'collects' all the sections over the whole scheme, and the tilde functor \(^{\sim}\), which constructs a sheaf of modules from an \(A\)-module.
Hom-Set Isomorphism
In category theory, a hom-set denotes the set of morphisms between two objects in a category. When dealing with the categories of modules, these morphisms are module homomorphisms. Isomorphisms between hom-sets are of particular interest as they reveal structural similarities between different categories.

In our exercise, the hom-set isomorphism refers to the equivalence between the set of \(A\)-module homomorphisms \(\operatorname{Hom}_{A}(M, \Gamma(X, \mathscr{F}))\) and the set of sheaf homomorphisms \(\operatorname{Hom}_{\mathscr{O}_X}(\widetilde{M}, \mathscr{F})\), for an \(A\)-module \(M\) and sheaf of modules \(\mathscr{F}\). This isomorphism is 'natural' in the sense that it is independent of the particular choice of \(M\) or \(\mathscr{F}\), a property that will be more thoroughly explored when we discuss natural transformations.
Natural Transformation and Adjoint Functors
A natural transformation provides a coherent way of transforming one functor into another, ensuring that the 'shape' of the categories involved is respected. If we have two functors, like \(^{\sim}\) and \(\Gamma\) from our exercise that relate categories, a natural transformation gives a systematic method to relate the outcomes of these functors on every object in one category to outcomes in the other category.

Natural transformations are pivotal when discussing adjoint functors like we see in this exercise. Adjoint functors provide a bridge between different worlds, like modules and sheaves, by showing how a construction in one category (like sending an \(A\)-module \(M\) to its corresponding sheaf \(\widetilde{M}\)) has a counterpart in another category (like taking global sections of a sheaf with \(\Gamma\)).

In our exercise, the adjunction between the functors \(^{\sim}\) and \(\Gamma\) manifests through the natural isomorphism between the hom-sets, which reflects the deep connection between algebraic structures (modules) and geometric structures (sheaves on schemes) in algebraic geometry.

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Most popular questions from this chapter

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.

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 \\]

A morphism \(f: X \rightarrow Y\) is quasi-finite if for every point \(y \in Y, f^{-1}(y)\) is a finite set. (a) Show that a finite morphism is quasi-finite. (b) Show that a finite morphism is closed, i.e., the image of any closed subset is closed. (c) Show by example that a surjective, finite-type, quasi-finite morphism need not be finite.

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 real importance of the notion of constructible subsets derives from the following theorem of Chevalley-see Cartan and Chevalley [1, expos茅 7] and see also Matsumura \([2, \mathrm{Ch} .2, \$ 6]:\) let \(f: X \rightarrow Y\) be a morphism of finite type of noetherian schemes. Then the image of any constructible subset of \(X\) is a constructible subset of \(Y\). In particular, \(f(X),\) which need not be either open or closed, is a constructible subset of \(Y\). Prove this theorem in the following steps. (a) Reduce to showing that \(f(X)\) itself is constructible, in the case where \(X\) and \(Y\) are affine, integral noetherian schemes, and \(f\) is a dominant morphism. (b) In that case, show that \(f(X)\) contains a nonempty open subset of \(Y\) by using the following result from commutative algebra: let \(A \subseteq B\) be an inclusion of noetherian integral domains, such that \(B\) is a finitely generated \(A\) -algebra. Then given a nonzero element \(b \in B,\) there is a nonzero element \(a \in A\) with the following property: if \(\varphi: A \rightarrow K\) is any homomorphism of \(A\) to an algebraically closed field \(K,\) such that \(\varphi(a) \neq 0,\) then \(\varphi\) extends to a homomorphism \(\varphi^{\prime}\) of \(B\) into \(K,\) such that \(\varphi^{\prime}(b) \neq 0 .[\) Hint: Prove this algebraic result by induction on the number of generators of \(B\) over \(A\). For the case of one generator, prove the result directly. In the application, take \(b=1 .]\) (c) Now use noetherian induction on \(Y\) to complete the proof. (d) Give some examples of morphisms \(f: X \rightarrow Y\) of varieties over an algebraically closed field \(k,\) to show that \(f(X)\) need not be either open or closed.
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