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Chapter 2: Problem 14

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.

Short Answer

Expert verified
The closed points of a scheme \(X\) of finite type over a field are dense because the underlying topological space can be covered by a finite number of affine schemes, and any open set of points contains a closed point. However, in general, for larger dimensional affine spaces, like \(A^n_{\mathbb{Z}}\), the closed points are not dense.

Step by step solution

01

Determine Dense Closed Points

First, remember that a scheme \(X\) of finite type over a field \(K\) has the property that the underlying topological space \(|X|\) can be covered by a finite number of affine schemes. If \(S\) is the set of closed points of \(X\), then for any open set \(U\), we must show that there is some closed point in \(S\) which also belongs to \(U\) to prove that \(S\) is dense. This can be done by finding the closed points of \(X\) (this includes all of the maximal ideals of the coordinate ring), and then showing that their closure in \(X\) is the entire scheme.
02

Provide Counter-Example

After establishing that the closed points of a scheme of finite type over a field are dense, the task now is to give an example to show that this assertion may not hold for arbitrary schemes. One of the simplest examples of such a scheme is the infinite dimensional affine space \(A^n_{\mathbb{Z}}\). In this case, the set of closed points is not dense because it cannot include any generic point of the space, which is in the closure of every nonempty open set.

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Key Concepts

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

Closed Points
In algebraic geometry, a closed point on a scheme represents a specific kind of point related to closed sets. Schemes can be thought of broadly as spaces that are made up of points that fulfill certain algebraic properties. These closed points are analogous to closed sets in topology but have additional algebraic significance. In the context of an affine scheme, closed points correspond to maximal ideals of the coordinate ring. This is because closed points in the Zariski topology are defined as those points whose corresponding prime ideal is maximal.

In practice, identifying closed points is critical because they help describe the structure of the scheme. For schemes over a field, these closed points form dense subsets. This means they can be found everywhere in the space in the sense that every open set in the scheme contains a closed point, which gives us insight into their encompassing nature.

To understand this better, consider an affine scheme over a field. Here, every maximal ideal relates to a closed point. Due to this relationship, the collection of all closed points interacts closely with the structure of the entire scheme and helps understand that structure deeply.
Density of Points
When we talk about the density of closed points in a scheme, we are referring to the idea that any open subset in the scheme will contain at least one closed point. This property is essential because it highlights how pervasive these points are in the fabric of the scheme. In practical terms, dense sets in topology are those that are so "spread out" that they cannot be avoided.

For schemes of finite type over a field, closed points are dense. This is based on the fact that the space can be covered by affine schemes, and in each affine scheme, closed points are abundant. By ensuring that every open set in the scheme has a closed point, we affirm the concept of density.

However, it's significant to note that density can change depending on the type of scheme. In finite type schemes, this density is a given, as illustrated by every open set containing closed points. But in schemes not necessarily of finite type, such as an infinite dimensional affine space, the condition does not hold, emphasizing the distinct behavior of non-finite type schemes.
Affine Schemes
An affine scheme can be seen as a fundamental building block in algebraic geometry. It is one of the simplest types of schemes and can be viewed as a geometric object that corresponds to a commutative ring. More precisely, an affine scheme is the spectrum of a ring, which is denoted as Spec(R), where R is the ring.

Affine schemes serve an essential role because any scheme locally looks like an affine scheme. This means that schemes can often be understood by examining their affine components, which are easier to study due to their algebraic structure. The connection between the geometry of a scheme and the algebraic properties of its ring is a cornerstone of algebraic geometry.

When a scheme is described as being of finite type, it indicates that the scheme can be covered by a finite number of these affine schemes. This finite type property ensures that the scheme's structure can be comprehensively described and analyzed through its components. Understanding affine schemes is key to grasping more complex scheme types and their properties, especially in terms of their points and how these points are distributed across the space.

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Most popular questions from this chapter

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.

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 \\]

Show that a morphism of sheaves is an isomorphism if and only if it is both injective and surjective.

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.]
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