91Ó°ÊÓ

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.

Step by step solution

01

Part a

To show \(\operatorname{sp}(X)\) is a Zariski space, first note that \(X\) as a noetherian scheme is a noetherian topological space. Next, recall that for a noetherian topological space, every nonempty closed subset can be expressed as a finite union of irreducible closed subsets. Then, each irreducible closed subset in a scheme has a unique generic point where the generic point of irreducible closed subset \(Y\) of a scheme \(X\) is the unique point \(y\) such that \(\overline{\{y\}} = Y\). Hence \(\operatorname{sp}(X)\) is a Zariski space.

02

Part b

By definition, in a Zariski space, every nonempty closed irreducible subset \(Y\) has one unique generic point. So, if we have a minimal nonempty closed subset, it must be irreducible and contains exactly one point, its own unique generic point.

03

Part c

The axiom \(T_{0}\) can be shown with the fact that the ZaRiski topology is defined by closed sets that are either the empty set or finite complements of singletons. Hence given two distinct points, one can always find an open set (or the complement of a closed set) that contains one of the points but not the other.

04

Part d

The statement can be proven by contradiction. Assume there exists a nonempty open subset \(O\) in \(X\) not containing the generic point \(g\). The complement of \(O\) in \(X\), being a closed subset, must be a finite union of irreducible closed subsets of \(X\). Since \(X\) is irreducible and the union does not contain \(g\), we get a contradiction. Hence the generic point must be in every nonempty open subset.

05

Part e

The 'specializes to' relation on a space induces a partial ordering on its points. The 'closed points' are minimal elements of this ordering and 'generic point' of an irreducible component are maximal elements. If \(x_{1} \leadsto x_{0}, x_{1}\) is in the closure of the singleton set \(\{x_{0}\}\), hence any closed subset containing \(x_{1}\) must contain \(x_{0}\). Thus a closed subset contains all specializations of its points.

06

Part f

The functor \(t\) introduced in 2.6 transforms \(X\) into its specialization preorder set equipped with the order topology. Since \(X\) is noetherian, so is its specialization preorder set because it has the same number of points. And each point can be associated with a unique closed set and thus a unique irreducible subset. Hence the \(t(X)\) is a Zariski space. \(X\) is a Zariski space itself if only if the condition of unique generic point is preserved under the homeomorphism \(\alpha\), i.e., \(\alpha\) maps generic points to generic points.

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Key Concepts

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

Noetherian Scheme

A Noetherian scheme is a foundational concept in algebraic geometry and topology. It characterizes a type of scheme with properties that make it manageable and predictable.

To understand Noetherian schemes, it's vital to know what it means for a scheme to be Noetherian.

• A scheme is Noetherian if its underlying topological space is Noetherian. This means every descending chain of closed subsets eventually stabilizes, or equivalently, any open subset is compact.
• In a Noetherian scheme, every non-empty closed subset can be represented as a finite union of irreducible closed subsets.
• Practically, Noetherian schemes simplify various proofs and constructions in algebraic geometry due to their compact nature.

This concept is essential when dealing with Zariski spaces, as it ensures each non-empty closed subset has a unique minimal decomposition.

Generic Point

The concept of a generic point is central when discussing Zariski spaces and plays a critical role in understanding the structure and behavior of irreducible subsets.

A generic point for an irreducible closed subset is a special type of point with unique properties:

• There is a unique point in each irreducible closed subset of a Noetherian topology, known as its generic point.
• This point is characterized by the closure of the singleton set containing it being the entire subset.
• In other words, when you have an irreducible closed subset \(Y\), and point \(y\) such that \(\overline{\{y\}} = Y\), then \(y\) is the generic point of \(Y\).

The importance of the generic point comes from its role in understanding minima in partially ordered spaces and establishing the uniqueness of certain minimal conditions.

Closed Subset

In the context of a Zariski space, closed subsets are crucial for defining its topological properties. Understanding closed subsets helps unravel the structure of such spaces.

Closed subsets have specific characteristics in a Zariski space:

• They are stable under specialization, meaning any specializations (points in the closure of a singleton set) within a space reside within the same closed subset that includes the point it specializes from.
• A minimal non-empty closed subset in a Zariski space contains exactly one point and is known as a closed point.
• Closed subsets, particularly under the Zariski topology, can provide insights into how the space's irreducible components interact and overlap.

Closed subsets effectively highlight the intricacies of Zariski spaces and facilitate easier handling of topological dimensions.

Specialization

Specialization is a key operation in the study of Zariski spaces, revealing how points are related and interact in topological spaces.

The specialization relation can be understood as follows:

• Given two points \(x_0\) and \(x_1\) within a topological space \(X\), \(x_1\) specializes to \(x_0\) (written as \(x_1 \leadsto x_0\)) if \(x_0\) is in the closure of \(\{x_1\}\).
• Specialization induces a partial ordering among the points of a space, linking minima and maxima to closed and generic points, respectively.
• Minimals with regards to specialization are closed points, while maximals are generic points in their irreducible components.

Understanding specialization helps in discerning the "flow" of points within a space under the given topological order, enhancing insight into the structure and behavior of Zariski spaces.