Separatedness, Completeness, Hausdorffness, And Compactness The Analogues In Topology And Algebraic Geometry

Understanding the core concepts in mathematics often involves drawing parallels between different fields. In algebraic geometry and general topology, the notions of separatedness and completeness frequently arise as analogues to the topological properties of Hausdorffness and compactness, respectively. This article aims to delve into the reasons behind these analogies, particularly in the context of algebraic varieties and their topological counterparts. We will explore these concepts, drawing primarily from Gathmann's notes on algebraic geometry, and discuss why separatedness and completeness serve as the “correct” analogues. This exploration will provide a deeper understanding of these fundamental concepts and their applications in both fields.

Understanding Hausdorffness and Compactness

In general topology, Hausdorffness and compactness are pivotal concepts that define important properties of topological spaces. A topological space is said to be Hausdorff (or T2) if, for any two distinct points, there exist disjoint open neighborhoods. This property ensures that points can be topologically distinguished, which is crucial for many constructions and proofs in analysis and topology. The Hausdorff property is essential because it allows us to define limits uniquely and ensures that continuous functions behave predictably. For example, in a non-Hausdorff space, a sequence might converge to multiple limits, which complicates the analysis significantly. Therefore, the Hausdorff condition provides a level of separation that is fundamental for many standard results in topology and analysis.

Compactness, on the other hand, is a property that, intuitively, captures the idea of a space being “small” or “finite” in some sense. More formally, a topological space is compact if every open cover has a finite subcover. This means that from any collection of open sets that cover the space, one can always choose a finite number of these open sets that still cover the space. Compactness is a powerful condition that has many important consequences. For instance, a continuous function from a compact space to a Hausdorff space maps closed sets to closed sets, and continuous real-valued functions on compact spaces attain their maximum and minimum values. The Heine-Borel theorem provides a concrete example: a subset of Euclidean space is compact if and only if it is closed and bounded. This theorem illustrates how compactness combines the notions of being “bounded” (akin to being finite in extent) and “closed” (containing its limit points).

The Significance of Hausdorffness and Compactness

These properties are not just abstract conditions; they have profound implications in various areas of mathematics. In real analysis, compactness is crucial for proving the existence of maxima and minima of continuous functions on closed and bounded intervals. In complex analysis, compactness plays a vital role in the study of holomorphic functions and their properties. In general topology, these concepts are foundational for defining more advanced topological structures and theorems.

For example, the extreme value theorem, a cornerstone of real analysis, relies heavily on the compactness of the domain. It states that if a function is continuous on a compact set, it attains its maximum and minimum values. This theorem is not generally true for non-compact spaces, highlighting the critical role compactness plays. Similarly, in the study of manifolds, the Hausdorff condition is essential for ensuring that the manifold has a well-defined topology and that topological operations, such as taking quotients, yield spaces with desirable properties.

Furthermore, the concept of compactness extends beyond mere sets of points; it applies to spaces of functions as well. The Arzelà-Ascoli theorem, for instance, provides conditions under which a set of continuous functions is compact in the space of all continuous functions. This result is fundamental in the study of differential equations and functional analysis, where compactness arguments are used to prove the existence of solutions.

In summary, Hausdorffness and compactness are fundamental topological properties that underpin a vast array of results in mathematics. They provide the necessary framework for defining limits, ensuring well-behaved functions, and proving existence theorems. Their analogues in algebraic geometry, separatedness and completeness, serve similar roles in the algebraic setting, ensuring that algebraic varieties have desirable properties and that constructions in algebraic geometry behave as expected.

Separatedness as an Analogue of Hausdorffness

In algebraic geometry, the concept of separatedness serves as an analogue to Hausdorffness in general topology. To understand why, it’s crucial to examine the motivation behind the Hausdorff property. As discussed, Hausdorffness ensures that distinct points in a topological space can be distinguished by disjoint open neighborhoods. This property is vital for the uniqueness of limits and the well-behavedness of continuous functions.

In the context of algebraic varieties, which are geometric objects defined by polynomial equations, a similar notion of “distinguishability” is essential. However, the Zariski topology, the natural topology in algebraic geometry, is much coarser than the familiar topologies on Euclidean spaces. This means that there are fewer open sets, making it more challenging to separate points. The Zariski topology on an algebraic variety is defined by taking the closed sets to be the algebraic subsets, which are the sets of solutions to systems of polynomial equations. Consequently, open sets are complements of these algebraic subsets, and they tend to be very large. This coarseness of the Zariski topology necessitates a different approach to defining a separation condition analogous to Hausdorffness.

The separatedness condition in algebraic geometry addresses this challenge. An algebraic variety X is said to be separated if the diagonal morphism Δ: X → X × X is a closed immersion. Let’s break this down: The product X × X is the Cartesian product of X with itself, and the diagonal morphism Δ maps a point x in X to the point (x, x) in X × X. A morphism is a map between algebraic varieties that is defined locally by polynomial functions. A closed immersion is a morphism that is both an immersion (locally injective) and a closed map (the image is closed). The separatedness condition essentially requires that the diagonal, which consists of all points (x, x), forms a closed subset of the product space X × X.

Why Separatedness Captures the Essence of Hausdorffness

At first glance, this definition may seem abstract, but it captures the essence of Hausdorffness in the algebraic setting. To see this, consider what it means for the diagonal to be closed. If the diagonal Δ(X) is a closed subset of X × X, then its complement is open. Points in the complement of the diagonal are pairs (x, y) with x ≠ y. The condition that the complement is open implies that we can “separate” distinct points in X in a suitable sense. Specifically, it means that there exists an open set in X × X containing (x, y) that does not intersect the diagonal. This is analogous to the existence of disjoint open neighborhoods in a Hausdorff space.

The separatedness condition has profound implications for the geometry of algebraic varieties. For instance, it ensures that the intersection of two algebraic subsets behaves predictably. It also guarantees the uniqueness of certain types of morphisms, which is crucial for many constructions in algebraic geometry. Without separatedness, the behavior of algebraic varieties can become pathological, with undesirable intersections and non-unique morphisms.

Examples and Consequences

Consider the example of affine space 𝔸n over a field k. Affine space is the set of all n-tuples of elements from k, and it is a fundamental object in algebraic geometry. It can be shown that affine space is separated. This is because the diagonal in 𝔸n × 𝔸n is defined by the equations xi = yi for i = 1, ..., n, which are polynomial equations. Thus, the diagonal is an algebraic subset and hence closed in the Zariski topology.

Non-separated schemes exist and are important for illustrating the necessity of the separatedness condition. One classic example is the affine line with a doubled origin. This space is constructed by gluing two copies of the affine line along their complements (i.e., all points except the origin). The resulting space is not separated because the two origins cannot be distinguished in the product space. This non-separatedness leads to several pathological behaviors, such as non-uniqueness of morphisms and unexpected intersections.

In summary, separatedness in algebraic geometry serves as a crucial analogue of Hausdorffness in general topology. It ensures that algebraic varieties exhibit predictable and well-behaved geometric properties. The condition that the diagonal morphism is a closed immersion provides a robust criterion for distinguishing points in the Zariski topology, mirroring the separation property provided by disjoint open neighborhoods in Hausdorff spaces.

Completeness as an Analogue of Compactness

The concept of completeness in algebraic geometry is often seen as the analogue of compactness in topology. This analogy stems from the shared intuition that both properties describe a certain “wholeness” or “closedness” of the space in question. While compactness ensures that every open cover has a finite subcover, completeness in algebraic geometry, particularly for varieties, implies that the variety “contains all its limit points” in a specific geometric sense. This section will delve into the definition of completeness, its connection to compactness, and why it serves as the correct analogue in the context of algebraic geometry.

Defining Completeness in Algebraic Geometry

The formal definition of completeness in algebraic geometry involves the concept of proper morphisms. A morphism f: X → Y between algebraic varieties is said to be proper if it is separated, of finite type, and universally closed. Let’s break down these components:

1. Separatedness: As discussed earlier, separatedness ensures that the diagonal morphism Δ: X → X × X is a closed immersion. This condition is crucial for the well-behavedness of the variety and its morphisms.
2. Finite Type: A morphism f: X → Y is of finite type if, for every affine open subset V of Y, the preimage f−1(V) can be covered by finitely many affine open subsets Ui such that the ring homomorphism associated with f from the coordinate ring of V to the coordinate ring of each Ui is finitely generated. This condition essentially means that the fibers of the morphism are “not too big” in an algebraic sense.
3. Universally Closed: This is the most critical condition for completeness. A morphism f: X → Y is said to be universally closed if, for any morphism Z → Y, the induced morphism X ×Y Z → Z is closed. A morphism is closed if it maps closed subsets to closed subsets. The universal closedness condition implies that not only does f map closed sets to closed sets, but it does so even after base change (i.e., after taking fibered products with other varieties over Y).

An algebraic variety X is then defined to be complete if the structure morphism X → Spec(k) is proper, where Spec(k) is the spectrum of the base field k (essentially a point in this context). In simpler terms, X is complete if the morphism from X to a point is proper. This condition ensures that the variety X has a certain “closedness” property in a geometric sense.

The Analogy with Compactness

The analogy between completeness and compactness becomes clearer when considering the topological implications of these properties. In topology, compactness ensures that every open cover has a finite subcover, which, intuitively, means that the space is “bounded” and “closed” in some sense. Completeness in algebraic geometry provides a similar notion of “boundedness” and “closedness” but in an algebraic context. The universal closedness condition is particularly important here.

Consider a morphism f: X → Y and a closed subset Z in X. If f is a proper morphism, then f(Z) is closed in Y. This is analogous to the property in topology that continuous functions from a compact space to a Hausdorff space map closed sets to closed sets. The universal closedness condition strengthens this by requiring that this property holds even after base change, which means that the “closedness” is preserved under various algebraic operations.

The Importance of Completeness

Completeness is a crucial property in algebraic geometry because it ensures that certain geometric constructions behave predictably. For instance, the Main Theorem of Elimination Theory states that the projection from a complete variety to another variety is a closed morphism. This theorem is fundamental for proving the existence of solutions to systems of polynomial equations and for studying the geometry of algebraic varieties.

One of the most significant examples of complete varieties is projective space ℙn. Projective space is constructed by taking the set of all lines through the origin in an (n+1)-dimensional vector space. It plays a central role in algebraic geometry, and its completeness has many important consequences. For example, the completeness of projective space is crucial for proving that projective varieties (which are closed subvarieties of projective space) are also complete.

Completeness and Compactness in Different Topologies

It’s worth noting that the analogy between completeness and compactness is most direct when considering the classical topology on complex varieties (i.e., the topology induced by the complex numbers). In this setting, a complex projective variety is complete if and only if it is compact in the classical topology. This result underscores the deep connection between the algebraic notion of completeness and the topological notion of compactness.

In summary, completeness in algebraic geometry serves as the analogue of compactness in topology by ensuring a certain “wholeness” and “closedness” of algebraic varieties. The definition involving proper morphisms, particularly the universal closedness condition, captures this essence. Completeness is crucial for ensuring that geometric constructions behave predictably and for proving fundamental theorems in algebraic geometry, such as the Main Theorem of Elimination Theory. The correspondence between completeness and compactness is particularly evident when considering complex projective varieties in the classical topology, highlighting the deep connection between these concepts.

Gathmann's Perspective on Compactness and Completeness

Andreas Gathmann's notes on algebraic geometry provide valuable insights into the relationship between compactness in classical topology and completeness in the context of algebraic varieties. On page 58, Gathmann emphasizes the significance of projective varieties, stating that their most important property is their compactness in the classical topology when considering varieties over the complex numbers. This compactness is not merely a topological curiosity; it is deeply connected to the algebraic properties of these varieties, particularly their completeness.

Gathmann’s discussion underscores the idea that completeness is the algebraic analogue of compactness. He highlights that projective varieties, which are fundamental objects in algebraic geometry, exhibit compactness when viewed as complex manifolds (i.e., in the classical topology). This compactness is a consequence of their algebraic structure and is reflected in the completeness property.

The Role of Projective Varieties

Projective varieties are defined as closed subvarieties of projective space ℙn. Projective space itself is constructed by considering the set of all lines through the origin in an (n+1)-dimensional vector space. This construction gives projective space a rich algebraic and geometric structure. One of the key results in algebraic geometry is that projective varieties are complete. This means that they satisfy the universal closedness condition discussed earlier, ensuring that morphisms from projective varieties behave predictably.

Gathmann’s notes likely delve into the proof that projective varieties are complete, which typically involves showing that the projection morphisms from projective space to other varieties are closed. This result is a cornerstone of elimination theory and has far-reaching implications for the study of algebraic varieties.

Connecting Classical Topology and Algebraic Geometry

The connection between classical topology and algebraic geometry becomes particularly apparent when considering varieties over the complex numbers. In this setting, the Zariski topology on a variety is much coarser than the classical topology, which is induced by the complex metric. However, the completeness property in algebraic geometry has a direct topological interpretation in the classical topology.

Specifically, a complex projective variety is complete if and only if it is compact in the classical topology. This equivalence is a powerful result that bridges the gap between algebraic and topological properties. It means that the algebraic condition of completeness, defined in terms of proper morphisms and universal closedness, corresponds precisely to the topological condition of compactness, defined in terms of open covers and finite subcovers.

Gathmann likely emphasizes this connection to motivate the definition of completeness in algebraic geometry. By showing that completeness corresponds to compactness in the classical topology for complex projective varieties, he provides a compelling reason to view completeness as the correct analogue of compactness in the algebraic setting.

Implications for Morphisms and Constructions

The completeness of projective varieties has significant implications for morphisms and constructions in algebraic geometry. As mentioned earlier, the Main Theorem of Elimination Theory states that the projection from a complete variety to another variety is a closed morphism. This theorem is crucial for proving the existence of solutions to systems of polynomial equations and for studying the geometry of algebraic varieties.

Furthermore, the completeness of projective varieties ensures that certain types of morphisms, such as those from a complete variety to an affine variety, have finite fibers. This property is essential for many constructions in algebraic geometry, such as the construction of moduli spaces and the study of birational geometry.

In summary, Gathmann’s perspective highlights the deep connection between compactness in classical topology and completeness in algebraic geometry. By emphasizing the compactness of projective varieties in the classical topology, Gathmann motivates the definition of completeness as the correct analogue of compactness in the algebraic setting. This connection underscores the importance of projective varieties in algebraic geometry and their well-behaved properties, which are crucial for many constructions and theorems in the field.

Conclusion

In conclusion, the analogy between separatedness and Hausdorffness, and between completeness and compactness, is not merely a superficial resemblance. These concepts serve as fundamental pillars in their respective fields, ensuring that spaces and morphisms exhibit predictable and desirable properties. Separatedness in algebraic geometry captures the essence of Hausdorffness by ensuring that distinct points can be distinguished in a meaningful way within the coarser Zariski topology. Completeness, on the other hand, mirrors compactness by guaranteeing a form of “wholeness” or “closedness” that is critical for geometric constructions and theorems.

Gathmann's notes emphasize the compactness of projective varieties in the classical topology, thereby reinforcing the idea that completeness is the appropriate algebraic analogue of compactness. The parallels between these concepts highlight the deep connections between topology and algebraic geometry, providing a richer understanding of both fields. By grasping the nuances of separatedness and completeness, mathematicians gain essential tools for navigating the intricacies of algebraic varieties and their properties. These concepts not only facilitate a deeper appreciation of algebraic geometry but also bridge the gap between algebraic and topological intuitions, enriching the broader landscape of mathematical thought.