Separatedness And Completeness As Analogues Of Hausdorffness And Compactness
Completeness and separatedness in algebraic geometry serve as crucial analogues to compactness and the Hausdorff property in classical topology. This article delves into why these concepts are considered the "correct" analogies, drawing insights from Gathmann's notes on algebraic geometry and expanding on the fundamental properties that make them so.
Separatedness as an Analogue of Hausdorffness
In the realm of separatedness, its analogy to the Hausdorff property in topology is a cornerstone concept. A topological space is termed Hausdorff if, for any two distinct points, there exist disjoint open neighborhoods. This property ensures that points can be distinguished from one another topologically. In algebraic geometry, separatedness plays a similar role, albeit in the context of varieties and schemes. A morphism X → Y is separated if the diagonal morphism X → X ×Y X is a closed immersion. This might seem like an abstract definition, but its implications are profound. Separatedness ensures that the image of the diagonal morphism is a closed subset, which is a geometric way of saying that distinct points in X that map to the same point in Y can be distinguished in a certain sense. The importance of separatedness becomes evident when considering pathological examples of non-separated schemes, which can exhibit bizarre behavior, such as non-unique limits of sequences. For instance, consider the affine line with a doubled origin. This space is formed by taking two copies of the affine line and identifying all points except for the origins. The resulting space is non-Hausdorff because the two origins cannot be separated by disjoint open sets. Similarly, in algebraic geometry, non-separated schemes can lead to difficulties in defining and working with intersections and other fundamental geometric constructions. Separatedness, therefore, is a natural condition to impose to avoid such pathologies and to ensure that the geometry behaves in a way that aligns with our intuition from classical topology. Moreover, separatedness is crucial for the uniqueness of morphisms. If f, g: X → Y are two morphisms from a scheme X to a separated scheme Y, then the set of points where f and g agree is a closed subset of X. This property is vital for many constructions in algebraic geometry, such as defining the graph of a morphism and ensuring that certain universal properties hold. In essence, separatedness provides a framework for working with schemes and varieties in a way that mirrors the well-behaved nature of Hausdorff spaces in topology, making it an indispensable concept in algebraic geometry.
Completeness as an Analogue of Compactness
The notion of completeness in algebraic geometry mirrors the concept of compactness in topology, serving as a critical property, particularly for projective varieties. In topology, a space is compact if every open cover has a finite subcover. This definition ensures that compact spaces are, in some sense, "small" or "bounded." In algebraic geometry, completeness captures a similar idea, but in a more geometric way. A variety X over a field k is said to be complete if for any variety Y over k, the projection morphism X ×k Y → Y is a closed map. This definition might appear technical, but its implications are far-reaching. A closed map is one that sends closed sets to closed sets. Thus, completeness implies that the image of a closed subset of X ×k Y under the projection map is a closed subset of Y. This property is a powerful form of properness, ensuring that geometric objects behave predictably under projections. The analogy to compactness becomes clearer when considering the classical topology of complex projective varieties. As Gathmann notes, projective varieties are compact in the classical topology. This compactness is a consequence of the fact that projective space itself is compact, and closed subvarieties of compact spaces are also compact. Completeness in algebraic geometry can be seen as an algebraic analogue of this topological compactness. One of the most important consequences of completeness is the valuative criterion of completeness. This criterion provides a way to check whether a variety is complete by considering morphisms from the spectrum of a discrete valuation ring (DVR). Specifically, a variety X is complete if and only if every morphism from the spectrum of the fraction field of a DVR to X can be extended to a morphism from the spectrum of the DVR itself. This criterion is particularly useful in proving that certain varieties are complete and in understanding the behavior of morphisms to complete varieties. Moreover, completeness is essential for the proper definition of intersection theory and for proving fundamental theorems such as the finiteness of cohomology groups for coherent sheaves on projective varieties. In summary, completeness in algebraic geometry serves as a powerful analogue of compactness in topology, providing a framework for working with varieties that exhibit a form of geometric "boundedness" and ensuring that certain fundamental properties hold, making it a cornerstone concept in the field.
The Importance of Properness
The concept of properness bridges the gap between separatedness and completeness, acting as a unifying theme. A morphism f: X → Y is proper if it is separated, of finite type, and universally closed. The universal closedness condition means that for any morphism Z → Y, the base change X ×Y Z → Z is a closed map. Properness, therefore, combines the desirable properties of separatedness and completeness, ensuring that the morphism behaves well geometrically. In classical topology, proper maps are those for which the preimage of any compact set is compact. This characterization highlights the close relationship between properness and compactness. In algebraic geometry, properness plays a similar role, ensuring that the morphism preserves a notion of "boundedness" or "finiteness." Proper morphisms are crucial for many fundamental results in algebraic geometry. For instance, the Proper Mapping Theorem states that the image of a proper morphism is closed and that the higher direct images of coherent sheaves under a proper morphism are coherent. This theorem is a cornerstone of modern algebraic geometry and has numerous applications in intersection theory, moduli theory, and the study of algebraic cycles. The connection between properness and completeness is particularly evident in the context of varieties over a field. A variety X is complete if and only if the structure morphism X → Spec k is proper. This equivalence underscores the fact that completeness is a property of the variety itself, while properness is a property of a morphism. In summary, properness serves as a central concept in algebraic geometry, unifying separatedness and completeness and providing a powerful framework for studying morphisms between varieties and schemes. Its analogy to compactness in topology further solidifies its importance, making it an indispensable tool for understanding the geometry of algebraic varieties.
Valuative Criteria: A Deeper Dive
To further understand completeness and separatedness, delving into the valuative criteria provides essential insights. These criteria offer a powerful means of verifying these properties using discrete valuation rings (DVRs). A DVR is a local ring with a unique maximal ideal and a valuation that measures the "size" of elements. The valuative criterion for separatedness states that a morphism f: X → Y is separated if for any DVR R with fraction field K, and any two morphisms Spec K → X that agree when composed with f, there is at most one morphism Spec R → X that extends both. This criterion captures the idea that separated morphisms have a uniqueness property for extensions of morphisms from generic points. Geometrically, this means that if two paths in X that agree in Y come arbitrarily close to each other, they must be the same path. The valuative criterion for completeness, on the other hand, states that a morphism f: X → Y is complete if for any DVR R with fraction field K, any morphism Spec K → X can be extended to a morphism Spec R → X. This criterion ensures that morphisms from generic points can be extended to the entire DVR, which is a form of "filling in" property. Geometrically, this means that if a curve in X has a generic point mapping into X, then the entire curve can be extended to lie in X. These valuative criteria are not only useful for verifying separatedness and completeness but also provide a deeper understanding of the geometric meaning of these properties. They connect the abstract algebraic definitions to more intuitive geometric notions, making them invaluable tools for working with varieties and schemes. In particular, the valuative criterion for completeness is often used to prove that projective varieties are complete, by showing that any morphism from the generic point of a curve can be extended to the entire curve. In summary, the valuative criteria for separatedness and completeness provide a powerful and practical means of verifying these properties, while also offering a deeper geometric understanding of their meaning. They highlight the close relationship between separatedness, completeness, and the behavior of morphisms from DVRs, making them essential tools in algebraic geometry.
Projective Varieties and Their Properties
Projective varieties serve as a prime example of varieties that exhibit both separatedness and completeness, reinforcing their significance in algebraic geometry. A projective variety is a closed subvariety of projective space, which is a fundamental object in algebraic geometry. Projective spaces are constructed by taking the set of lines through the origin in a vector space and endowing it with a suitable algebraic structure. Projective varieties inherit many desirable properties from projective space, including separatedness and completeness. The separatedness of projective varieties follows from the fact that projective space itself is separated. This can be shown using the valuative criterion for separatedness or by directly verifying that the diagonal morphism is a closed immersion. The completeness of projective varieties is a deeper result, but it is a cornerstone of algebraic geometry. As Gathmann notes, projective varieties are compact in the classical topology when the base field is the complex numbers. This compactness is a consequence of the fact that complex projective space is compact, and closed subvarieties of compact spaces are compact. In algebraic geometry, the completeness of projective varieties is typically proven using the valuative criterion for completeness. This involves showing that any morphism from the generic point of a curve to a projective variety can be extended to the entire curve. The completeness of projective varieties has numerous important consequences. For instance, it implies that the intersection of two projective varieties in projective space is always non-empty, provided their dimensions add up to at least the dimension of the ambient projective space. This result is a fundamental theorem in intersection theory and has far-reaching applications in algebraic geometry and related fields. Moreover, the completeness of projective varieties is crucial for the study of moduli spaces, which are spaces that parameterize families of geometric objects. Many moduli spaces are constructed as quotients of projective varieties, and the completeness of the underlying projective varieties ensures that the moduli spaces have desirable properties. In summary, projective varieties serve as a central example of varieties that are both separated and complete, highlighting the importance of these properties in algebraic geometry. Their completeness has numerous important consequences, making them a fundamental object of study in the field.
In conclusion, separatedness and completeness stand as the algebraic geometric counterparts to Hausdorffness and compactness in topology, respectively. Their definitions, though abstract, ensure the well-behaved nature of varieties and morphisms, mirroring the intuitive properties of their topological analogues. Through the lens of valuative criteria and the example of projective varieties, the profound implications of separatedness and completeness become evident, solidifying their role as fundamental concepts in algebraic geometry.