Hausdorff Premanifolds & Separated Morphisms Explained

Let's dive into a fascinating topic that bridges differential geometry and algebraic geometry, specifically concerning premanifolds, Hausdorff spaces, and closed immersions. This exploration aims to provide a solid understanding of why the Hausdorff property of a premanifold is deeply connected to the nature of its diagonal morphism in the context of locally $\mathbb{R}$-ringed spaces. We will also discuss the motivation behind separated morphisms in scheme theory.

Understanding Premanifolds and Hausdorff Spaces

Premanifolds are, in essence, spaces that locally resemble Euclidean space. Think of them as a generalization of manifolds, where we relax some of the stricter conditions typically imposed. More formally, a premanifold can be defined through an atlas of charts, similar to a manifold, but without necessarily requiring the Hausdorff condition or second countability. This relaxation opens up a broader class of spaces to study, which can be useful in various contexts.

A Hausdorff space, on the other hand, is a topological space where distinct points have disjoint neighborhoods. In simpler terms, if you have two different points in a Hausdorff space, you can always find regions around each point that don't overlap. This property ensures that points are well-separated and that limits are unique, which is crucial for many analytical and geometrical arguments.

The connection between premanifolds and Hausdorff spaces becomes interesting when we consider how the Hausdorff property can be characterized using morphisms in a category of spaces. Specifically, we'll look at locally $\mathbb{R}$-ringed spaces, which are spaces equipped with a sheaf of rings that locally look like rings of functions on open subsets of $\mathbb{R}^n$.

The Diagonal Morphism

The diagonal morphism is a map that takes a point in a space to the corresponding point in the product of the space with itself. More formally, for a space $X$, the diagonal morphism $\Delta: X \to X \times X$ is defined by $\Delta(x) = (x, x)$. This seemingly simple map encodes a lot of information about the space $X$. In the context of topological spaces, the diagonal morphism is closely related to the Hausdorff property.

For instance, a topological space $X$ is Hausdorff if and only if the diagonal $\Delta(X)$ is a closed subset of $X \times X$ with the product topology. This characterization provides a powerful way to check whether a space is Hausdorff by examining the properties of its diagonal.

Closed Immersions and Locally $\mathbb{R}$-ringed Spaces

Now, let's move to the concept of a closed immersion. In the context of locally $\mathbb{R}$-ringed spaces (or more generally, schemes), a closed immersion is a morphism $f: Z \to X$ such that $f$ is a topological embedding and the induced map of sheaves $O_X \to f_*O_Z$ is surjective. Intuitively, this means that $Z$ is a closed subset of $X$, and the functions on $Z$ are precisely the restrictions of functions on $X$.

The statement that a premanifold is Hausdorff if and only if its diagonal morphism is a closed immersion in the category of locally $\mathbb{R}$-ringed spaces is a deep result. It connects a topological property (Hausdorffness) with an algebraic one (closed immersion). This connection allows us to use tools from algebraic geometry to study topological spaces and vice versa.

To understand this better, consider a premanifold $X$. If $X$ is Hausdorff, then the diagonal $\Delta: X \to X \times X$ is a closed subset. Moreover, the morphism $\Delta$ is a closed immersion because the ideal sheaf defining the diagonal is locally generated by differences of coordinate functions, ensuring the surjectivity condition for closed immersions is met. Conversely, if the diagonal morphism is a closed immersion, it implies that the diagonal is a closed subset, which in turn implies that $X$ is Hausdorff.

Motivations for Separated Morphisms in Scheme Theory

The concept of separated morphisms in scheme theory is deeply rooted in the idea of generalizing the Hausdorff property from topological spaces to schemes. In scheme theory, a morphism $f: X \to S$ is said to be separated if the diagonal morphism $\Delta_{X/S}: X \to X \times_S X$ is a closed immersion. Here, $X \times_S X$ denotes the fiber product of $X$ with itself over $S$.

The primary motivation for introducing separated morphisms is to ensure that schemes behave in a manner analogous to Hausdorff spaces. The Hausdorff property is crucial in many areas of topology and analysis because it guarantees the uniqueness of limits and allows for well-behaved constructions. In the context of schemes, separatedness plays a similar role.

Why Separated Morphisms Matter

1. Uniqueness of Limits: Separatedness ensures that limits of morphisms into a scheme are unique. This is particularly important when dealing with algebraic varieties and their function fields. Without separatedness, one might encounter situations where a sequence of points converges to multiple different points, which can lead to pathological behavior.
2. Well-Behaved Fiber Products: Separated morphisms lead to better-behaved fiber products. The fiber product $X \times_S Y$ represents the intersection of $X$ and $Y$ over $S$. If $X$ and $Y$ are separated over $S$, then their fiber product has nicer properties, making it easier to work with in various constructions and proofs.
3. Valuative Criterion for Separatedness: The valuative criterion for separatedness provides a powerful tool for checking whether a morphism is separated. It states that a morphism $f: X \to S$ is separated if and only if for every valuation ring $R$ with fraction field $K$, and for every commutative diagram:

   Spec(K) --> X
     |        |
     v        v
   Spec(R) --> S
   

   there exists at most one morphism $Spec(R) \to X$ making the entire diagram commute. This criterion is particularly useful in proving that certain morphisms are separated and in understanding the geometric implications of separatedness.
4. Applications in Moduli Theory: Separatedness is a crucial condition in moduli theory, where one studies the parameter spaces of certain geometric objects. Requiring moduli spaces to be separated ensures that the objects they parameterize have well-defined moduli, preventing multiple points in the moduli space from representing the same object.

In summary, the concept of separated morphisms in scheme theory is a generalization of the Hausdorff property, providing essential properties that ensure schemes behave in a geometrically intuitive and analytically tractable manner. By requiring the diagonal morphism to be a closed immersion, separated morphisms guarantee uniqueness of limits, well-behaved fiber products, and a robust framework for studying algebraic varieties and their moduli spaces.

Examples and Further Insights

To solidify your understanding, consider some examples. Projective space $\mathbb{P}^n$ over a field $k$ is a classic example of a separated scheme over $k$. The diagonal morphism $\Delta: \mathbb{P}^n \to \mathbb{P}^n \times_k \mathbb{P}^n$ is indeed a closed immersion, reflecting the fact that projective space behaves like a Hausdorff space in this algebraic setting.

On the other hand, consider an affine line with a doubled origin. This space is non-Hausdorff because the two origins cannot be separated by disjoint open sets. Consequently, the diagonal morphism for this space is not a closed immersion, illustrating the connection between the Hausdorff property and the nature of the diagonal morphism.

The statement about premanifolds highlights a fundamental principle: topological properties can often be characterized using algebraic conditions on morphisms. This interplay between topology and algebra is a recurring theme in modern geometry and provides powerful tools for studying complex spaces.

Conclusion

In conclusion, the characterization of a premanifold being Hausdorff if and only if its diagonal morphism is a closed immersion in the category of locally $\mathbb{R}$-ringed spaces is a profound result. It showcases the deep connections between topology and algebra, and it underscores the importance of the Hausdorff property in ensuring well-behaved spaces. Understanding separated morphisms in scheme theory, motivated by the desire to generalize the Hausdorff property, is crucial for anyone delving into algebraic geometry and its applications. By ensuring uniqueness of limits and well-behaved fiber products, separated morphisms provide a robust foundation for studying algebraic varieties and their moduli spaces. So, next time you're pondering the subtleties of schemes and morphisms, remember the humble Hausdorff space and its far-reaching implications.