Valuative Criterion: Why All DVRs Matter?
Hey guys! Let's dive into the fascinating world of algebraic geometry, specifically focusing on a tricky concept: the valuative criterion of properness. This criterion is super important when we're trying to figure out if a morphism (a fancy term for a map between schemes) has the property of being proper. Properness, in a nutshell, is a notion of compactness in the algebraic world, and it has some really nice implications.
When we talk about properness, the valuative criterion comes into play. It essentially gives us a way to check for properness by looking at what happens with discrete valuation rings (DVRs). Now, I know that might sound like a mouthful, but we'll break it down. A DVR is a special kind of ring that's really useful for studying the behavior of schemes “at infinity” or near singularities. The criterion essentially states that a morphism f from X to Y is proper if and only if for every DVR R, a certain condition P(R) holds. That condition P(R) involves looking at maps from the spectrum of the fraction field of R into X and seeing if they can be extended to maps from the spectrum of R itself into X. It’s a bit abstract, but don’t worry, we’ll get there!
The heart of this discussion, though, is a particular question: Can we find an example where we absolutely need to check all DVRs to determine properness? Are there cases where looking at just a few DVRs won't cut it? This is a crucial question because, in practice, checking the valuative criterion for every single DVR can be a daunting task. If we could get away with checking only a specific set of DVRs, that would be a huge win. But, alas, the world of math is rarely that simple. There are indeed situations where we need to consider the whole shebang, the entire universe of DVRs, to get the correct answer. We're going to explore one such example in detail, so buckle up!
Okay, let's break this down further. To truly understand why we sometimes need all DVRs, we first need to solidify our understanding of the valuative criterion itself. Imagine we have a morphism f: X → Y, where X and Y are Noetherian schemes of finite type (basically, geometric objects with some nice finiteness properties). The valuative criterion of properness gives us a powerful tool to determine if f is a proper morphism.
The criterion states that f is proper if and only if the following condition holds: For every discrete valuation ring (DVR) R with fraction field K, and for every commutative diagram:
Spec(K) --> X
| /|
| /
v /
f Y
where the map Spec(K) → Y is induced by a map Spec(K) → X, there exists a unique morphism Spec(R) → X such that the following diagram commutes:
Spec(K) --> X <-- Spec(R)
| /| ^
| / | /
v / | /
f Y <-- Spec(R)
Let's unpack this a bit. A DVR, as we mentioned earlier, is a special type of integral domain with a unique maximal ideal and a valuation that measures the “size” of elements. Think of it as a ring that captures the behavior of functions near a point or along a divisor. The spectrum of a ring, denoted Spec(R), is a geometric object associated with the ring. Points in Spec(R) correspond to prime ideals of R, and it gives us a way to visualize the algebraic structure of the ring geometrically.
The fraction field K of R is the field of fractions of R, obtained by inverting all non-zero elements. The spectrum Spec(K) represents the “generic point” of Spec(R). So, the diagram above is essentially saying: If we have a map from the “generic point” of a DVR into X that factors through Y, can we extend this map to the entire DVR? And is this extension unique?
Properness, in this context, can be thought of as a condition that ensures that “limits exist” in a certain sense. If f is proper, then any map from the “generic point” of a DVR into X that “almost” lifts to X (i.e., it factors through Y) can be uniquely extended to a map from the entire DVR into X. This is a powerful idea, and it’s closely related to the topological notion of compactness. The uniqueness part is equally crucial; it ensures that the extension is well-defined and that there aren't multiple ways to “fill in the gap.”
Why do we need to check all DVRs? Well, the valuative criterion is an if and only if statement. This means that if f is proper, the condition with DVRs must hold for every DVR. Conversely, if the condition holds for every DVR, then f is proper. So, if there's even a single DVR for which the condition fails, then f is not proper. This is why we can't just pick and choose a few DVRs; we need to consider the entire landscape to be sure.
Alright, let's get to the juicy part: an example that demonstrates the necessity of checking all DVRs. This is where things get concrete, and we'll see the valuative criterion in action.
Consider the morphism f: X → ℙ¹, where ℙ¹ is the projective line (think of it as the Riemann sphere in complex geometry) and X is obtained from ℙ¹ × ℙ¹ (the product of two projective lines) by blowing up a single point. This is a classic example in algebraic geometry, and it's packed with interesting properties.
Let's break down what this means. The projective line ℙ¹ can be thought of as the set of all lines through the origin in a 2-dimensional vector space. It’s a fundamental object in algebraic geometry, and it’s often used as a building block for more complicated varieties. The product ℙ¹ × ℙ¹ is simply the Cartesian product of two projective lines, and it's a smooth, irreducible surface.
Blowing up a point is a geometric operation that replaces a point on a variety with a projective space (in this case, a projective line). It's a way of resolving singularities or modifying the geometry of a space. In our case, blowing up a point on ℙ¹ × ℙ¹ creates a new surface X that is still smooth, but it has a different structure near the blown-up point.
Now, the morphism f: X → ℙ¹ is the projection map from X onto one of the factors of the original product ℙ¹ × ℙ¹. It's a natural map that forgets the other coordinate. The claim is that this morphism f is not proper.
To see why f is not proper, we need to find a DVR for which the valuative criterion fails. In other words, we need to find a DVR R and a map Spec(K) → X (where K is the fraction field of R) that factors through ℙ¹ but does not extend to a map Spec(R) → X. This is where things get a little technical, but bear with me.
The key is to consider a DVR R that “sees” the exceptional divisor created by the blow-up. The exceptional divisor is the projective line that replaces the blown-up point. It’s a crucial feature of the blow-up, and it’s the reason why f fails to be proper.
Let's say we blew up the point (0, 0) in ℙ¹ × ℙ¹. The exceptional divisor E is then isomorphic to ℙ¹. Now, we can choose a DVR R whose spectrum intersects the exceptional divisor E. We can construct a map Spec(K) → X that maps the generic point of Spec(R) to a point on E. This map factors through f: X → ℙ¹, but it cannot be extended to a map Spec(R) → X. The reason is that any such extension would have to map the special point of Spec(R) (the closed point) to a point on X, but there’s no consistent way to do this while preserving the commutativity of the diagram. The exceptional divisor acts as an obstruction to extending the map.
This example demonstrates that f is not proper because the valuative criterion fails for at least one DVR. But here’s the crucial point: It might be tempting to think that we could have checked only DVRs that don’t “see” the exceptional divisor and concluded that f is “almost proper” in some sense. However, this is not the case. To truly determine that f is not proper, we need to consider DVRs that are sensitive to the blow-up, DVRs that “see” the exceptional divisor. This highlights the necessity of considering all DVRs in the valuative criterion.
So, why is this whole discussion about needing all DVRs important? It's not just an abstract mathematical curiosity; it has significant implications for how we work with properness in algebraic geometry.
The valuative criterion of properness is a cornerstone tool for proving that morphisms are proper. Properness, as we've mentioned, is a fundamental concept that has far-reaching consequences. Proper morphisms have many desirable properties; for instance, they preserve finiteness, and they are closed maps (meaning they map closed sets to closed sets). These properties are crucial in many geometric arguments and constructions.
The fact that we sometimes need to check all DVRs to verify properness underscores the subtle and global nature of the concept. It tells us that properness is not just a local property; it depends on the behavior of the morphism at “infinity” or near singularities, which are precisely the regions that DVRs are designed to probe. This means we can't just look at a few points or a few local rings; we need to consider the entire spectrum of DVRs to get the full picture.
In practical terms, this means that proving properness can be challenging. There's no magic bullet, no single set of DVRs that will always work. We need to be mindful of the geometry of the situation and choose our DVRs carefully. This often involves a deep understanding of the specific morphism and the schemes involved.
However, the necessity of checking all DVRs also has some positive implications. It forces us to think carefully about the geometry of our schemes and morphisms. It encourages us to develop a more nuanced understanding of properness and its relationship to other geometric concepts. By grappling with the complexities of the valuative criterion, we gain a deeper appreciation for the richness and subtlety of algebraic geometry.
Furthermore, this discussion highlights the power and importance of DVRs in algebraic geometry. DVRs are not just technical tools; they are essential probes that allow us to explore the behavior of schemes in detail. They provide a bridge between algebra and geometry, allowing us to translate algebraic properties of rings into geometric properties of schemes. Understanding DVRs is crucial for anyone working in algebraic geometry, and the valuative criterion of properness is a prime example of their utility.
In conclusion, the example we've discussed demonstrates that there are indeed situations where the valuative criterion of properness necessitates checking all DVRs. The blow-up example shows that the behavior of a morphism near exceptional divisors can be subtle, and we need to be careful to consider DVRs that “see” these features. This highlights the global nature of properness and the importance of DVRs as probes in algebraic geometry.
While the need to check all DVRs might seem daunting, it's also a testament to the power and depth of the valuative criterion. It forces us to think critically about the geometry of our schemes and morphisms, and it ultimately leads to a richer understanding of properness and its implications. So, the next time you're wrestling with a properness problem, remember the blow-up example and the importance of considering all the DVRs!
