Closed Morphism Of Schemes: Locally Affine? A Tricky Question

Hey guys! Ever stumbled upon a seemingly straightforward question in algebraic geometry that turns out to be a real head-scratcher? That's exactly what happened with Exercise 12.3 in Görtz and Wedhorn's Algebraic Geometry. The question asks whether a closed morphism of schemes is locally affine. It sounds simple enough, but after lingering on Math Stack Exchange for five years without a complete answer, it's clear this problem has some serious depth. So, let's dive in and explore this fascinating question together!

Unpacking the Question

Before we get too far, let's make sure we're all on the same page with the terminology. In the world of algebraic geometry, a scheme is a geometric object that generalizes the notion of an algebraic variety. Think of it as a space built from gluing together affine schemes, which are basically the geometric counterparts of rings. A morphism of schemes is a map between schemes, and a closed morphism is one that behaves nicely with respect to closed subsets. Specifically, the image of any closed subset under a closed morphism is also closed. Now, the crucial term here is locally affine. A morphism f: X -> Y is locally affine if, for every affine open subset V of Y, the preimage f⁻¹(V) is also affine. In simpler terms, if you zoom in on an affine piece of the target scheme, the corresponding piece of the source scheme should also be affine.

The question, therefore, is: if we have a morphism that "preserves closedness" (a closed morphism), does it automatically preserve affineness locally? Intuitively, you might think this should be true. Closed morphisms are well-behaved in many ways, and affineness is a fundamental property in scheme theory. However, as we'll see, the answer isn't so straightforward. This exercise challenges our understanding of the relationship between topological properties of morphisms (like being closed) and geometric properties (like being locally affine). The fact that it has remained unanswered for so long highlights the subtle complexities within algebraic geometry and the importance of careful consideration of definitions and examples. It forces us to think critically about the underlying structures and how they interact. We need to carefully examine the definitions of closed morphisms and locally affine morphisms, and consider potential counterexamples. This exploration will lead us to a deeper appreciation of the nuances of scheme theory and the importance of rigorous proofs in mathematics.

Why the Difficulty?

So, what makes this question so tricky? Well, the challenge lies in finding a morphism that is closed but not locally affine. To prove that a statement is false, all we need is a single counterexample. But constructing such a counterexample in this case requires some cleverness. We need a morphism where the preimage of an affine open set is not affine, even though the morphism itself preserves closed sets. This is where the intuition starts to break down a bit. We're used to closed morphisms having nice properties, so finding one that violates this local affineness condition is surprisingly difficult. One of the main hurdles is the abstract nature of schemes themselves. They are built from rings and their spectra, which can be quite abstract to visualize. This makes it challenging to construct concrete examples and to reason about their properties. Furthermore, the definition of a locally affine morphism involves looking at preimages of affine open sets, which can be difficult to track down in general. We need to understand how the morphism interacts with the underlying topological spaces and the structure sheaves of the schemes involved. This requires a solid understanding of the foundations of scheme theory and a willingness to explore different possibilities. The difficulty of this problem also underscores the importance of having a strong arsenal of examples and counterexamples in algebraic geometry. These examples serve as a testing ground for our intuition and help us to refine our understanding of the concepts involved. Without concrete examples, it's easy to get lost in the abstract definitions and theorems.

Potential Approaches and Suspicions

My suspicion is that the statement is indeed false. To tackle this, we need to think about what can go wrong when taking preimages. Affineness is a fairly restrictive condition, and it's possible to imagine a situation where the preimage of an affine scheme becomes "too complicated" and loses its affineness. Maybe it picks up some extra irreducible components, or maybe its global sections don't form a finitely generated algebra. These are just some potential avenues to explore. One possible strategy is to look for morphisms that "collapse" certain parts of the source scheme, making the preimage of an affine open set non-affine. Another approach is to consider morphisms between projective schemes, where the geometry can be more intricate. Projective schemes are built from homogeneous coordinates, which can lead to interesting behaviors when taking preimages. We might also consider using some known counterexamples in related areas of algebraic geometry as inspiration. For instance, there are examples of proper morphisms that are not projective, which suggests that there might be a similar phenomenon at play here. The key is to carefully analyze the properties of the morphism and the schemes involved, and to look for situations where the preimage of an affine open set could fail to be affine. This might involve constructing a specific example, or it might involve proving a general result that implies the existence of such examples. It's a challenging problem, but one that is sure to deepen our understanding of scheme theory.

Looking for a Counterexample

The hunt for a counterexample is on! We need two schemes, X and Y, and a closed morphism f: X -> Y such that there exists an affine open V ⊆ Y with f⁻¹(V) not affine. This is where things get interesting. We need to think outside the box and consider some less "well-behaved" schemes. Smooth, projective varieties are often too nice for counterexamples. We need something with a bit more… singularity, perhaps? Or maybe some non-reduced structure? These are the kinds of questions we should be asking ourselves. One idea is to consider a morphism that collapses a non-affine scheme onto an affine scheme. For instance, we could try to find a non-affine scheme X and a closed immersion X -> Aⁿ, where Aⁿ is affine space. Then, the projection map Aⁿ -> Spec(k) (where k is the base field) is affine, and the composition X -> Aⁿ -> Spec(k) might be a good candidate for a closed morphism that is not locally affine. Another approach is to think about blowing up schemes. Blowing up is a fundamental operation in algebraic geometry that can change the geometry of a scheme in a controlled way. It's possible that blowing up an affine scheme along a non-affine subscheme could lead to a non-affine scheme that maps closedly onto an affine scheme. This is a more sophisticated idea, but it's worth exploring. The key is to be creative and to try different constructions until we find one that works. It's a process of trial and error, but it's also a process of learning and discovery. Each attempt, even if it fails, can provide valuable insights into the problem and lead us closer to a solution.

Community Input and the Quest for a Solution

It's fascinating that this question has lingered on Math Stack Exchange for so long. It really highlights the collaborative nature of mathematics. Often, the toughest problems require a collective effort, with different people contributing ideas and perspectives. The fact that no one has provided a complete answer in five years suggests that this problem is not trivial. It also underscores the importance of persistence and the willingness to grapple with difficult questions. Sometimes, the most rewarding discoveries come from tackling problems that have resisted solution for a long time. The discussions on Math Stack Exchange likely contain valuable insights and partial solutions. It would be worthwhile to carefully examine these discussions and see if there are any clues or leads that we can follow. Perhaps someone has made a suggestion that hasn't been fully explored, or perhaps there is a subtle error in one of the arguments that has prevented a solution from being found. The community aspect of mathematics is also important for motivation and encouragement. When you're stuck on a problem, it can be helpful to know that others are also struggling with it. It can also be inspiring to see how other mathematicians approach the problem and the different techniques they use. Ultimately, the quest for a solution to this problem is a testament to the enduring power of mathematical curiosity and the joy of collaborative problem-solving.

Open to Suggestions!

I'm throwing this out to the community – have you guys encountered a similar problem? Any insights or potential counterexamples that come to mind? Let's crack this together! Maybe someone out there has a brilliant idea that can finally put this question to rest. Or perhaps this problem will continue to challenge us for years to come, a reminder of the depth and beauty of algebraic geometry. Either way, it's a fascinating journey, and I'm excited to see where it leads. This kind of open-ended problem is what makes mathematics so engaging. It's not just about finding the right answer; it's about the process of exploration, the struggle with difficult concepts, and the satisfaction of finally understanding something new. So, let's keep the discussion going, share our ideas, and hopefully, we can shed some light on this intriguing question. Who knows, maybe our collective efforts will lead to a new theorem or a deeper understanding of the foundations of algebraic geometry. The possibilities are endless!

Conclusion (For Now…)

So, is a closed morphism of schemes locally affine? The jury's still out! But the journey of exploring this question has already been incredibly insightful. It reminds us that even seemingly basic questions in algebraic geometry can be surprisingly deep and that sometimes, the best learning comes from grappling with the unknown. Let's keep digging, keep questioning, and keep exploring the fascinating world of schemes and morphisms! And hey, if you have any ideas, don't hesitate to share!