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Counterexample in section 'The category of schemes'
> But all proper closed subsets of Spec (Z[X]) are finite.
Well-behavedness of the category of schemes
The "motivation" section declares that "admitting arbitrary schemes makes the whole category of schemes better-behaved." Could somebody more knowledgeable than myself makes this more precise? The category of rings, and therefore the category of affine schemes, is already complete and cocomplete... and I don't know enough of this to know what nicer properties there are. 22.214.171.124 (talk) 01:15, 26 October 2009 (UTC)
- This has special meaning, and I think it has to do with the things you'd find at nLab. However, if you consider fields to be the algebras for affine space, and you consider affine space to be the Euclidean-like geometries, then something locally covered by affine schemes is a manifold. Like making manifolds, you want to be able to glue affine spaces together to describe good but nontrivial geometries. Hence the importance of projective spaces not being affine schemes. It may be complete and cocomplete, but connected sums are neither limits nor colimits. And god forbid you try and glue one to itself. ᛭ LokiClock (talk) 07:02, 19 November 2012 (UTC)
- While it's true that the category of affine schemes is complete and cocomplete, that's basically the wrong question. The right question is, after we take the topology into account, do we have enough affine schemes? And now the answer is a strong no: Affine schemes have Zariski open subsets, these subsets really are useful, and they are not captured by affine schemes. For example, I may be interested in A2 − (0,0), which is a Zariski open subset that does not correspond to an affine scheme. There are plenty of good reasons to be interested in this set; for instance, it's the complement of the origin, and the origin is interesting. But if I insist on staying within the framework of affine schemes, I have no language to describe A2 − (0,0).
- That may sound like a technical point, but technical considerations can be important: You don't know for sure if something's true until you prove it, and if your technical setup is too weak the statement you want to prove may be out of reach. For example, I understand that this was the case in algebraic K-theory in the 70s: It was believed that algebraic K-theory had certain obvious-looking functorial properties with respect to open immersions of schemes, but these were not known in general even for ring maps of the form R → Rf. There was simply not enough technique. Finally the situation was resolved by Thomason, and the relevant exact sequence was established for any quasi-compact and quasi-separated scheme (as a side effect, also establishing it for general affine schemes); and his techniques were fundamentally global and did not amount to reducing to the affine case.
- If we want to take the topology on the category of affine schemes into account, then we shouldn't look in the category of affine schemes. Completeness or cocompleteness of that category is therefore the wrong question. Instead, we look at the category of sheaves on the category of affine schemes, i.e., set-valued contravariant functors from affine schemes to sets, or equivalently set-valued covariant functors from commutative rings to sets, subject to the gluing axiom. Such sheaves represent global data. Pullback of sheaves is like restriction, and so an object which is locally an affine scheme is like a globalized version of an affine scheme; this is a scheme. If we do the same construction with the étale topology, then we get the category of algebraic spaces. This idea also tells us how to construct manifolds (topological, PL, smooth, analytic, or complex depending upon the maps we allow) out of open subsets of Euclidean space. Also it does real and complex analytic spaces, and I think rigid analytic spaces, too (though that's not a theory I'm very familiar with). But all of these work only because we pay attention to the topology.
- Stacks come out of more general considerations on what it means to talk about local behavior. A stack is essentially a sheaf of categories. The point is that, if you take seriously the viewpoint that all of the interesting data about a space can be described using its category of sheaves, then instead of associating to each object a set like a sheaf does, we should associate a whole category. These categories should be compatible with each other in appropriate ways. This definition is hugely general; it is hard to find a geometric object in "the real world" which is not a stack. In algebraic geometry, the situation is even better, because most stacks that turn up in geometric problems are very nice: They are Deligne–Mumford or Artin stacks. These are special stacks that are quite close to being schemes (for Deligne–Mumford stacks) or algebraic spaces (for Artin stacks). But the fundamental idea is still that they are put together from local data, just in a nice way. Ozob (talk) 01:14, 11 September 2014 (UTC)