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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. 126.96.36.199 (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)