Infinite projective space - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T23:33:53Z http://mathoverflow.net/feeds/question/19791 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/19791/infinite-projective-space Infinite projective space Colin Tan 2010-03-30T03:37:01Z 2010-03-30T15:09:14Z <p>Is infinite (say complex) projective space a scheme? More generally, can schemes have infinite cardinal dimension? It seems that infinite dimensional projective space is not a manifold, since it is not locally Euclidean for any R^n.</p> <p>Related question. If inifinite projective space is a scheme, then take a nonclosed point. Taking the closure of this nonclosed point, can we get infinite dimensional subschemes? Sorry for I'm quite foreign to schemes.</p> http://mathoverflow.net/questions/19791/infinite-projective-space/19792#19792 Answer by Mariano Suárez-Alvarez for Infinite projective space Mariano Suárez-Alvarez 2010-03-30T03:44:11Z 2010-03-30T03:44:11Z <p>Since rings can have infinite Krull dimension, affine schemes can have infinite dimension.</p> http://mathoverflow.net/questions/19791/infinite-projective-space/19794#19794 Answer by solbap for Infinite projective space solbap 2010-03-30T03:49:05Z 2010-03-30T05:14:35Z <p>You can define $Proj S$ for any graded ring $S$ and this is certainly a scheme; this is in Hartshorne II.2. Infinite projective space is $Proj S$ where $S = k[x_0, x_1, ....]$ and $k$ is the base field.</p> <p>Regarding your second question, if you take any homogeneous element $f \in S$, then the vanishing of this should define a closed subscheme of codimension 1 (in particular still infinite dimensional).</p> <p>Maybe I should say $Proj S$ is the algebraic analogue of infinite projective space. As a topological space it is very different from $\mathbb{C}^{\infty} - 0$/scaling. But this is even true in the finite dimensional case (Zariski topology is not the same as topology considered as a real or complex manifold).</p> http://mathoverflow.net/questions/19791/infinite-projective-space/19795#19795 Answer by Peter McNamara for Infinite projective space Peter McNamara 2010-03-30T03:57:59Z 2010-03-30T15:09:14Z <p>Starting with the affine case, if you try to define infinite dimensional affine space as Spec of k{x<sub>1</sub>,x<sub>2</sub>,...], then you realise that this is not a vector space of countable dimension, but something much larger. If you want a vector space over k of countable dimension, then this will not be a scheme, but instead will be an ind-scheme. A similar description should hold in the projective case.</p> <p>Edit: Regarding why I am saying that Spec(k[x<sub>1</sub>,x<sub>2</sub>,...]) is too big: A (k-)point of Spec(k[x<sub>1</sub>,x<sub>2</sub>,...]) is an infinite sequence a<sub>1</sub>,a<sub>2</sub>,... of elements of k. If I wanted a vector space of countable dimension, then I should be asking for sequences a<sub>1</sub>,a<sub>2</sub>,... of elements of k, only finitely many of which are non-zero. This latter space is the inductive limit of affine n-space as n tends to infinity.</p>