Infinite projective space 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.
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.
 A: Since rings can have infinite Krull dimension, affine schemes can have infinite dimension.
A: 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.
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).
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).
A: Starting with the affine case, if you try to define infinite dimensional affine space as Spec of k{x1,x2,...], 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.
Edit: Regarding why I am saying that Spec(k[x1,x2,...]) is too big: A (k-)point of Spec(k[x1,x2,...]) is an infinite sequence a1,a2,... of elements of k. If I wanted a vector space of countable dimension, then I should be asking for sequences a1,a2,... 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.
