# Infinite dimensional algebraic geometry

Hi!

I am looking for basic references about infinite dimensional algebraic geometry, in particular about the $\textrm{Proj}$ of an infinite dimensional graded commutative algebra.

I have a specific problem as well. I have two projective systems of finite dimensional $\mathbb{C}$-algebras, $A_n$ and $B_n$ ($n$ runs overt the positive integers), both projective limits are infinite dimensional. I have surjective (with a big kernel!) maps $T_n$ from $A_n$ to $B_n$ which are compatible with the projective limit. Moreover, the map $T_{\infty}$ is an isomorphism. Does this mean that the varieties "$\textrm{Proj }A_{\infty}$" and "$\textrm{Proj }B_{\infty}$" are isomorphic?

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If I remember correctly, in the book "Infinite Grassmannians and moduli spaces of G-bundles" by S.Kumar there's a chapter or at least a paragraph on Ind-varieties. –  Qfwfq Apr 4 '13 at 14:56
The good reference for ind-varieties is Chapter IV of S. Kumar, "Kac-Moody groups, their flag varieties and representation theory." Progress in Mathematics, 204. Birkhäuser Boston, Inc., Boston, MA, 2002. –  Jérémy Blanc Apr 4 '13 at 17:43
"Infinite-dimensional graded commutative algebra" doesn't sound like the term you want; presumably you mean infinitely-generated. –  Qiaochu Yuan Apr 4 '13 at 18:55
In fact, my reference is mostly for affine ind-varieties, not projective ones. I do not know what is the good reference for Proj of a limit of algebras, and on how the grading of the limit comes from the grading of the $A_n$ (in a unique way??) –  Jérémy Blanc Apr 5 '13 at 8:03