ind-variety

Let $𝕂$ be a field. An ind-variety over $𝕂$ is a set $X$ along with afiltration:

$$X_0\subset X_1\subset \mathrm{\cdots }{X}_n\subset \mathrm{\cdots }$$

such that

1. $X=\bigcup _{j\ge 0}{X}_j$

2. Each $X_i$ is a finite dimensional algebraic variety over $𝕂$

3. The inclusions $i_j:X_j\to X_{j+1}$ are closedembeddings of algebraic varieties

The ring of regular functions on an ind-variety $X$ is defined to be$𝕂[X]:=\underleftarrow{\lim }𝕂[X_j]$ where the limit is takenwith respect to the family of maps ${\{i_j^{*}:𝕂[X_{j+1}]\to 𝕂[X_j]\}}_{j\ge 0}$.

This ring is given the structure of a topological ring by letting each$𝕂[X_j]$ have the discrete topology and $𝕂[X]$ have the inducedinverse limit topology, i.e. the topology induced from the canonicalinclusion $\underleftarrow{\lim }𝕂[X_j]\subset {\prod }_j𝕂[X_j]$ and theproduct topology on ${\prod }_j𝕂[X_j]$.

An ind-variety is called affine (resp. projective) if each$X_j$ is affine (resp. projective).

The notion of an ind-variety goes back to Igor Shafarevich in [3] and [4].

Examples

Let $𝒦:=𝕂((t))$ be the ring of formal Laurantseries over $𝕂$ and $𝒪:=𝕂[[t]]$ be itsring of integers, the formal Taylor series. Let $V={𝕂}^n$. Then the set $X$ of $𝒪$-lattices ($𝒪$-submodules of maximal rank) in $V{\otimes }_{𝕂}𝒦$ is an example of a (non-finitedimensional) projective ind-variety using the filtration

$$X_i:=\{L\in X\mid t^i{L}_0\subset L\subset t^{-i}{L}_0,{dim}_{𝕂}L/t^i{L}_0=in\}$$

where $L_0:=V{\otimes }_{𝕂}𝒪$.

(cf. [1] section 11, or [2] appendix C part 7)