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# ind-variety

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

$X_{0}\subset X_{1}\subset\cdots X_{n}\subset\cdots$ |

such that

1. $X=\bigcup\limits_{{j\geq 0}}X_{j}$

2. Each $X_{i}$ is a finite dimensional algebraic variety over $\K$

3. The inclusions $i_{j}\colon X_{j}\to X_{{j+1}}$ are closed embeddings 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 taken with respect to the family of maps $\left\{i_{j}^{*}\colon\K[X_{{j+1}}]\to\K[X_{j}]\right\}_{{j\geq 0}}$.

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 induced inverse limit topology, i.e. the topology induced from the canonical inclusion $\underleftarrow{\lim}\K[X_{j}]\subset\prod_{j}\K[X_{j}]$ and the product topology on $\prod_{j}\K[X_{j}]$.

An ind-variety is called *affine* (resp. *projective*) if each
$X_{j}$ is affine (resp. projective).

# Examples

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

$X_{i}:=\left\{L\in X\mid t^{i}L_{0}\subset L\subset t^{{-i}}L_{0},\dim_{{\K}}L% /t^{i}L_{0}=in\right\}$ |

where $L_{0}:=V\otimes_{{\K}}\mathcal{O}$.

# References

- 1
George Lusztig,
*Singularities, character formulas, and a q-analog of weight multiplicities*, Astérisque 101-102 (1983), pp. 208-229. - 2
Shrawan Kumar,
*Kac-Moody Groups, their Flag Varieties and Representation Theory*. Progress in Mathematics Vol. 204. Birkhauser, 2002. - 3
Igor Shafarevich,
*On some infinite-dimensional groups. II*Math USSR Izvestija 18 (1982), pp. 185 - 194. - 4
Igor Shafarevich,
*Letter to the editors: ”On some infinite-dimensional groups. II“*Izv. Ross. Akad. Nauk. Ser. Mat. 59 (1995), pp. 224 - 224.

## Mathematics Subject Classification

14A10*no label found*14L15

*no label found*

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