ind-variety

Let $\operatorname{\mathbb{K}}$ be a field. An ind-variety over $\operatorname{\mathbb{K}}$ 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 $\operatorname{\mathbb{K}}$
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 $\operatorname{\mathbb{K}}[X]:=\varprojlim\operatorname{\mathbb{K}}[X_{j}]$ where the limit is taken with respect to the family of maps $\left\{i_{j}^{*}\colon\operatorname{\mathbb{K}}[X_{j+1}]\to\operatorname{% \mathbb{K}}[X_{j}]\right\}_{j\geq 0}$ .

This ring is given the structure of a topological ring by letting each $\operatorname{\mathbb{K}}[X_{j}]$ have the discrete topology and $\operatorname{\mathbb{K}}[X]$ have the induced inverse limit topology, i.e. the topology induced from the canonical inclusion $\varprojlim\operatorname{\mathbb{K}}[X_{j}]\subset\prod_{j}\operatorname{% \mathbb{K}}[X_{j}]$ and the product topology on $\prod_{j}\operatorname{\mathbb{K}}[X_{j}]$ .

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 $\mathcal{K}:=\operatorname{\mathbb{K}}((t))$ be the ring of formal Laurant series over $\operatorname{\mathbb{K}}$ and $\mathcal{O}:=\operatorname{\mathbb{K}}[[t]]$ be its ring of integers, the formal Taylor series. Let $V=\operatorname{\mathbb{K}}^{n}$ . Then the set $X$ of $\mathcal{O}$ -lattices ( $\mathcal{O}$ -submodules of maximal rank) in $V\otimes_{\operatorname{\mathbb{K}}}\mathcal{K}$ 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_{% \operatorname{\mathbb{K}}}L/t^{i}L_{0}=in\right\}

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

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

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.

Title	ind-variety
Canonical name	Indvariety
Date of creation	2013-03-22 15:30:56
Last modified on	2013-03-22 15:30:56
Owner	benjaminfjones (879)
Last modified by	benjaminfjones (879)
Numerical id	7
Author	benjaminfjones (879)
Entry type	Definition
Classification	msc 14A10
Classification	msc 14L15
Defines	ind-variety