ind-variety
Let π be a field. An ind-variety over π is a set X along with a
filtration:
X0βX1ββ―Xnββ― |
such that
-
1.
X=βjβ₯0Xj
-
2.
Each Xi is a finite dimensional algebraic variety over π
-
3.
The inclusions ij:XjβXj+1 are closed embeddings
of algebraic varieties
The ring of regular functions on an ind-variety X is defined to be
π[X]:= where the limit is taken
with respect to the family of maps .
This ring is given the structure of a topological ring by letting each
have the discrete topology and have the induced
inverse limit topology
, i.e. the topology induced from the canonical
inclusion and the
product topology on .
An ind-variety is called affine (resp. projective) if each is affine (resp. projective).
Examples
Let be the ring of formal Laurant
series over and be its
ring of integers, the formal Taylor series. Let . Then the set of -lattices (-submodules of maximal rank) in is an example of a (non-finite
dimensional) projective ind-variety using the filtration
where .
References
- 1 George Lusztig, Singularities, character formulas, and a q-analog of weight multiplicities, AstΓ©risque 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: βOn some infinite-dimensional groups. IIβ 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 |