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


such that

  1. 1.


  2. 2.

    Each Xi is a finite dimensional algebraic variety over 𝕂

  3. 3.

    The inclusions ij:Xj→Xj+1 are closed embeddingsMathworldPlanetmathPlanetmath of algebraic varieties

The ring of regular functionsMathworldPlanetmath on an ind-variety X is defined to be 𝕂⁑[X]:=lim←⁑𝕂⁑[Xj] where the limit is taken with respect to the family of maps {ij*:𝕂⁑[Xj+1]→𝕂⁑[Xj]}jβ‰₯0.

This ring is given the structureMathworldPlanetmath of a topological ring by letting each 𝕂⁑[Xj] have the discrete topology and 𝕂⁑[X] have the induced inverse limit topologyMathworldPlanetmathPlanetmath, i.e. the topology induced from the canonical inclusion lim←⁑𝕂⁑[Xj]βŠ‚βˆj𝕂⁑[Xj] and the product topology on ∏j𝕂⁑[Xj].

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

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


Let 𝒦:=𝕂⁑((t)) be the ring of formal Laurant series over 𝕂 and π’ͺ:=𝕂⁑[[t]] be its ring of integers, the formal Taylor seriesMathworldPlanetmath. Let V=𝕂n. Then the set X of π’ͺ-lattices (π’ͺ-submodules of maximal rank) in VβŠ—π•‚π’¦ is an example of a (non-finite dimensional) projective ind-variety using the filtration


where L0:=VβŠ—π•‚π’ͺ.

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


  • 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: ”On some infinite-dimensional groups. IIβ€œ 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