Definition

An ind-scheme over a field kk is a functor on kk-algebras admitting a presentation

XlimiXiX\simeq\varinjlim_i X_i

by schemes XiX_i whose transition maps are closed immersions. It is ind-projective if the XiX_i can be chosen projective.

The presentation is not part of the object: two filtered systems define the same ind-scheme when their colimit functors are isomorphic.

References
  1. Alexander Beilinson and Vladimir Drinfeld, Chiral Algebras, American Mathematical Society, 2004, §7.11.