algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, an ind-scheme is a set-valued

functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...

that can be written (represented) as a

direct limit In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any cate ...

(i.e., inductive limit) of closed embedding of schemes.

Examples

\mathbbP^ = \varinjlim \mathbbP^N

is an ind-scheme. *Perhaps the most famous example of an ind-scheme is an infinite grassmannian (which is a quotient of the loop group of an

algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...

''G''.)

References

*A. Beilinson, Vladimir Drinfel'd, Quantization of Hitchin’s integrable system and Hecke eigensheaves on Hitchin system, preliminary versio

*V.Drinfeld, Infinite-dimensional vector bundles in algebraic geometry, notes of the talk at the `Unity of Mathematics' conference
Expanded version
*http://ncatlab.org/nlab/show/ind-scheme Algebraic geometry {{algebraic-geometry-stub

Examples

See also

References