In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the term affine Grassmannian has two distinct meanings. In one meaning the affine Grassmannian (manifold)

Affine Grassmannian (manifold)

In mathematics, there are two distinct meanings of the term affine Grassmannian. In one it is the manifold of all k-dimensional affine subspaces of Rn , while in the other the affine Grassmannian is a quotient of a group-ring based on formal Laurent series.-Formal definition:Given a...

 is the manifold of all k-dimensional affine subspaces of a finite dimensional vector space, while (described here) the affine Grassmannian of an algebraic group

Algebraic group

In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...

 G over a field k is defined in one of two ways:

• As the coset space G(K)/G(O), where K = k((t)) is the field of formal Laurent series over k and O = kt is the ring of formal power series
  Formal power series
  In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates...
  
  ;
• As the ind-scheme Gr_G which is described as a functor by the following data: to every k-algebra A, Gr_G(A) is the set of isomorphism classes of pairs (E, φ), where E is a principal homogeneous space
  Principal homogeneous space
  In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G such that the stabilizer subgroup of any point is trivial...
  
   for G over Spec At and φ is an isomorphism, defined over Spec A((t)), of E with the trivial G-bundle G × Spec A((t)). By the Beauville–Laszlo theorem
  Beauville–Laszlo theorem
  In mathematics, the Beauville–Laszlo theorem is a result in commutative algebra and algebraic geometry that allows one to "glue" two sheaves over an infinitesimal neighborhood of a point on an algebraic curve. It was proved by .-The theorem:...
  
  , it is also possible to specify this data by fixing an algebraic curve
  Algebraic curve
  In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
  
   X over k, a k-point x on X, and taking E to be a G-bundle on X_A and φ a trivialization on (X − x)_A.

That Gr_G is an ind-scheme does not follow trivially from the definition. Its k-points can be identified with the coset space in the first definition, by choosing a trivialization of E over all of Spec O. When G is a reductive group

Reductive group

In mathematics, a reductive group is an algebraic group G over an algebraically closed field such that the unipotent radical of G is trivial . Any semisimple algebraic group is reductive, as is any algebraic torus and any general linear group...

, Gr_G is in fact ind-projective, i.e., an inductive limit of projective schemes.