Grassmannian - The Grassmannian As A Scheme

The Grassmannian As A Scheme

In the realm of algebraic geometry, the Grassmannian can be constructed as a scheme by expressing it as a representable functor.

Let be a quasi-coherent sheaf on a scheme S. Fix a positive integer r. Then the Grassmannian functor associates to each S-scheme T the set of quotient modules of locally free of rank r on T.

This functor is representable by a separated S-scheme . The latter is projective if is finitely generated. When S is the spectrum of a field k, then the sheaf is given by a vector space V and we recover the usual Grassmannian variety of the dual space of V.

By construction, the Grassmannian scheme is compatible with base changes: for any S-scheme S′, we have a canonical isomorphism

In particular, for any point s of S, the canonical morphism {s} = Spec(k(s)) → S, induces an isomorphism from the fiber to the usual Grassmannian over the residue field k(s).

Universal family As the Grassmannian scheme represents a functor, it comes with a universal object which is an object of for, therefore a quotient module of locally free of rank r over . The quotient homomorphism induces a closed immersion from the projective bundle

For any morphism of S-schemes, this closed immersion induces a closed immersion

.

Conversely, any such closed immersion comes from a surjective homomorphism of -modules from to a locally free module of rank r. Therefore, the elements of are exactly the projective subbundles of rank r in . Under this identification, when T = S is the spectrum of a field k and is given by a vector space V, the set of rational points correspond to the projective linear subspaces of dimension r−1 in P(V), and the image of in is the set

.