In algebraic geometry , the parameterized Grassmann scheme the dimensional subspaces of arbitrary rings .
![d](https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
definition
The Graßmann functor
![{\ displaystyle {\ mathcal {G}} (d, n) \ colon \ left \ {{\ mathcal {Rings}} \ right \} \ to \ left \ {{\ mathcal {Sets}} \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b41acc62fa2aff2fd14af0f8b9969985941ccca)
forms a ring on the set of direct summands of the rank of .
![d](https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab)
![R ^ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ebce0f18894435d56a8fe182e22f135aa8ed07)
The Graßmann scheme is a scheme representing this functor . So it should apply
![{\ displaystyle {\ mathcal {G}} (d, n) (R) = Mor (Spec (R), G (d, n))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31bede2ce4e4edcf2291d21b30991e0482e556d7)
for each ring .
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
From Yoneda's lemma it follows that is uniquely determined. The construction given below shows that it actually exists.
![{\ displaystyle G (d, n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f094bd67482d619bc7248eeba204008b5933e15)
construction
The Graßmann scheme is constructed as follows:
-
,
where the index set of the variables runs through the various -element subsets of and is the ideal generated by the Graßmann-Plücker relations .
![I.](https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f)
![{\ displaystyle \ left ({\ begin {array} {c} n \\ d \ end {array}} \ right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8b4af588f67dc9cd28a1c943819579e00759f3e)
![d](https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab)
![{\ displaystyle \ left \ {1, \ ldots, n \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ec03c2c0a2e6e2620359728e2e54d39be8775b2)
![{\ displaystyle I_ {d, n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed58000f1cf2b818a37feaafe59bfe98908b5dc0)
For (or more generally a body ) the set of closed points is of the classical Graßmann variety .
![k = \ mathbb {R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bf4d723f5664de728ebf62b66da9f83cb55d4fe)
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
![{\ displaystyle G (d, n) (k)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2e60ba7bedf3d61a14db564d68a5e1a5e87afc4)
literature
- Eisenbud-Harris: The Geometry of Schemes. Lecture Notes in Mathematics 197, Springer-Verlag New York. on-line