Graßmann scheme

In algebraic geometry , the parameterized Grassmann scheme the dimensional subspaces of arbitrary rings . ${\displaystyle d}$ ${\displaystyle R^{n}}$ ${\displaystyle R}$

definition

The Graßmann functor

${\displaystyle {\mathcal {G}}(d,n)\colon \left\{{\mathcal {Rings}}\right\}\to \left\{{\mathcal {Sets}}\right\}}$

forms a ring on the set of direct summands of the rank of . ${\displaystyle R}$ ${\displaystyle d}$ ${\displaystyle R^{n}}$

The Graßmann scheme is a scheme representing this functor . So it should apply ${\displaystyle G(d,n)}$

${\displaystyle {\mathcal {G}}(d,n)(R)=Mor(Spec(R),G(d,n))}$

for each ring . ${\displaystyle R}$

From Yoneda's lemma it follows that is uniquely determined. The construction given below shows that it actually exists. ${\displaystyle G(d,n)}$

construction

The Graßmann scheme is constructed as follows:

${\displaystyle G(d,n)=Proj(\mathbb {Z} \left[x_{I}\right]/I_{d,n})}$,

where the index set of the variables runs through the various -element subsets of and is the ideal generated by the Graßmann-Plücker relations . ${\displaystyle I}$ ${\displaystyle \left({\begin{array}{c}n\\d\end{array}}\right)}$ ${\displaystyle d}$ ${\displaystyle \left\{1,\ldots ,n\right\}}$ ${\displaystyle I_{d,n}}$

For (or more generally a body ) the set of closed points is of the classical Graßmann variety . ${\displaystyle k=\mathbb {R} }$ ${\displaystyle k}$ ${\displaystyle G(d,n)(k)}$

literature

• Eisenbud-Harris: The Geometry of Schemes. Lecture Notes in Mathematics 197, Springer-Verlag New York. on-line