Point functor

In mathematics , the point functor is a term from algebraic geometry . It makes it possible to speak of points in abstractly defined schemes and thus to generalize the classic concept of points of a variety .

definition

A point functor is associated with a scheme ${\displaystyle X}$

${\displaystyle h_{X}\colon \left\{Schemes\right\}^{op}\to \left\{Sets\right\}}$

by

${\displaystyle h_{X}(Y)=Mor(Y,X)}$,

i.e. by assigning the set of morphisms from to to a scheme . ${\displaystyle Y}$ ${\displaystyle Y}$ ${\displaystyle X}$

The mapping defined by is assigned to each morphism . ${\displaystyle f\colon Y\to Z}$ ${\displaystyle g\mapsto g\circ f}$ ${\displaystyle h_{X}(Z)\to h_{X}(Y)}$

The elements of the set are (according to Grothendieck ) called -valent points of . In particular, for a ring with a spectrum, the -value points are referred to as -value points of . ${\displaystyle h_{X}(Y)}$ ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle T}$ ${\displaystyle Spec(T)}$ ${\displaystyle Spec(T)}$ ${\displaystyle T}$ ${\displaystyle X}$

example

Watch with ${\displaystyle SL_{2}:=Spec(R)}$

${\displaystyle R=\mathbb {Z} \left[x_{11},x_{12},x_{21},x_{22}\right]/(x_{11}x_{22}-x_{12}x_{21}-1)}$.

Then, corresponding to th significant points of the elements of which -valent points correspond to the elements of the -valent points correspond to the elements of and the -valent points of the elements of . ${\displaystyle \mathbb {C} }$ ${\displaystyle SL_{2}}$ ${\displaystyle SL(2,\mathbb {C} )}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle SL_{2}}$ ${\displaystyle SL(2,\mathbb {R} )}$ ${\displaystyle F_{q}}$ ${\displaystyle SL_{2}}$ ${\displaystyle SL(2,F_{q})}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle SL_{2}}$ ${\displaystyle SL(2,\mathbb {Z} )}$

On the other hand, not all points would correspond to elements from , because there are also maximal ideals in this ring which correspond to pairs of complex conjugate matrices from . ${\displaystyle Spec(\mathbb {R} \left[x_{11},x_{12},x_{21},x_{22}\right]/(x_{11}x_{22}-x_{12}x_{21}-1))}$ ${\displaystyle SL(2,\mathbb {R} )}$ ${\displaystyle SL(2,\mathbb {C} )}$

Uniqueness

From Yoneda's lemma it follows that the point functor uniquely determines the scheme . In fact, a scheme over a commutative ring is already uniquely determined by the values ​​of on affine schemes over . ${\displaystyle h_{X}}$ ${\displaystyle X}$ ${\displaystyle R}$ ${\displaystyle h_{X}}$ ${\displaystyle R}$

Rational points

For a scheme over a body (i.e. a scheme with a morphism ), -value points are those morphisms whose composition is the identity mapping . ${\displaystyle K}$ ${\displaystyle X}$ ${\displaystyle f\colon X\to Spec(K)}$ ${\displaystyle K}$ ${\displaystyle Spec(K)\to X}$ ${\displaystyle f}$

The -valent points are then exactly the K-rational , closed points of . (A point is called -rational if the quotient field of the local ring is isomorphic to its maximum ideal .) ${\displaystyle K}$ ${\displaystyle X}$ ${\displaystyle K}$ ${\displaystyle K}$

For example, the schema has no -valent points, while the schema has two -valent points. ${\displaystyle Spec(\mathbb {R} [x]/(x^{2}+1))}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {C} }$

literature

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