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 .
i.e. by assigning the set of morphisms from to to a scheme .
The mapping defined by is assigned to each morphism .
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 .
example
Watch with
.
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 .
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 .
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 .
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 .
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 .)
For example, the schema has no -valent points, while the schema has two -valent points.
literature
Eisenbud-Harris: The Geometry of Schemes. Lecture Notes in Mathematics 197, Springer-Verlag New York. on-line