Set of homomorphisms between two affine schemes

For schemes \(X\) and \(Y\), this module implements the set of morphisms \(Hom(X,Y)\). This is done by SchemeHomset_generic.

As a special case, the Hom-sets can also represent the points of a scheme. Recall that the \(K\)-rational points of a scheme \(X\) over \(k\) can be identified with the set of morphisms \(Spec(K) \to X\). In Sage the rational points are implemented by such scheme morphisms. This is done by SchemeHomset_points and its subclasses.

Note

You should not create the Hom-sets manually. Instead, use the Hom() method that is inherited by all schemes.

AUTHORS:

• William Stein (2006): initial version.

class sage.schemes.affine.affine_homset.SchemeHomset_points_affine(X, Y, category=None, check=True, base=Integer Ring)

Bases: sage.schemes.generic.homset.SchemeHomset_points

Set of rational points of an affine variety.

INPUT:

See SchemeHomset_generic.

EXAMPLES:

sage: from sage.schemes.affine.affine_homset import SchemeHomset_points_affine
sage: SchemeHomset_points_affine(Spec(QQ), AffineSpace(ZZ,2))
Set of rational points of Affine Space of dimension 2 over Rational Field

points(B=0)

Return some or all rational points of an affine scheme.

INPUT:

• B – integer (optional, default: 0). The bound for the height of the coordinates.

OUTPUT:

• If the base ring is a finite field: all points of the scheme, given by coordinate tuples.
• If the base ring is \(\QQ\) or \(\ZZ\): the subset of points whose coordinates have height B or less.

EXAMPLES: The bug reported at #11526 is fixed:

sage: A2 = AffineSpace(ZZ,2)
sage: F = GF(3)
sage: A2(F).points()
[(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)]

sage: R = ZZ
sage: A.<x,y> = R[]
sage: I = A.ideal(x^2-y^2-1)
sage: V = AffineSpace(R,2)
sage: X = V.subscheme(I)
sage: M = X(R)
sage: M.points(1)
[(-1, 0), (1, 0)]

class sage.schemes.affine.affine_homset.SchemeHomset_points_spec(X, Y, category=None, check=True, base=None)

Bases: sage.schemes.generic.homset.SchemeHomset_generic

Set of rational points of an affine variety.

INPUT:

See SchemeHomset_generic.

EXAMPLES:

sage: from sage.schemes.affine.affine_homset import SchemeHomset_points_spec
sage: SchemeHomset_points_spec(Spec(QQ), Spec(QQ))
Set of rational points of Spectrum of Rational Field