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# 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

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