r"""
Scheme implementation overview

Various parts of schemes were implemented by
Volker Braun,
David Joyner,
David Kohel,
Andrey Novoseltsev,
and
William Stein.

AUTHORS:

- David Kohel (2006-01-03): initial version
- William Stein (2006-01-05)
- William Stein (2006-01-20)
- Andrey Novoseltsev (2010-09-24): update due to addition of toric varieties.

.. comment seperator

- **Scheme:**
  A scheme whose datatype might not be defined in terms
  of algebraic equations: e.g. the Jacobian of a curve may be
  represented by means of a Scheme.

- **AlgebraicScheme:**
  A scheme defined by means of polynomial equations, which may be
  reducible or defined over a ring other than a field.
  In particular, the defining ideal need not be a radical ideal,
  and an algebraic scheme may be defined over Spec(R).

- **AmbientSpaces:**
  Most effective models of algebraic scheme will be
  defined not by generic gluings, but by embeddings in some fixed
  ambient space.

- **AffineSpace:**
  Affine spaces and their affine subschemes form the most important
  universal objects from which algebraic schemes are built.
  The affine spaces form universal objects in the sense that a morphism
  is uniquely determined by the images of its coordinate functions and
  any such images determine a well-defined morphism.

  By default affine spaces will embed in some ordinary projective space,
  unless it is created as an affine patch of another object.

- **ProjectiveSpace:**
  Projective spaces are the most natural ambient spaces for most
  projective objects.  They are locally universal objects.

- **ProjectiveSpace_ordinary (not implemented)**
  The ordinary projective spaces have the standard weights `[1,..,1]`
  on their coefficients.

- **ProjectiveSpace_weighted (not implemented):**
  A special subtype for non-standard weights.

- **ToricVariety:**
  Toric varieties are (partial) compactifications of algebraic tori
  `\left(\CC^*\right)^n` compatible with torus action. Affine and projective
  spaces are examples of toric varieties, but it is not envisioned that these
  special cases should inherit from ``ToricVariety``.

- **AlgebraicScheme_subscheme_affine:**
  An algebraic scheme defined by means of an embedding in a
  fixed ambient affine space.

- **AlgebraicScheme_subscheme_projective:**
  An algebraic scheme defined by means of an embedding in a fixed ambient
  projective space.

- **QuasiAffineScheme (not yet implemented):**
  An open subset `U = X \setminus Z` of a closed subset `X` of affine space;
  note that this is mathematically a quasi-projective scheme, but its
  ambient space is an affine space and its points are represented by
  affine rather than projective points.

  .. NOTE::

    AlgebraicScheme_quasi is implemented, as a base class for this.

- **QuasiProjectiveScheme (not yet implemented):**
  An open subset of a closed subset of projective space; this datatype
  stores the defining polynomial, polynomials, or ideal defining the
  projective closure `X` plus the closed subscheme `Z` of `X` whose complement
  `U = X \setminus Z` is the quasi-projective scheme.

  .. NOTE::

    The quasi-affine and quasi-projective datatype lets one create schemes
    like the multiplicative group scheme
    `\mathbb{G}_m = \mathbb{A}^1\setminus \{(0)\}`
    and the non-affine scheme `\mathbb{A}^2\setminus \{(0,0)\}`.  The latter
    is not affine and is not of the form `\mathrm{Spec}(R)`.

TODO List
---------

- **PointSets and points over a ring:**
  For algebraic schemes `X/S` and `T/S` over `S`, one can form
  the point set `X(T)` of morphisms from `T\to X` over `S`.

  ::

        sage: PP.<X,Y,Z> = ProjectiveSpace(2, QQ)
        sage: PP
        Projective Space of dimension 2 over Rational Field

  The first line is an abuse of language -- returning the generators
  of the coordinate ring by ``gens()``.

  A projective space object in the category of schemes is a locally
  free object -- the images of the generator functions *locally*
  determine a point.  Over a field, one can choose one of the standard
  affine patches by the condition that a coordinate function `X_i \ne 0`

  ::

        sage: PP(QQ)
        Set of rational points of Projective Space
        of dimension 2 over Rational Field
        sage: PP(QQ)([-2,3,5])
        (-2/5 : 3/5 : 1)

  Over a ring, this is not true, e.g. even over an integral domain which is not
  a PID, there may be no *single* affine patch which covers a point.

  ::

        sage: R.<x> = ZZ[]
        sage: S.<t> = R.quo(x^2+5)
        sage: P.<X,Y,Z> = ProjectiveSpace(2, S)
        sage: P(S)
        Set of rational points of Projective Space of dimension 2 over
        Univariate Quotient Polynomial Ring in t over Integer Ring with
        modulus x^2 + 5

  In order to represent the projective point `(2:1+t) = (1-t:3)` we
  note that the first representative is not well-defined at the
  prime `pp = (2,1+t)` and the second element is not well-defined at
  the prime `qq = (1-t,3)`, but that `pp + qq = (1)`, so globally the
  pair of coordinate representatives is well-defined.

  ::

        sage: P( [2, 1+t] )
        (2 : t + 1 : 1)

  In fact, we need a test ``R.ideal([2,1+t]) == R.ideal([1])`` in order
  to make this meaningful.

"""

