Scheme implementation overview¶

Various parts of schemes were implemented by David Kohel, David Joyner, and William Stein.

AUTHORS:

David Kohel (2006-01-03): initial version
William Stein (2006-01-05)
William Stein (2006-01-20)

Scheme: A scheme whose datatype might be 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: The 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.
ToricSpace (not implemented): This defines a projective toric variety, which defines a space equipped with a toral action and certain defining data. These generalise projective spaces, but it is not envisioned that the latter should inherit from the ToricSpace type.
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 last 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] )
...
NotImplementedError
```
In fact, we need a test R.ideal([2,1+t]) == R.ideal([1]) in order to make this meaningful.