Make Spec into a functor

[pjbruin]

Sage's Spec command currently produces a Spec object that derives from, but is not the same as, an AffineScheme. The goal of this ticket is

• merge the existing Spec with AffineScheme by moving all existing methods of Spec to AffineScheme;
• upgrade Spec to a functor from CommutativeRings to Schemes (or Schemes(A) if a base ring A is specified), returning objects of type AffineScheme.

Example of the new functionality:

sage: A.<x,y> = QQ[]
sage: Spec(A)
Spectrum of Multivariate Polynomial Ring in x, y over Rational Field
sage: type(Spec(A))
<class 'sage.schemes.generic.scheme.AffineScheme_with_category'>
sage: B.<t> = QQ[]
sage: f = A.hom((t^2, t^3))
sage: Spec(f)
Affine Scheme morphism:
  From: Spectrum of Univariate Polynomial Ring in t over Rational Field
  To:   Spectrum of Multivariate Polynomial Ring in x, y over Rational Field
  Defn: Ring morphism:
          From: Multivariate Polynomial Ring in x, y over Rational Field
          To:   Univariate Polynomial Ring in t over Rational Field
          Defn: x |--> t^2
                y |--> t^3

Two small user-visible changes had to be made to accommodate the new situation:

• If S = Spec(A) is an affine scheme, then the syntax S(a_1, ..., a_n) to construct the topological point of S defined by the prime ideal P = (a₁, ..., a_n) of A is no longer supported. The syntax S(A.ideal(a_1, ..., a_n)) now has to be used instead. This is because it conflicts with the much more useful application of this syntax to construct the point with coordinates (a₁, ..., a_n) if S is (a subscheme of) an affine space Aⁿ.
• Given S = Spec(A) and another scheme X, the result of X(A) is the same as before (a point homset), but X(S), which used to be identical to this, now returns the standard scheme homset. To get the point homset, one now has to type X(A) or X(S.coordinate_ring()). This seems the "principle of least surprise" convention to me, and it is consistent with the fact that X.point_homset() only accepts rings, not affine schemes.

More improvements to affine schemes are made in #7946.

Depends on #15990
Depends on #16156
Depends on #16680

CC: @nthiery @vbraun

Component: algebraic geometry

Keywords: Spec functor affine scheme

Author: Peter Bruin

Branch/Commit: 185b49c

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/16158

[pjbruin]

Branch: u/pbruin/16158-Spec_functor

[pjbruin]

comment:1

For easier reviewing (until #15990 is merged), it is probably helpful to look at the individual commits or at the output of git diff bf9f3f4a 68f25537. [Edit: not necessary anymore.]

[pjbruin]

Commit: 9d4ddc6

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

d87d5a8	Merge branch 'develop' into ticket/16158-Spec_functor
cb82f01	fix raise syntax and a line of documentation

[sagetrac-git]

Changed commit from 9d4ddc6 to cb82f01

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

dcf608f

Merge branch 'develop' into ticket/16158-Spec_functor

[sagetrac-git]

Changed commit from cb82f01 to dcf608f

[pjbruin]

Description changed:

--- 
+++ 
@@ -25,3 +25,5 @@
 Two small user-visible changes had to be made to accommodate the new situation:
 - If *S* = Spec(*A*) is an affine scheme, then the syntax `S(a_1, ..., a_n)` to construct the topological point of *S* defined by the prime ideal *P* = (*a*<sub>1</sub>, ..., *a<sub>n</sub>*) of *A* is no longer supported.  The syntax `S(A.ideal(a_1, ..., a_n))` now has to be used instead.  This is because it conflicts with the much more useful application of this syntax to construct the point with coordinates (*a*<sub>1</sub>, ..., *a<sub>n</sub>*) if *S* is (a subscheme of) an affine space **A**<sup>*n*</sup>.
 - Given *S* = Spec(*A*) and another scheme *X*, the result of `X(A)` is the same as before (a point homset), but `X(S)`, which used to be identical to this, now returns the standard scheme homset.  To get the point homset, one now has to type `X(A)` or `X(S.coordinate_ring())`.  This seems the "principle of least surprise" convention to me, and it is consistent with the fact that `X.point_homset()` only accepts rings, not affine schemes.
+
+More improvements to affine schemes are made in #7946.