Fix TestSuite failures for schemes

[nthiery]

Consider the following situation:

sage: S = Spec(ZZ)
sage: x = S.an_element()

Running TestSuite(S) and TestSuite(x) yields several failures. A related problem is

sage: S
Spectrum of Integer Ring
sage: parent(x)
Set of rational points of Spectrum of Integer Ring

whereas we expect parent(x) is S.

Here is the complete TestSuite report:

sage: TestSuite(S).run()
Failure in _test_an_element:
Traceback (most recent call last):
...
NotImplementedError
------------------------------------------------------------
  Failure in _test_category:
  Traceback (most recent call last):
  ...
  AssertionError: False is not true
  ------------------------------------------------------------
  Failure in _test_pickling:
  Traceback (most recent call last):
  ...
  AssertionError: Point on Spectrum of Integer Ring defined by the Principal ideal (991) of Integer Ring != Point on Spectrum of Integer Ring defined by the Principal ideal (991) of Integer Ring
  ------------------------------------------------------------
  The following tests failed: _test_category, _test_pickling
Failure in _test_elements
Failure in _test_some_elements:
Traceback (most recent call last):
...
NotImplementedError
------------------------------------------------------------
The following tests failed: _test_an_element, _test_elements, _test_some_elements

sage: TestSuite(x).run()
Failure in _test_category:
Traceback (most recent call last):
...
AssertionError: False is not true
------------------------------------------------------------
Failure in _test_pickling:
Traceback (most recent call last):
...
AssertionError: Point on Spectrum of Integer Ring defined by the Principal ideal (991) of Integer Ring != Point on Spectrum of Integer Ring defined by the Principal ideal (991) of Integer Ring
------------------------------------------------------------
The following tests failed: _test_category, _test_pickling

(Note: the NotImplementedError that one gets after applying #16158 is a different one than before.)

Depends on #16158

CC: @vbraun

Component: algebraic geometry

Author: Peter Bruin

Branch/Commit: 5fd92ee

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/7946

[nthiery]

Description changed:

--- 
+++ 
@@ -1,3 +1,4 @@
+Spec(...) does not specify its category:
 
 ```
 sage: C = Spec(ZZ)
@@ -8,3 +9,42 @@
 ```
 
 Caught with #7921; please write patch on top of it to avoid conflicts.
+
+Here is the complete TestSuite report, which also complains about
+an_element, ...::
+
+```
+sage: TestSuite(Spec(ZZ)).run()
+running ._test_an_element() . . . fail
+Traceback (most recent call last):
+...
+AssertionError: self.an_element() is not in self
+------------------------------------------------------------
+running ._test_category() . . . fail
+Traceback (most recent call last):
+...
+AssertionError: category of self improperly initialized
+------------------------------------------------------------
+running ._test_elements() . . . 
+  Failure in _test_category:
+  Traceback (most recent call last):
+  ...
+  AssertionError
+  ------------------------------------------------------------
+  running ._test_not_implemented_methods() . . . pass
+  running ._test_pickling() . . . fail
+  Traceback (most recent call last):
+  ...
+  RuntimeError: ...BUG in map, returned None Point on Spectrum of Integer Ring defined by the Principal ideal (2) of Integer Ring <type 'sage.categories.morphism.CallMorphism'> Call morphism:
+  ------------------------------------------------------------
+  The following tests failed: _test_category, _test_pickling
+running ._test_not_implemented_methods() . . . pass
+running ._test_pickling() . . . pass
+running ._test_some_elements() . . . fail
+Traceback (most recent call last):
+...
+AssertionError: the object Point on Spectrum of Integer Ring defined by the Principal ideal (2) of Integer Ring in self.some_elements() is not in self
+------------------------------------------------------------
+The following tests failed: _test_an_element, _test_category, _test_elements, _test_some_elements
+```
+

[novoselt]

comment:3

#11599 claims to fix this, but I am not entirely sure what would it take to fix this ticket - the reported category is now schemes and only 3 of the TestSuite tests fail.

[pjbruin]

comment:6

What is Spec intended to mean in the first place? The definition currently starts with

class Spec(AffineScheme):
    ....

This suggests that Spec is a special kind of affine scheme. But aren't "the spectrum of a ring" and "an affine scheme" exactly the same thing?

This code probably predates the category framework. A more modern design would perhaps be something like the following:

• define the category AffineSchemes (or Schemes().Affine()) of affine schemes;
• designate AffineScheme as the "canonical" type for objects in this category, and move functionality from Spec to AffineScheme as appropriate;
• convert Spec into a functor from CommutativeRings to AffineSchemes.
  (Mathematically speaking, Spec [as a functor from commutative rings to schemes, or even locally ringed spaces] can be defined as the adjoint functor of the global sections functor X -> O_X(X) from schemes to commutative rings, and this gives an anti-equivalence of categories from commutative rings to affine schemes; I wonder if this could somehow be formalised in Sage's category framework, but that's another story...)

[Edit: the above idea is now essentially #16158, although there is no category of affine schemes yet.]

[pjbruin]

Dependencies: #16158

[pjbruin]

comment:7

To show that the first part of the original ticket is fixed:

sage: S = Spec(ZZ)
sage: S.category()
Category of Schemes
sage: isinstance(S, S.category().parent_class) 
True

[pjbruin]

Description changed:

--- 
+++ 
@@ -1,50 +1,51 @@
-Spec(...) does not specify its category:
+Here is the complete `TestSuite` report:
 
 ```
-sage: C = Spec(ZZ)
-sage: C.category()
-Category of sets
-sage: isinstance(C, C.category().element_class)
-False
+sage: S=Spec(ZZ)
+sage: x=S.an_element()
 ```
 
-Caught with #7921; please write patch on top of it to avoid conflicts.
-
-Here is the complete TestSuite report, which also complains about
-an_element, ...::
-
 ```
-sage: TestSuite(Spec(ZZ)).run()
-running ._test_an_element() . . . fail
+sage: TestSuite(S).run()
+Failure in _test_an_element:
 Traceback (most recent call last):
 ...
-AssertionError: self.an_element() is not in self
+NotImplementedError
 ------------------------------------------------------------
-running ._test_category() . . . fail
-Traceback (most recent call last):
-...
-AssertionError: category of self improperly initialized
-------------------------------------------------------------
-running ._test_elements() . . . 
   Failure in _test_category:
   Traceback (most recent call last):
   ...
-  AssertionError
+  AssertionError: False is not true
   ------------------------------------------------------------
-  running ._test_not_implemented_methods() . . . pass
-  running ._test_pickling() . . . fail
+  Failure in _test_pickling:
   Traceback (most recent call last):
   ...
-  RuntimeError: ...BUG in map, returned None Point on Spectrum of Integer Ring defined by the Principal ideal (2) of Integer Ring <type 'sage.categories.morphism.CallMorphism'> Call morphism:
+  AssertionError: Point on Spectrum of Integer Ring defined by the Principal ideal (991) of Integer Ring != Point on Spectrum of Integer Ring defined by the Principal ideal (991) of Integer Ring
   ------------------------------------------------------------
   The following tests failed: _test_category, _test_pickling
-running ._test_not_implemented_methods() . . . pass
-running ._test_pickling() . . . pass
-running ._test_some_elements() . . . fail
+Failure in _test_elements
+Failure in _test_some_elements:
 Traceback (most recent call last):
 ...
-AssertionError: the object Point on Spectrum of Integer Ring defined by the Principal ideal (2) of Integer Ring in self.some_elements() is not in self
+NotImplementedError
 ------------------------------------------------------------
-The following tests failed: _test_an_element, _test_category, _test_elements, _test_some_elements
+The following tests failed: _test_an_element, _test_elements, _test_some_elements
 ```
 
+```
+sage: TestSuite(x).run()
+Failure in _test_category:
+Traceback (most recent call last):
+...
+AssertionError: False is not true
+------------------------------------------------------------
+Failure in _test_pickling:
+Traceback (most recent call last):
+...
+AssertionError: Point on Spectrum of Integer Ring defined by the Principal ideal (991) of Integer Ring != Point on Spectrum of Integer Ring defined by the Principal ideal (991) of Integer Ring
+------------------------------------------------------------
+The following tests failed: _test_category, _test_pickling
+```
+
+(Note: the `NotImplementedError` that one gets after applying #16158 is a different one than before.)
+