gh-38018: move Frobenius and derivations to category of Commutative R…

…ings

    
move three methods from the class `CommutativeRing` to the category of
commutative rings

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
    
URL: #38018
Reported by: Frédéric Chapoton
Reviewer(s): Travis Scrimshaw

src/doc/en/thematic_tutorials/coercion_and_categories.rst

@@ -126,13 +126,10 @@ This base class provides a lot more methods than a general parent::
      '_zero_ideal',
      'algebraic_closure',
      'base_extend',
-     'derivation',
-     'derivation_module',
      'divides',
      'epsilon',
      'extension',
      'fraction_field',
-     'frobenius_endomorphism',
      'gen',
      'gens',
      'ideal',

src/sage/categories/commutative_rings.py

@@ -13,6 +13,7 @@
 
 from sage.categories.category_with_axiom import CategoryWithAxiom
 from sage.categories.cartesian_product import CartesianProductsCategory
+from sage.structure.sequence import Sequence
 
 
 class CommutativeRings(CategoryWithAxiom):
@@ -267,6 +268,206 @@ def over(self, base=None, gen=None, gens=None, name=None, names=None):
                 gens = (gen,)
             return RingExtension(self, base, gens, names)
 
+        def frobenius_endomorphism(self, n=1):
+            """
+            Return the Frobenius endomorphism.
+
+            INPUT:
+
+            - ``n`` -- a nonnegative integer (default: 1)
+
+            OUTPUT:
+
+            The `n`-th power of the absolute arithmetic Frobenius
+            endomorphism on this commutative ring.
+
+            EXAMPLES::
+
+                sage: K.<u> = PowerSeriesRing(GF(5))
+                sage: Frob = K.frobenius_endomorphism(); Frob
+                Frobenius endomorphism x |--> x^5 of Power Series Ring in u
+                 over Finite Field of size 5
+                sage: Frob(u)
+                u^5
+
+            We can specify a power::
+
+                sage: f = K.frobenius_endomorphism(2); f
+                Frobenius endomorphism x |--> x^(5^2) of Power Series Ring in u
+                 over Finite Field of size 5
+                sage: f(1+u)
+                1 + u^25
+            """
+            from sage.rings.morphism import FrobeniusEndomorphism_generic
+            return FrobeniusEndomorphism_generic(self, n)
+
+        def derivation_module(self, codomain=None, twist=None):
+            r"""
+            Return the module of derivations over this ring.
+
+            INPUT:
+
+            - ``codomain`` -- an algebra over this ring or a ring homomorphism
+              whose domain is this ring or ``None`` (default: ``None``); if it
+              is a morphism, the codomain of derivations will be the codomain
+              of the morphism viewed as an algebra over ``self`` through the
+              given morphism; if ``None``, the codomain will be this ring
+
+            - ``twist`` -- a morphism from this ring to ``codomain``
+              or ``None`` (default: ``None``); if ``None``, the coercion
+              map from this ring to ``codomain`` will be used
+
+            .. NOTE::
+
+                A twisted derivation with respect to `\theta` (or a
+                `\theta`-derivation for short) is an additive map `d`
+                satisfying the following axiom for all `x, y` in the domain:
+
+                .. MATH::
+
+                    d(xy) = \theta(x) d(y) + d(x) y.
+
+            EXAMPLES::
+
+                sage: R.<x,y,z> = QQ[]
+                sage: M = R.derivation_module(); M                                          # needs sage.modules
+                Module of derivations over
+                 Multivariate Polynomial Ring in x, y, z over Rational Field
+                sage: M.gens()                                                              # needs sage.modules
+                (d/dx, d/dy, d/dz)
+
+            We can specify a different codomain::
+
+                sage: K = R.fraction_field()
+                sage: M = R.derivation_module(K); M                                         # needs sage.modules
+                Module of derivations
+                 from Multivariate Polynomial Ring in x, y, z over Rational Field
+                   to Fraction Field of
+                      Multivariate Polynomial Ring in x, y, z over Rational Field
+                sage: M.gen() / x                                                           # needs sage.modules
+                1/x*d/dx
+
+            Here is an example with a non-canonical defining morphism::
+
+                sage: ev = R.hom([QQ(0), QQ(1), QQ(2)])
+                sage: ev
+                Ring morphism:
+                  From: Multivariate Polynomial Ring in x, y, z over Rational Field
+                  To:   Rational Field
+                  Defn: x |--> 0
+                        y |--> 1
+                        z |--> 2
+                sage: M = R.derivation_module(ev)                                           # needs sage.modules
+                sage: M                                                                     # needs sage.modules
+                Module of derivations
+                 from Multivariate Polynomial Ring in x, y, z over Rational Field
+                   to Rational Field
+
+            Elements in `M` acts as derivations at `(0,1,2)`::
+
+                sage: # needs sage.modules
+                sage: Dx = M.gen(0); Dx
+                d/dx
+                sage: Dy = M.gen(1); Dy
+                d/dy
+                sage: Dz = M.gen(2); Dz
+                d/dz
+                sage: f = x^2 + y^2 + z^2
+                sage: Dx(f)  # = 2*x evaluated at (0,1,2)
+                0
+                sage: Dy(f)  # = 2*y evaluated at (0,1,2)
+                2
+                sage: Dz(f)  # = 2*z evaluated at (0,1,2)
+                4
+
+            An example with a twisting homomorphism::
+
+                sage: theta = R.hom([x^2, y^2, z^2])
+                sage: M = R.derivation_module(twist=theta); M                               # needs sage.modules
+                Module of twisted derivations over Multivariate Polynomial Ring in x, y, z
+                 over Rational Field (twisting morphism: x |--> x^2, y |--> y^2, z |--> z^2)
+
+            .. SEEALSO::
+
+                :meth:`derivation`
+
+            """
+            from sage.rings.derivation import RingDerivationModule
+            if codomain is None:
+                codomain = self
+            return RingDerivationModule(self, codomain, twist)
+
+        def derivation(self, arg=None, twist=None):
+            r"""
+            Return the twisted or untwisted derivation over this ring
+            specified by ``arg``.
+
+            .. NOTE::
+
+                A twisted derivation with respect to `\theta` (or a
+                `\theta`-derivation for short) is an additive map `d`
+                satisfying the following axiom for all `x, y` in the domain:
+
+                .. MATH::
+
+                    d(xy) = \theta(x) d(y) + d(x) y.
+
+            INPUT:
+
+            - ``arg`` -- (optional) a generator or a list of coefficients
+              that defines the derivation
+
+            - ``twist`` -- (optional) the twisting homomorphism
+
+            EXAMPLES::
+
+                sage: R.<x,y,z> = QQ[]
+                sage: R.derivation()                                                        # needs sage.modules
+                d/dx
+
+            In that case, ``arg`` could be a generator::
+
+                sage: R.derivation(y)                                                       # needs sage.modules
+                d/dy
+
+            or a list of coefficients::
+
+                sage: R.derivation([1,2,3])                                                 # needs sage.modules
+                d/dx + 2*d/dy + 3*d/dz
+
+            It is not possible to define derivations with respect to a
+            polynomial which is not a variable::
+
+                sage: R.derivation(x^2)                                                     # needs sage.modules
+                Traceback (most recent call last):
+                ...
+                ValueError: unable to create the derivation
+
+            Here is an example with twisted derivations::
+
+                sage: R.<x,y,z> = QQ[]
+                sage: theta = R.hom([x^2, y^2, z^2])
+                sage: f = R.derivation(twist=theta); f                                      # needs sage.modules
+                0
+                sage: f.parent()                                                            # needs sage.modules
+                Module of twisted derivations over Multivariate Polynomial Ring in x, y, z
+                 over Rational Field (twisting morphism: x |--> x^2, y |--> y^2, z |--> z^2)
+
+            Specifying a scalar, the returned twisted derivation is the
+            corresponding multiple of `\theta - id`::
+
+                sage: R.derivation(1, twist=theta)                                          # needs sage.modules
+                [x |--> x^2, y |--> y^2, z |--> z^2] - id
+                sage: R.derivation(x, twist=theta)                                          # needs sage.modules
+                x*([x |--> x^2, y |--> y^2, z |--> z^2] - id)
+
+            """
+            if isinstance(arg, (list, tuple)):
+                codomain = Sequence([self(0)] + list(arg)).universe()
+            else:
+                codomain = self
+            return self.derivation_module(codomain, twist=twist)(arg)
+
     class ElementMethods:
         pass