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src/sage/algebras/commutative_dga.py: Remove obsolete '# needs sage.r…
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…ings.finite_rings'
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@mkoeppe
mkoeppe committed Nov 12, 2023
1 parent e1874f9 commit bd006df
Showing 1 changed file with 30 additions and 32 deletions.
62 changes: 30 additions & 32 deletions src/sage/algebras/commutative_dga.py
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Original file line number Diff line number Diff line change
Expand Up @@ -949,9 +949,9 @@ def __classcall__(cls, base, names=None, degrees=None, R=None, I=None, category=
TESTS::
sage: A1 = GradedCommutativeAlgebra(GF(2), 'x,y', (3, 6)) # needs sage.rings.finite_rings
sage: A2 = GradedCommutativeAlgebra(GF(2), ['x', 'y'], [3, 6]) # needs sage.rings.finite_rings
sage: A1 is A2 # needs sage.rings.finite_rings
sage: A1 = GradedCommutativeAlgebra(GF(2), 'x,y', (3, 6))
sage: A2 = GradedCommutativeAlgebra(GF(2), ['x', 'y'], [3, 6])
sage: A1 is A2
True
Testing the single generator case (:trac:`25276`)::
Expand All @@ -962,8 +962,8 @@ def __classcall__(cls, base, names=None, degrees=None, R=None, I=None, category=
sage: A4.<z> = GradedCommutativeAlgebra(QQ, degrees=[4])
sage: z**2 == 0
False
sage: A5.<z> = GradedCommutativeAlgebra(GF(2)) # needs sage.rings.finite_rings
sage: z**2 == 0 # needs sage.rings.finite_rings
sage: A5.<z> = GradedCommutativeAlgebra(GF(2))
sage: z**2 == 0
False
"""
if names is None:
Expand Down Expand Up @@ -1207,7 +1207,6 @@ def quotient(self, I, check=True):
EXAMPLES::
sage: # needs sage.rings.finite_rings
sage: A.<x,y,z,t> = GradedCommutativeAlgebra(GF(5), degrees=(2, 2, 3, 4))
sage: I = A.ideal([x*t+z^2, x*y - t])
sage: B = A.quotient(I); B
Expand Down Expand Up @@ -1910,7 +1909,6 @@ def degree(self, total=False):
EXAMPLES::
sage: # needs sage.rings.finite_rings
sage: A.<a,b,c> = GradedCommutativeAlgebra(GF(2),
....: degrees=((1,0), (0,1), (1,1)))
sage: (a**2*b).degree()
Expand Down Expand Up @@ -2390,23 +2388,23 @@ def cohomology_generators(self, max_degree):
In contrast, the corresponding algebra in characteristic `p`
has finitely generated cohomology::
sage: A3.<a,x,y> = GradedCommutativeAlgebra(GF(3), degrees=(1,2,2)) # needs sage.rings.finite_rings
sage: B3 = A3.cdg_algebra(differential={y: a*x}) # needs sage.rings.finite_rings
sage: B3.cohomology_generators(16) # needs sage.rings.finite_rings
sage: A3.<a,x,y> = GradedCommutativeAlgebra(GF(3), degrees=(1,2,2))
sage: B3 = A3.cdg_algebra(differential={y: a*x})
sage: B3.cohomology_generators(16)
{1: [a], 2: [x], 3: [a*y], 5: [a*y^2], 6: [y^3]}
This method works with both singly graded and multi-graded algebras::
sage: Cs.<a,b,c,d> = GradedCommutativeAlgebra(GF(2), degrees=(1,2,2,3)) # needs sage.rings.finite_rings
sage: Ds = Cs.cdg_algebra({a:c, b:d}) # needs sage.rings.finite_rings
sage: Ds.cohomology_generators(10) # needs sage.rings.finite_rings
sage: Cs.<a,b,c,d> = GradedCommutativeAlgebra(GF(2), degrees=(1,2,2,3))
sage: Ds = Cs.cdg_algebra({a:c, b:d})
sage: Ds.cohomology_generators(10)
{2: [a^2], 4: [b^2]}
sage: Cm.<a,b,c,d> = GradedCommutativeAlgebra(GF(2), # needs sage.rings.finite_rings
sage: Cm.<a,b,c,d> = GradedCommutativeAlgebra(GF(2),
....: degrees=((1,0), (1,1),
....: (0,2), (0,3)))
sage: Dm = Cm.cdg_algebra({a:c, b:d}) # needs sage.rings.finite_rings
sage: Dm.cohomology_generators(10) # needs sage.rings.finite_rings
sage: Dm = Cm.cdg_algebra({a:c, b:d})
sage: Dm.cohomology_generators(10)
{2: [a^2], 4: [b^2]}
TESTS:
Expand Down Expand Up @@ -3509,9 +3507,9 @@ def GradedCommutativeAlgebra(ring, names=None, degrees=None, max_degree=None,
We can construct multi-graded rings as well. We work in characteristic 2
for a change, so the algebras here are honestly commutative::
sage: C.<a,b,c,d> = GradedCommutativeAlgebra(GF(2), # needs sage.rings.finite_rings
sage: C.<a,b,c,d> = GradedCommutativeAlgebra(GF(2),
....: degrees=((1,0), (1,1), (0,2), (0,3)))
sage: D = C.cdg_algebra(differential={a: c, b: d}); D # needs sage.rings.finite_rings
sage: D = C.cdg_algebra(differential={a: c, b: d}); D
Commutative Differential Graded Algebra with generators ('a', 'b', 'c', 'd')
in degrees ((1, 0), (1, 1), (0, 2), (0, 3)) over Finite Field of size 2
with differential:
Expand All @@ -3524,46 +3522,46 @@ def GradedCommutativeAlgebra(ring, names=None, degrees=None, max_degree=None,
Use tuples, lists, vectors, or elements of additive
abelian groups to specify degrees::
sage: D.basis(3) # basis in total degree 3 # needs sage.rings.finite_rings
sage: D.basis(3) # basis in total degree 3
[a^3, a*b, a*c, d]
sage: D.basis((1,2)) # basis in degree (1,2) # needs sage.rings.finite_rings
sage: D.basis((1,2)) # basis in degree (1,2)
[a*c]
sage: D.basis([1,2]) # needs sage.rings.finite_rings
sage: D.basis([1,2])
[a*c]
sage: D.basis(vector([1,2])) # needs sage.rings.finite_rings
sage: D.basis(vector([1,2]))
[a*c]
sage: G = AdditiveAbelianGroup([0,0]); G
Additive abelian group isomorphic to Z + Z
sage: D.basis(G(vector([1,2]))) # needs sage.rings.finite_rings
sage: D.basis(G(vector([1,2])))
[a*c]
At this point, ``a``, for example, is an element of ``C``. We can
redefine it so that it is instead an element of ``D`` in several
ways, for instance using :meth:`gens` method::
sage: a, b, c, d = D.gens() # needs sage.rings.finite_rings
sage: a.differential() # needs sage.rings.finite_rings
sage: a, b, c, d = D.gens()
sage: a.differential()
c
Or the :meth:`inject_variables` method::
sage: D.inject_variables() # needs sage.rings.finite_rings
sage: D.inject_variables()
Defining a, b, c, d
sage: (a*b).differential() # needs sage.rings.finite_rings
sage: (a*b).differential()
b*c + a*d
sage: (a*b*c**2).degree() # needs sage.rings.finite_rings
sage: (a*b*c**2).degree()
(2, 5)
Degrees are returned as elements of additive abelian groups::
sage: (a*b*c**2).degree() in G # needs sage.rings.finite_rings
sage: (a*b*c**2).degree() in G
True
sage: (a*b*c**2).degree(total=True) # total degree # needs sage.rings.finite_rings
sage: (a*b*c**2).degree(total=True) # total degree
7
sage: D.cohomology(4) # needs sage.rings.finite_rings
sage: D.cohomology(4)
Free module generated by {[a^4], [b^2]} over Finite Field of size 2
sage: D.cohomology((2,2)) # needs sage.rings.finite_rings
sage: D.cohomology((2,2))
Free module generated by {[b^2]} over Finite Field of size 2
Graded algebra with maximal degree::
Expand Down

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