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Matthias Koeppe
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src/sage/algebras/commutative_dga.py: Remove obsolete '# needs sage.rings.finite_rings'
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‎src/sage/algebras/commutative_dga.py

Lines changed: 30 additions & 32 deletions
Original file line numberDiff line numberDiff line change
@@ -949,9 +949,9 @@ def __classcall__(cls, base, names=None, degrees=None, R=None, I=None, category=
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TESTS::
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952-
sage: A1 = GradedCommutativeAlgebra(GF(2), 'x,y', (3, 6)) # needs sage.rings.finite_rings
953-
sage: A2 = GradedCommutativeAlgebra(GF(2), ['x', 'y'], [3, 6]) # needs sage.rings.finite_rings
954-
sage: A1 is A2 # needs sage.rings.finite_rings
952+
sage: A1 = GradedCommutativeAlgebra(GF(2), 'x,y', (3, 6))
953+
sage: A2 = GradedCommutativeAlgebra(GF(2), ['x', 'y'], [3, 6])
954+
sage: A1 is A2
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True
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Testing the single generator case (:trac:`25276`)::
@@ -962,8 +962,8 @@ def __classcall__(cls, base, names=None, degrees=None, R=None, I=None, category=
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sage: A4.<z> = GradedCommutativeAlgebra(QQ, degrees=[4])
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sage: z**2 == 0
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False
965-
sage: A5.<z> = GradedCommutativeAlgebra(GF(2)) # needs sage.rings.finite_rings
966-
sage: z**2 == 0 # needs sage.rings.finite_rings
965+
sage: A5.<z> = GradedCommutativeAlgebra(GF(2))
966+
sage: z**2 == 0
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False
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"""
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if names is None:
@@ -1207,7 +1207,6 @@ def quotient(self, I, check=True):
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EXAMPLES::
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1210-
sage: # needs sage.rings.finite_rings
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sage: A.<x,y,z,t> = GradedCommutativeAlgebra(GF(5), degrees=(2, 2, 3, 4))
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sage: I = A.ideal([x*t+z^2, x*y - t])
12131212
sage: B = A.quotient(I); B
@@ -1910,7 +1909,6 @@ def degree(self, total=False):
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EXAMPLES::
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1913-
sage: # needs sage.rings.finite_rings
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sage: A.<a,b,c> = GradedCommutativeAlgebra(GF(2),
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....: degrees=((1,0), (0,1), (1,1)))
19161914
sage: (a**2*b).degree()
@@ -2390,23 +2388,23 @@ def cohomology_generators(self, max_degree):
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In contrast, the corresponding algebra in characteristic `p`
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has finitely generated cohomology::
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2393-
sage: A3.<a,x,y> = GradedCommutativeAlgebra(GF(3), degrees=(1,2,2)) # needs sage.rings.finite_rings
2394-
sage: B3 = A3.cdg_algebra(differential={y: a*x}) # needs sage.rings.finite_rings
2395-
sage: B3.cohomology_generators(16) # needs sage.rings.finite_rings
2391+
sage: A3.<a,x,y> = GradedCommutativeAlgebra(GF(3), degrees=(1,2,2))
2392+
sage: B3 = A3.cdg_algebra(differential={y: a*x})
2393+
sage: B3.cohomology_generators(16)
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{1: [a], 2: [x], 3: [a*y], 5: [a*y^2], 6: [y^3]}
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This method works with both singly graded and multi-graded algebras::
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2400-
sage: Cs.<a,b,c,d> = GradedCommutativeAlgebra(GF(2), degrees=(1,2,2,3)) # needs sage.rings.finite_rings
2401-
sage: Ds = Cs.cdg_algebra({a:c, b:d}) # needs sage.rings.finite_rings
2402-
sage: Ds.cohomology_generators(10) # needs sage.rings.finite_rings
2398+
sage: Cs.<a,b,c,d> = GradedCommutativeAlgebra(GF(2), degrees=(1,2,2,3))
2399+
sage: Ds = Cs.cdg_algebra({a:c, b:d})
2400+
sage: Ds.cohomology_generators(10)
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{2: [a^2], 4: [b^2]}
24042402
2405-
sage: Cm.<a,b,c,d> = GradedCommutativeAlgebra(GF(2), # needs sage.rings.finite_rings
2403+
sage: Cm.<a,b,c,d> = GradedCommutativeAlgebra(GF(2),
24062404
....: degrees=((1,0), (1,1),
24072405
....: (0,2), (0,3)))
2408-
sage: Dm = Cm.cdg_algebra({a:c, b:d}) # needs sage.rings.finite_rings
2409-
sage: Dm.cohomology_generators(10) # needs sage.rings.finite_rings
2406+
sage: Dm = Cm.cdg_algebra({a:c, b:d})
2407+
sage: Dm.cohomology_generators(10)
24102408
{2: [a^2], 4: [b^2]}
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24122410
TESTS:
@@ -3509,9 +3507,9 @@ def GradedCommutativeAlgebra(ring, names=None, degrees=None, max_degree=None,
35093507
We can construct multi-graded rings as well. We work in characteristic 2
35103508
for a change, so the algebras here are honestly commutative::
35113509
3512-
sage: C.<a,b,c,d> = GradedCommutativeAlgebra(GF(2), # needs sage.rings.finite_rings
3510+
sage: C.<a,b,c,d> = GradedCommutativeAlgebra(GF(2),
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....: degrees=((1,0), (1,1), (0,2), (0,3)))
3514-
sage: D = C.cdg_algebra(differential={a: c, b: d}); D # needs sage.rings.finite_rings
3512+
sage: D = C.cdg_algebra(differential={a: c, b: d}); D
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Commutative Differential Graded Algebra with generators ('a', 'b', 'c', 'd')
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in degrees ((1, 0), (1, 1), (0, 2), (0, 3)) over Finite Field of size 2
35173515
with differential:
@@ -3524,46 +3522,46 @@ def GradedCommutativeAlgebra(ring, names=None, degrees=None, max_degree=None,
35243522
Use tuples, lists, vectors, or elements of additive
35253523
abelian groups to specify degrees::
35263524
3527-
sage: D.basis(3) # basis in total degree 3 # needs sage.rings.finite_rings
3525+
sage: D.basis(3) # basis in total degree 3
35283526
[a^3, a*b, a*c, d]
3529-
sage: D.basis((1,2)) # basis in degree (1,2) # needs sage.rings.finite_rings
3527+
sage: D.basis((1,2)) # basis in degree (1,2)
35303528
[a*c]
3531-
sage: D.basis([1,2]) # needs sage.rings.finite_rings
3529+
sage: D.basis([1,2])
35323530
[a*c]
3533-
sage: D.basis(vector([1,2])) # needs sage.rings.finite_rings
3531+
sage: D.basis(vector([1,2]))
35343532
[a*c]
35353533
sage: G = AdditiveAbelianGroup([0,0]); G
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Additive abelian group isomorphic to Z + Z
3537-
sage: D.basis(G(vector([1,2]))) # needs sage.rings.finite_rings
3535+
sage: D.basis(G(vector([1,2])))
35383536
[a*c]
35393537
35403538
At this point, ``a``, for example, is an element of ``C``. We can
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redefine it so that it is instead an element of ``D`` in several
35423540
ways, for instance using :meth:`gens` method::
35433541
3544-
sage: a, b, c, d = D.gens() # needs sage.rings.finite_rings
3545-
sage: a.differential() # needs sage.rings.finite_rings
3542+
sage: a, b, c, d = D.gens()
3543+
sage: a.differential()
35463544
c
35473545
35483546
Or the :meth:`inject_variables` method::
35493547
3550-
sage: D.inject_variables() # needs sage.rings.finite_rings
3548+
sage: D.inject_variables()
35513549
Defining a, b, c, d
3552-
sage: (a*b).differential() # needs sage.rings.finite_rings
3550+
sage: (a*b).differential()
35533551
b*c + a*d
3554-
sage: (a*b*c**2).degree() # needs sage.rings.finite_rings
3552+
sage: (a*b*c**2).degree()
35553553
(2, 5)
35563554
35573555
Degrees are returned as elements of additive abelian groups::
35583556
3559-
sage: (a*b*c**2).degree() in G # needs sage.rings.finite_rings
3557+
sage: (a*b*c**2).degree() in G
35603558
True
35613559
3562-
sage: (a*b*c**2).degree(total=True) # total degree # needs sage.rings.finite_rings
3560+
sage: (a*b*c**2).degree(total=True) # total degree
35633561
7
3564-
sage: D.cohomology(4) # needs sage.rings.finite_rings
3562+
sage: D.cohomology(4)
35653563
Free module generated by {[a^4], [b^2]} over Finite Field of size 2
3566-
sage: D.cohomology((2,2)) # needs sage.rings.finite_rings
3564+
sage: D.cohomology((2,2))
35673565
Free module generated by {[b^2]} over Finite Field of size 2
35683566
35693567
Graded algebra with maximal degree::

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