Homomorphisms of finitely presented modules over the Steenrod algebra#

This class implements construction and basic manipulation of homomorphisms between finitely presented graded modules over the mod \(p\) Steenrod algebra.

AUTHORS:

Robert R. Bruner, Michael J. Catanzaro (2012): Initial version.
Sverre Lunoee–Nielsen and Koen van Woerden (2019-11-29): Updated the original software to Sage version 8.9.
Sverre Lunoee–Nielsen (2020-07-01): Refactored the code and added new documentation and tests.

class sage.modules.fp_graded.steenrod.morphism.SteenrodFPModuleMorphism(parent, values, check=True)#

Bases: FPModuleMorphism

cokernel_projection(verbose=False)#

Compute the map to the cokernel of self.

INPUT:

verbose – (default: False) whether log messages are printed

OUTPUT:

The natural projection from the codomain of this homomorphism to its cokernel.

EXAMPLES:

sage: from sage.modules.fp_graded.steenrod.module import SteenrodFPModule
sage: A1 = SteenrodAlgebra(2, profile=(2,1))
sage: M = SteenrodFPModule(A1, [0], [[Sq(2)]])
sage: F = SteenrodFPModule(A1, [0])

sage: r = Hom(F, M)([A1.Sq(1)*M.generator(0)])
sage: co = r.cokernel_projection(); co
Module morphism:
  From: Finitely presented left module on 1 generator and 1 relation over sub-Hopf algebra of mod 2 Steenrod algebra, milnor basis, profile function [2, 1]
  To:   Finitely presented left module on 1 generator and 2 relations over sub-Hopf algebra of mod 2 Steenrod algebra, milnor basis, profile function [2, 1]
  Defn: g[0] |--> g[0]

sage: co.domain().is_trivial()
False

image(top_dim=None, verbose=False)#

Return the image of self.

INPUT:

top_dim – integer (optional); used by this function to stop the computation at the given degree
verbose – (default: False) whether log messages are printed

OUTPUT:

An injective homomorphism into the codomain of self which is onto the image of self.

EXAMPLES:

sage: from sage.modules.fp_graded.steenrod.module import SteenrodFPModule
sage: A = SteenrodAlgebra(2)
sage: M = SteenrodFPModule(A, [0,1], [[Sq(2),Sq(1)], [0,Sq(2)]])
sage: S = SteenrodFPModule(A, [0], [[Sq(2)]])
sage: f = Hom(S, M)([M([0,1])])
sage: f.is_injective()
True
sage: i = f.image(); i
Module morphism:
  From: Finitely presented left module on 1 generator and 1 relation over mod 2 Steenrod algebra, milnor basis
  To:   Finitely presented left module on 2 generators and 2 relations over mod 2 Steenrod algebra, milnor basis
  Defn: g[1] |--> g[1]
sage: i.codomain() is M
True

Lift the map f over the inclusion i:

sage: f_ = f.lift(i)
sage: f_.is_injective()
True
sage: f_.is_surjective()
True

sage: g = Hom(S, M)([M([Sq(3),Sq(2)])])
sage: j = g.image(); j
Module morphism:
  From: Free graded left module on 0 generators over mod 2 Steenrod algebra, milnor basis
  To:   Finitely presented left module on 2 generators and 2 relations over mod 2 Steenrod algebra, milnor basis

So g had to be trivial:

sage: g.is_zero()
True

is_injective(top_dim=None, verbose=False)#

Return True if self is injective.

INPUT:

top_dim – (optional) stop the computation at this degree; if not specified, this is determined using profile()
verbose – (default: False) whether log messages are printed

EXAMPLES:

sage: from sage.modules.fp_graded.steenrod.module import SteenrodFPModule
sage: A = SteenrodAlgebra(2)
sage: M = SteenrodFPModule(A, [0,1], [[Sq(2),Sq(1)], [0,Sq(2)]])
sage: S = SteenrodFPModule(A, [0], [[Sq(2)]])
sage: f = Hom(S, M)([M([0,1])])
sage: f.is_injective()
True
sage: g = Hom(S, M).zero()
sage: g.is_injective()
False
sage: z = Hom(SteenrodFPModule(A, []), M).zero()
sage: z.is_injective()
True
sage: z.is_zero()
True

kernel_inclusion(top_dim=None, verbose=False)#

Return the kernel of self as a morphism.

INPUT:

top_dim – (optional) stop the computation at this degree; if not specified, this is determined using profile()
verbose – (default: False) whether log messages are printed

OUTPUT: An injective homomorphism into the domain self which is onto the kernel of this homomorphism.

EXAMPLES:

sage: from sage.modules.fp_graded.steenrod.module import SteenrodFPModule
sage: A = SteenrodAlgebra(2)
sage: M = SteenrodFPModule(A, [0,1], [[Sq(2),Sq(1)], [0,Sq(2)]])
sage: S = SteenrodFPModule(A, [0], [[Sq(2)]])
sage: f = Hom(S, M)([M([0,1])])
sage: f.is_injective()
True
sage: k = f.kernel_inclusion()
sage: k == 0
True

Since k is both trivial and injective, its domain should be the zero module:

sage: k.domain().is_trivial()
True

sage: g = Hom(S, M)([M([Sq(3),Sq(2)])])
sage: h = g.kernel_inclusion()
sage: h.is_identity()
True
sage: ker = h.domain();
sage: ker is S
True

So \(g\) had to be trivial:

sage: g.is_zero()
True

profile()#

Return a finite profile over which self can be defined.

This is in some ways the key method for these morphisms. As discussed in the “Theoretical background” section of sage.modules.fp_graded.steenrod.module, any homomorphism of finitely presented modules over the Steenrod algebra can be defined over a finite-dimensional sub-Hopf algebra, and this method identifies such a sub-Hopf algebra and returns its profile function.

EXAMPLES:

sage: from sage.modules.fp_graded.steenrod.module import SteenrodFPModule
sage: A = SteenrodAlgebra(2)
sage: M = SteenrodFPModule(A, [0,1], [[Sq(2),Sq(1)], [0,Sq(2)]])
sage: one = Hom(M,M).identity()
sage: one.profile()
(2, 1)
sage: zero = Hom(M,M).zero()
sage: zero.profile()
(2, 1)
sage: A_fin = SteenrodAlgebra(2, profile=(2,1))
sage: M_fin = M.change_ring(A_fin)

Change the ring of the module M:

sage: M_fin.change_ring(A) is M
True

We can change rings to the finite sub-Hopf algebra defined by the profile we just computed:

sage: one_fin = one.change_ring(A_fin)
sage: one_fin.domain()
Finitely presented left module on 2 generators and 2 relations over
 sub-Hopf algebra of mod 2 Steenrod algebra, milnor basis, profile function [2, 1]

If we change back to the full Steenrod algebra, we are back where we started:

sage: one_fin.change_ring(A) == one
True

class sage.modules.fp_graded.steenrod.morphism.SteenrodFreeModuleMorphism(parent, values)#: Bases: FreeGradedModuleMorphism, SteenrodFPModuleMorphism