Steenrod Algebra Modules¶
Author: Robert R. Bruner, Michael J. Catanzaro, Sverre Lunoee–Nielsen, and Koen van Woerden
Let \(p\) be a prime number. The mod \(p\) Steenrod algebra \(A\) is a connected algebra over \(\GF{p}\), the finite field of \(p\) elements. All modules presented here will be defined over \(A\) or one of its sub-Hopf algebras. E.g.:
sage: A = SteenrodAlgebra(p=2)
>>> from sage.all import *
>>> A = SteenrodAlgebra(p=Integer(2))
A = SteenrodAlgebra(p=2)
The constructor of the module class takes as arguments an ordered tuple of degrees and the algebra over which the module is defined, together with an optional set of relations:
sage: from sage.modules.fp_graded.steenrod.module import SteenrodFPModule
sage: F = SteenrodFPModule(A, [0, 1, 7]); F
Free graded left module on 3 generators over mod 2 Steenrod algebra, milnor basis
>>> from sage.all import *
>>> from sage.modules.fp_graded.steenrod.module import SteenrodFPModule
>>> F = SteenrodFPModule(A, [Integer(0), Integer(1), Integer(7)]); F
Free graded left module on 3 generators over mod 2 Steenrod algebra, milnor basis
from sage.modules.fp_graded.steenrod.module import SteenrodFPModule F = SteenrodFPModule(A, [0, 1, 7]); F
Denote the module generators of an \(A\)-module \(M\) by \(g_{d_1}, \ldots, g_{d_N}\), where subscripts denote their degrees. A homogeneous relation of degree \(n\) has the form
\[\sum_{i=1}^N a_i\cdot g_{d_i} = 0,\]
where the homogeneous coefficients \(a_1, \ldots, a_N\) lie in \(A\), such that \(\deg(a_i) + \deg(g_{d_i}) = n\) for \(i = 1, \ldots, N\). To create a module with relations, the coefficients for each relation is given:
sage: r1 = [Sq(8), Sq(7), 0] # First relation
sage: r2 = [Sq(7), 0, 1] # Second relation
sage: M = SteenrodFPModule(A, [0, 1, 7], relations=[r1, r2]); M
Finitely presented left module on 3 generators and 2 relations
over mod 2 Steenrod algebra, milnor basis
>>> from sage.all import *
>>> r1 = [Sq(Integer(8)), Sq(Integer(7)), Integer(0)] # First relation
>>> r2 = [Sq(Integer(7)), Integer(0), Integer(1)] # Second relation
>>> M = SteenrodFPModule(A, [Integer(0), Integer(1), Integer(7)], relations=[r1, r2]); M
Finitely presented left module on 3 generators and 2 relations
over mod 2 Steenrod algebra, milnor basis
r1 = [Sq(8), Sq(7), 0] # First relation r2 = [Sq(7), 0, 1] # Second relation M = SteenrodFPModule(A, [0, 1, 7], relations=[r1, r2]); M
The resulting module will have three generators in the degrees we gave them:
sage: M.generator_degrees()
(0, 1, 7)
>>> from sage.all import *
>>> M.generator_degrees()
(0, 1, 7)
M.generator_degrees()
The connectivity of a module over a connected graded algebra is the minimum degree of all its module generators. Thus, if the module is non-trivial, the connectivity is an integer:
sage: M.connectivity()
0
>>> from sage.all import *
>>> M.connectivity()
0
M.connectivity()
Each module is defined over a Steenrod algebra or some sub-Hopf algebra of it, given by its base ring:
sage: M.base_ring()
mod 2 Steenrod algebra, milnor basis
sage: SteenrodFPModule(SteenrodAlgebra(p=2,profile=(3,2,1)), [0]).base_ring()
sub-Hopf algebra of mod 2 Steenrod algebra, milnor basis, profile function
[3, 2, 1]
>>> from sage.all import *
>>> M.base_ring()
mod 2 Steenrod algebra, milnor basis
>>> SteenrodFPModule(SteenrodAlgebra(p=Integer(2),profile=(Integer(3),Integer(2),Integer(1))), [Integer(0)]).base_ring()
sub-Hopf algebra of mod 2 Steenrod algebra, milnor basis, profile function
[3, 2, 1]
M.base_ring() SteenrodFPModule(SteenrodAlgebra(p=2,profile=(3,2,1)), [0]).base_ring()
Note
Calling algebra() will not return the desired algebra. Users should
use the base_ring() method.
Module elements¶
Module elements are displayed in terms of generators, which by default
are called g[degree]:
sage: M.an_element(n=5)
Sq(2,1)*g[0] + Sq(4)*g[1]
sage: e = M.an_element(n=15); e
Sq(0,0,0,1)*g[0] + Sq(1,2,1)*g[1] + Sq(8)*g[7]
sage: e.dense_coefficient_list()
[Sq(0,0,0,1), Sq(1,2,1), Sq(8)]
>>> from sage.all import *
>>> M.an_element(n=Integer(5))
Sq(2,1)*g[0] + Sq(4)*g[1]
>>> e = M.an_element(n=Integer(15)); e
Sq(0,0,0,1)*g[0] + Sq(1,2,1)*g[1] + Sq(8)*g[7]
>>> e.dense_coefficient_list()
[Sq(0,0,0,1), Sq(1,2,1), Sq(8)]
M.an_element(n=5) e = M.an_element(n=15); e e.dense_coefficient_list()
The generators are themselves elements of the module:
sage: gens = M.generators(); gens
(g[0], g[1], g[7])
sage: gens[0] in M
True
>>> from sage.all import *
>>> gens = M.generators(); gens
(g[0], g[1], g[7])
>>> gens[Integer(0)] in M
True
gens = M.generators(); gens gens[0] in M
One can produce an element from a given set of algebra coefficients:
sage: coeffs = [Sq(15), Sq(10)*Sq(1,1), Sq(8)]
sage: x = M(coeffs); x
Sq(15)*g[0] + (Sq(4,1,1)+Sq(7,0,1)+Sq(11,1))*g[1] + Sq(8)*g[7]
>>> from sage.all import *
>>> coeffs = [Sq(Integer(15)), Sq(Integer(10))*Sq(Integer(1),Integer(1)), Sq(Integer(8))]
>>> x = M(coeffs); x
Sq(15)*g[0] + (Sq(4,1,1)+Sq(7,0,1)+Sq(11,1))*g[1] + Sq(8)*g[7]
coeffs = [Sq(15), Sq(10)*Sq(1,1), Sq(8)] x = M(coeffs); x
The module action produces new elements:
sage: Sq(2) * x
Sq(14,1)*g[0] + (Sq(7,1)+Sq(10))*g[7]
>>> from sage.all import *
>>> Sq(Integer(2)) * x
Sq(14,1)*g[0] + (Sq(7,1)+Sq(10))*g[7]
Sq(2) * x
Each non-zero homogeneous element has a well-defined degree:
sage: x.degree()
15
>>> from sage.all import *
>>> x.degree()
15
x.degree()
but the zero element does not:
sage: zero = M.zero(); zero
0
sage: zero.degree()
Traceback (most recent call last):
...
ValueError: the zero element does not have a well-defined degree
>>> from sage.all import *
>>> zero = M.zero(); zero
0
>>> zero.degree()
Traceback (most recent call last):
...
ValueError: the zero element does not have a well-defined degree
zero = M.zero(); zero zero.degree()
At this point it may be useful to point out that elements are not reduced to a minimal representation when they are created. A normalization can be enforced, however:
sage: g7 = M([0, 0, 1]); g7
g[7]
sage: g7.normalize()
Sq(7)*g[0]
sage: g7 == g7.normalize()
True
sage: m = M([Sq(7), 0, 0])
sage: s = m + g7; s # m and g7 are related by m = Sq(7)*g[0] = g[7],
Sq(7)*g[0] + g[7]
sage: s == 0 # so their sum should zero.
True
sage: s.normalize() # Its normalized form is more revealing.
0
>>> from sage.all import *
>>> g7 = M([Integer(0), Integer(0), Integer(1)]); g7
g[7]
>>> g7.normalize()
Sq(7)*g[0]
>>> g7 == g7.normalize()
True
>>> m = M([Sq(Integer(7)), Integer(0), Integer(0)])
>>> s = m + g7; s # m and g7 are related by m = Sq(7)*g[0] = g[7],
Sq(7)*g[0] + g[7]
>>> s == Integer(0) # so their sum should zero.
True
>>> s.normalize() # Its normalized form is more revealing.
0
g7 = M([0, 0, 1]); g7 g7.normalize() g7 == g7.normalize() m = M([Sq(7), 0, 0]) s = m + g7; s # m and g7 are related by m = Sq(7)*g[0] = g[7], s == 0 # so their sum should zero. s.normalize() # Its normalized form is more revealing.
For every integer \(n\), the set of module elements of degree \(n\) form a vector space over the ground field \(\GF{p}\). A basis for this vector space can be computed:
sage: M.basis_elements(7)
(Sq(0,0,1)*g[0],
Sq(1,2)*g[0],
Sq(4,1)*g[0],
Sq(7)*g[0],
Sq(0,2)*g[1],
Sq(3,1)*g[1],
Sq(6)*g[1])
>>> from sage.all import *
>>> M.basis_elements(Integer(7))
(Sq(0,0,1)*g[0],
Sq(1,2)*g[0],
Sq(4,1)*g[0],
Sq(7)*g[0],
Sq(0,2)*g[1],
Sq(3,1)*g[1],
Sq(6)*g[1])
M.basis_elements(7)
Note that the third generator \(g_7\) of degree 7 is apparently missing from the basis above. This is because of the relation \(\operatorname{Sq}^7(g_0) = g_7\).
A vector space presentation can be produced:
sage: M.vector_presentation(5)
Vector space quotient V/W of dimension 4 over Finite Field of size 2 where
V: Vector space of dimension 4 over Finite Field of size 2
W: Vector space of degree 4 and dimension 0 over Finite Field of size 2
Basis matrix:
[]
>>> from sage.all import *
>>> M.vector_presentation(Integer(5))
Vector space quotient V/W of dimension 4 over Finite Field of size 2 where
V: Vector space of dimension 4 over Finite Field of size 2
W: Vector space of degree 4 and dimension 0 over Finite Field of size 2
Basis matrix:
[]
M.vector_presentation(5)
Given any element, its coordinates with respect to this basis can be computed:
sage: x = M.an_element(7); x
Sq(0,0,1)*g[0] + Sq(3,1)*g[1] + g[7]
sage: v = x.vector_presentation(); v
(1, 0, 0, 1, 0, 1, 0)
>>> from sage.all import *
>>> x = M.an_element(Integer(7)); x
Sq(0,0,1)*g[0] + Sq(3,1)*g[1] + g[7]
>>> v = x.vector_presentation(); v
(1, 0, 0, 1, 0, 1, 0)
x = M.an_element(7); x v = x.vector_presentation(); v
Going the other way, any element can be constructed by specifying its coordinates:
sage: x_ = M.element_from_coordinates((1, 0, 0, 1, 0, 1, 0), 7)
sage: x_
(Sq(0,0,1)+Sq(7))*g[0] + Sq(3,1)*g[1]
sage: x_ == x
True
>>> from sage.all import *
>>> x_ = M.element_from_coordinates((Integer(1), Integer(0), Integer(0), Integer(1), Integer(0), Integer(1), Integer(0)), Integer(7))
>>> x_
(Sq(0,0,1)+Sq(7))*g[0] + Sq(3,1)*g[1]
>>> x_ == x
True
x_ = M.element_from_coordinates((1, 0, 0, 1, 0, 1, 0), 7) x_ x_ == x
Module homomorphisms¶
Homomorphisms of \(A\)-modules \(M\to N\) are linear maps of their underlying \(\GF{p}\)-vector spaces which commute with the \(A\)-module structure. Homomorphisms are required to be homogeneneous but need not be degree zero.
To create a homomorphism, first create the object modeling the set of all
such homomorphisms using the function Hom:
sage: Hko = SteenrodFPModule(A, [0], [[Sq(2)], [Sq(1)]])
sage: homspace = Hom(Hko, Hko); homspace
Set of Morphisms
from Finitely presented left module on 1 generator and 2 relations
over mod 2 Steenrod algebra, milnor basis
to Finitely presented left module on 1 generator and 2 relations
over mod 2 Steenrod algebra, milnor basis
in Category of finitely presented graded modules
over mod 2 Steenrod algebra, milnor basis
>>> from sage.all import *
>>> Hko = SteenrodFPModule(A, [Integer(0)], [[Sq(Integer(2))], [Sq(Integer(1))]])
>>> homspace = Hom(Hko, Hko); homspace
Set of Morphisms
from Finitely presented left module on 1 generator and 2 relations
over mod 2 Steenrod algebra, milnor basis
to Finitely presented left module on 1 generator and 2 relations
over mod 2 Steenrod algebra, milnor basis
in Category of finitely presented graded modules
over mod 2 Steenrod algebra, milnor basis
Hko = SteenrodFPModule(A, [0], [[Sq(2)], [Sq(1)]]) homspace = Hom(Hko, Hko); homspace
Just as with module elements, homomorphisms are created using the homspace. The only argument is a list of elements in the codomain, giving the images of the module generators of the domain:
sage: gen = Hko.generator(0) # the generator of the codomain module
sage: values = [Sq(0, 0, 1) * gen]; values
[Sq(0,0,1)*g[0]]
sage: f = homspace(values)
>>> from sage.all import *
>>> gen = Hko.generator(Integer(0)) # the generator of the codomain module
>>> values = [Sq(Integer(0), Integer(0), Integer(1)) * gen]; values
[Sq(0,0,1)*g[0]]
>>> f = homspace(values)
gen = Hko.generator(0) # the generator of the codomain module values = [Sq(0, 0, 1) * gen]; values f = homspace(values)
The resulting homomorphism is the one sending the \(i\)-th generator of the domain to the \(i\)-th codomain value given:
sage: f
Module endomorphism of Finitely presented left module on 1 generator
and 2 relations over mod 2 Steenrod algebra, milnor basis
Defn: g[0] |--> Sq(0,0,1)*g[0]
>>> from sage.all import *
>>> f
Module endomorphism of Finitely presented left module on 1 generator
and 2 relations over mod 2 Steenrod algebra, milnor basis
Defn: g[0] |--> Sq(0,0,1)*g[0]
f
Homomorphisms can be evaluated on elements of the domain module:
sage: v1 = f(Sq(4)*gen); v1
Sq(4,0,1)*g[0]
sage: v2 = f(Sq(2)*Sq(4)*gen); v2
(Sq(3,1,1)+Sq(6,0,1))*g[0]
>>> from sage.all import *
>>> v1 = f(Sq(Integer(4))*gen); v1
Sq(4,0,1)*g[0]
>>> v2 = f(Sq(Integer(2))*Sq(Integer(4))*gen); v2
(Sq(3,1,1)+Sq(6,0,1))*g[0]
v1 = f(Sq(4)*gen); v1 v2 = f(Sq(2)*Sq(4)*gen); v2
and they respect the module action:
sage: f(Sq(4)*gen) == Sq(4)*f(gen)
True
sage: f(Sq(2)*Sq(4)*gen) == Sq(2)*Sq(4)*f(gen)
True
>>> from sage.all import *
>>> f(Sq(Integer(4))*gen) == Sq(Integer(4))*f(gen)
True
>>> f(Sq(Integer(2))*Sq(Integer(4))*gen) == Sq(Integer(2))*Sq(Integer(4))*f(gen)
True
f(Sq(4)*gen) == Sq(4)*f(gen) f(Sq(2)*Sq(4)*gen) == Sq(2)*Sq(4)*f(gen)
Convenience methods exist for creating the trivial morphism:
sage: x = Sq(4)*Sq(7)*gen
sage: x == 0
False
sage: zero_map = homspace.zero(); zero_map
Module endomorphism of Finitely presented left module on 1 generator
and 2 relations over mod 2 Steenrod algebra, milnor basis
Defn: g[0] |--> 0
sage: zero_map(x)
0
sage: zero_map(x).is_zero()
True
>>> from sage.all import *
>>> x = Sq(Integer(4))*Sq(Integer(7))*gen
>>> x == Integer(0)
False
>>> zero_map = homspace.zero(); zero_map
Module endomorphism of Finitely presented left module on 1 generator
and 2 relations over mod 2 Steenrod algebra, milnor basis
Defn: g[0] |--> 0
>>> zero_map(x)
0
>>> zero_map(x).is_zero()
True
x = Sq(4)*Sq(7)*gen x == 0 zero_map = homspace.zero(); zero_map zero_map(x) zero_map(x).is_zero()
as well as the identity endomorphism:
sage: one = Hom(Hko, Hko).identity(); one
Module endomorphism of Finitely presented left module on 1 generator
and 2 relations over mod 2 Steenrod algebra, milnor basis
Defn: g[0] |--> g[0]
sage: one.is_endomorphism()
True
sage: one(x) == x
True
sage: one.is_identity()
True
>>> from sage.all import *
>>> one = Hom(Hko, Hko).identity(); one
Module endomorphism of Finitely presented left module on 1 generator
and 2 relations over mod 2 Steenrod algebra, milnor basis
Defn: g[0] |--> g[0]
>>> one.is_endomorphism()
True
>>> one(x) == x
True
>>> one.is_identity()
True
one = Hom(Hko, Hko).identity(); one one.is_endomorphism() one(x) == x one.is_identity()
Any non-trivial homomorphism has a well defined degree:
sage: f.degree()
7
>>> from sage.all import *
>>> f.degree()
7
f.degree()
but just as for module elements, the trivial homomorphism does not:
sage: zero_map = homspace.zero()
sage: zero_map.degree()
Traceback (most recent call last):
...
ValueError: the zero morphism does not have a well-defined degree
>>> from sage.all import *
>>> zero_map = homspace.zero()
>>> zero_map.degree()
Traceback (most recent call last):
...
ValueError: the zero morphism does not have a well-defined degree
zero_map = homspace.zero() zero_map.degree()
Any two homomorphisms can be added as long as they are of the same degree:
sage: f1 = homspace([Hko([Sq(0,0,3) + Sq(0,2,0,1)])])
sage: f2 = homspace([Hko([Sq(8,2,1)])])
sage: (f1 + f2).is_zero()
False
sage: f1 + f2
Module endomorphism of Finitely presented left module on 1 generator
and 2 relations over mod 2 Steenrod algebra, milnor basis
Defn: g[0] |--> (Sq(0,0,3)+Sq(0,2,0,1)+Sq(8,2,1))*g[0]
>>> from sage.all import *
>>> f1 = homspace([Hko([Sq(Integer(0),Integer(0),Integer(3)) + Sq(Integer(0),Integer(2),Integer(0),Integer(1))])])
>>> f2 = homspace([Hko([Sq(Integer(8),Integer(2),Integer(1))])])
>>> (f1 + f2).is_zero()
False
>>> f1 + f2
Module endomorphism of Finitely presented left module on 1 generator
and 2 relations over mod 2 Steenrod algebra, milnor basis
Defn: g[0] |--> (Sq(0,0,3)+Sq(0,2,0,1)+Sq(8,2,1))*g[0]
f1 = homspace([Hko([Sq(0,0,3) + Sq(0,2,0,1)])]) f2 = homspace([Hko([Sq(8,2,1)])]) (f1 + f2).is_zero() f1 + f2
or when at least one of them is zero:
sage: f + zero_map == f
True
>>> from sage.all import *
>>> f + zero_map == f
True
f + zero_map == f
but not if they have different degrees:
sage: F = SteenrodFPModule(A, [0])
sage: b4 = Hom(F, F)([Sq(4) * F.generator(0)])
sage: b8 = Hom(F, F)([Sq(8) * F.generator(0)])
sage: b4 + b8
Traceback (most recent call last):
...
ValueError: morphisms do not have the same degree
>>> from sage.all import *
>>> F = SteenrodFPModule(A, [Integer(0)])
>>> b4 = Hom(F, F)([Sq(Integer(4)) * F.generator(Integer(0))])
>>> b8 = Hom(F, F)([Sq(Integer(8)) * F.generator(Integer(0))])
>>> b4 + b8
Traceback (most recent call last):
...
ValueError: morphisms do not have the same degree
F = SteenrodFPModule(A, [0]) b4 = Hom(F, F)([Sq(4) * F.generator(0)]) b8 = Hom(F, F)([Sq(8) * F.generator(0)]) b4 + b8
Finally, additive inverses exist:
sage: (f - f) == 0
True
>>> from sage.all import *
>>> (f - f) == Integer(0)
True
(f - f) == 0
The restriction of a homomorphism to the vector space of \(n\)-dimensional module elements is a linear transformation:
sage: f_21 = f.vector_presentation(21); f_21
Vector space morphism represented by the matrix:
[1 0 0 0 0 0]
[0 0 0 0 0 0]
[1 0 0 0 0 0]
Domain: Vector space quotient V/W of dimension 3 over Finite Field of size 2 where
V: Vector space of dimension 20 over Finite Field of size 2
W: Vector space of degree 20 and dimension 17 over Finite Field of size 2
Basis matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1]
[0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 1]
[0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1]
[0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1]
[0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1]
Codomain: Vector space quotient V/W of dimension 6 over Finite Field of size 2 where
V: Vector space of dimension 35 over Finite Field of size 2
W: Vector space of degree 35 and dimension 29 over Finite Field of size 2
Basis matrix:
29 x 35 dense matrix over Finite Field of size 2
>>> from sage.all import *
>>> f_21 = f.vector_presentation(Integer(21)); f_21
Vector space morphism represented by the matrix:
[1 0 0 0 0 0]
[0 0 0 0 0 0]
[1 0 0 0 0 0]
Domain: Vector space quotient V/W of dimension 3 over Finite Field of size 2 where
V: Vector space of dimension 20 over Finite Field of size 2
W: Vector space of degree 20 and dimension 17 over Finite Field of size 2
Basis matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1]
[0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 1]
[0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1]
[0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1]
[0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1]
Codomain: Vector space quotient V/W of dimension 6 over Finite Field of size 2 where
V: Vector space of dimension 35 over Finite Field of size 2
W: Vector space of degree 35 and dimension 29 over Finite Field of size 2
Basis matrix:
29 x 35 dense matrix over Finite Field of size 2
f_21 = f.vector_presentation(21); f_21
This is compatible with the vector presentations of its domain and codomain modules:
sage: f.domain() is Hko
True
sage: f.codomain() is Hko
True
sage: f_21.domain() is Hko.vector_presentation(21)
True
sage: f_21.codomain() is Hko.vector_presentation(21 + f.degree())
True
>>> from sage.all import *
>>> f.domain() is Hko
True
>>> f.codomain() is Hko
True
>>> f_21.domain() is Hko.vector_presentation(Integer(21))
True
>>> f_21.codomain() is Hko.vector_presentation(Integer(21) + f.degree())
True
f.domain() is Hko f.codomain() is Hko f_21.domain() is Hko.vector_presentation(21) f_21.codomain() is Hko.vector_presentation(21 + f.degree())
Elements in the preimage of a homomorphism can be found:
sage: f.solve(Sq(2)*Sq(4)*Sq(7)*gen)
Sq(0,2)*g[0]
sage: f.solve(Sq(8)*gen) is None
True
>>> from sage.all import *
>>> f.solve(Sq(Integer(2))*Sq(Integer(4))*Sq(Integer(7))*gen)
Sq(0,2)*g[0]
>>> f.solve(Sq(Integer(8))*gen) is None
True
f.solve(Sq(2)*Sq(4)*Sq(7)*gen) f.solve(Sq(8)*gen) is None
Homomorphisms can be composed as expected:
sage: g = homspace([Sq(0, 0, 0, 1)*gen]); g
Module endomorphism of Finitely presented left module on 1 generator
and 2 relations over mod 2 Steenrod algebra, milnor basis
Defn: g[0] |--> Sq(0,0,0,1)*g[0]
sage: g*f
Module endomorphism of Finitely presented left module on 1 generator
and 2 relations over mod 2 Steenrod algebra, milnor basis
Defn: g[0] |--> Sq(0,0,1,1)*g[0]
sage: one = homspace.identity()
sage: f*one == f
True
>>> from sage.all import *
>>> g = homspace([Sq(Integer(0), Integer(0), Integer(0), Integer(1))*gen]); g
Module endomorphism of Finitely presented left module on 1 generator
and 2 relations over mod 2 Steenrod algebra, milnor basis
Defn: g[0] |--> Sq(0,0,0,1)*g[0]
>>> g*f
Module endomorphism of Finitely presented left module on 1 generator
and 2 relations over mod 2 Steenrod algebra, milnor basis
Defn: g[0] |--> Sq(0,0,1,1)*g[0]
>>> one = homspace.identity()
>>> f*one == f
True
g = homspace([Sq(0, 0, 0, 1)*gen]); g g*f one = homspace.identity() f*one == f
Homological algebra¶
The category of modules over \(A\) is Abelian, so kernels, images and
cokernels all exist and can be computed using the morphisms.
Note
Methods for morphisms like
compute sub- and quotient modules related to homomorphisms, but they do not return instances of the module class. Rather, they return the natural homomorphisms which connect these modules to the modules that gave rise to them.
For example the function
kernel_inclusion()
returns an injective homomorphism which is onto the kernel
submodule we asked it to compute, while the function
cokernel_projection()
provides a surjective homomorphism onto the cokernel module.
In each case, getting a reference to the module instance requires calling
domain()
or
codomain()
on the returned homomorphism, depending on the case.
Refer to each function’s documentation for specific details.
Cokernels¶
In the following example, we define a cyclic module \(H\ZZ\) with one relation in two ways: first explicitly, and then as the cokernel of a homomorphism of free modules. We then construct a candidate for an isomorphism and check that it is both injective and surjective:
sage: HZ = SteenrodFPModule(A, [0], [[Sq(1)]]); HZ
Finitely presented left module on 1 generator and 1 relation
over mod 2 Steenrod algebra, milnor basis
sage: F = SteenrodFPModule(A, [0])
sage: j = Hom(F, F)([Sq(1)*F.generator(0)])
sage: coker = j.cokernel_projection() # the natural quotient homomorphism onto the cokernel.
sage: hz = coker.codomain(); hz
Finitely presented left module on 1 generator and 1 relation
over mod 2 Steenrod algebra, milnor basis
sage: a = Hom(HZ, hz)([hz.generator(0)])
sage: a.is_injective()
True
sage: a.is_surjective()
True
>>> from sage.all import *
>>> HZ = SteenrodFPModule(A, [Integer(0)], [[Sq(Integer(1))]]); HZ
Finitely presented left module on 1 generator and 1 relation
over mod 2 Steenrod algebra, milnor basis
>>> F = SteenrodFPModule(A, [Integer(0)])
>>> j = Hom(F, F)([Sq(Integer(1))*F.generator(Integer(0))])
>>> coker = j.cokernel_projection() # the natural quotient homomorphism onto the cokernel.
>>> hz = coker.codomain(); hz
Finitely presented left module on 1 generator and 1 relation
over mod 2 Steenrod algebra, milnor basis
>>> a = Hom(HZ, hz)([hz.generator(Integer(0))])
>>> a.is_injective()
True
>>> a.is_surjective()
True
HZ = SteenrodFPModule(A, [0], [[Sq(1)]]); HZ F = SteenrodFPModule(A, [0]) j = Hom(F, F)([Sq(1)*F.generator(0)]) coker = j.cokernel_projection() # the natural quotient homomorphism onto the cokernel. hz = coker.codomain(); hz a = Hom(HZ, hz)([hz.generator(0)]) a.is_injective() a.is_surjective()
Kernels¶
When computing the kernel of a homomorphism \(f\), the result is an injective homomorphism into the domain of \(f\):
sage: k = f.kernel_inclusion(); k
Module morphism:
From: Finitely presented left module on 1 generator and 3 relations
over mod 2 Steenrod algebra, milnor basis
To: Finitely presented left module on 1 generator and 2 relations
over mod 2 Steenrod algebra, milnor basis
Defn: g[7] |--> Sq(0,0,1)*g[0]
sage: k.codomain() == f.domain()
True
sage: k.is_injective()
True
sage: ker = k.domain()
sage: ker
Finitely presented left module on 1 generator and 3 relations
over mod 2 Steenrod algebra, milnor basis
>>> from sage.all import *
>>> k = f.kernel_inclusion(); k
Module morphism:
From: Finitely presented left module on 1 generator and 3 relations
over mod 2 Steenrod algebra, milnor basis
To: Finitely presented left module on 1 generator and 2 relations
over mod 2 Steenrod algebra, milnor basis
Defn: g[7] |--> Sq(0,0,1)*g[0]
>>> k.codomain() == f.domain()
True
>>> k.is_injective()
True
>>> ker = k.domain()
>>> ker
Finitely presented left module on 1 generator and 3 relations
over mod 2 Steenrod algebra, milnor basis
k = f.kernel_inclusion(); k k.codomain() == f.domain() k.is_injective() ker = k.domain() ker
We can check that the injective image of \(k\) is the kernel of \(f\) by showing that \(f\) factors as \(h\circ c\), where \(c\) is the quotient map to the cokernel of \(k\), and \(h\) is injective:
sage: K = k.codomain() # We want to check that this really is the kernel of f.
sage: coker = k.cokernel_projection() # coker is the natural map: Hko -> coker(f) with kernel K.
sage: h = Hom(coker.codomain(), Hko)(f.values())
sage: h*coker == f # Is K contained in ker(f) ?
True
sage: h.is_injective() # Is ker(f) contained in K ?
True
>>> from sage.all import *
>>> K = k.codomain() # We want to check that this really is the kernel of f.
>>> coker = k.cokernel_projection() # coker is the natural map: Hko -> coker(f) with kernel K.
>>> h = Hom(coker.codomain(), Hko)(f.values())
>>> h*coker == f # Is K contained in ker(f) ?
True
>>> h.is_injective() # Is ker(f) contained in K ?
True
K = k.codomain() # We want to check that this really is the kernel of f. coker = k.cokernel_projection() # coker is the natural map: Hko -> coker(f) with kernel K. h = Hom(coker.codomain(), Hko)(f.values()) h*coker == f # Is K contained in ker(f) ? h.is_injective() # Is ker(f) contained in K ?
Images¶
The method
image()
behaves similarly, returning an injective homomorphism with image
equal to the submodule \(\operatorname{im}(f)\):
sage: i = f.image(); i
Module morphism:
From: Finitely presented left module on 1 generator and 3 relations
over mod 2 Steenrod algebra, milnor basis
To: Finitely presented left module on 1 generator and 2 relations
over mod 2 Steenrod algebra, milnor basis
Defn: g[7] |--> Sq(0,0,1)*g[0]
sage: i.codomain() == f.codomain()
True
sage: i.is_injective()
True
>>> from sage.all import *
>>> i = f.image(); i
Module morphism:
From: Finitely presented left module on 1 generator and 3 relations
over mod 2 Steenrod algebra, milnor basis
To: Finitely presented left module on 1 generator and 2 relations
over mod 2 Steenrod algebra, milnor basis
Defn: g[7] |--> Sq(0,0,1)*g[0]
>>> i.codomain() == f.codomain()
True
>>> i.is_injective()
True
i = f.image(); i i.codomain() == f.codomain() i.is_injective()
We can check that the injective image of \(i\) is the image of \(f\) by lifting \(f\) over \(i\), and showing that the lift is surjective:
sage: f_ = f.lift(i); f_
Module morphism:
From: Finitely presented left module on 1 generator and 2 relations
over mod 2 Steenrod algebra, milnor basis
To: Finitely presented left module on 1 generator and 3 relations
over mod 2 Steenrod algebra, milnor basis
Defn: g[0] |--> g[7]
sage: i*f_ == f # Is im(i) contained in im(f) ?
True
sage: f_.is_surjective() # Is im(f) contained in im(i) ?
True
>>> from sage.all import *
>>> f_ = f.lift(i); f_
Module morphism:
From: Finitely presented left module on 1 generator and 2 relations
over mod 2 Steenrod algebra, milnor basis
To: Finitely presented left module on 1 generator and 3 relations
over mod 2 Steenrod algebra, milnor basis
Defn: g[0] |--> g[7]
>>> i*f_ == f # Is im(i) contained in im(f) ?
True
>>> f_.is_surjective() # Is im(f) contained in im(i) ?
True
f_ = f.lift(i); f_ i*f_ == f # Is im(i) contained in im(f) ? f_.is_surjective() # Is im(f) contained in im(i) ?
When a pair of composable homomorphisms \(g\circ f: M\to N\to L\) satisfy the condition \(g\circ f = 0\), the sub-quotient \(\ker(g) / \operatorname{im}(f)\) can be computed and is given by the natural quotient homomorphism with domain \(\ker(g)\):
sage: f * f == 0 # Does the kernel of f contain the image of f ?
True
sage: K = f.kernel_inclusion() # k: ker(f) -> Hko
sage: h = f.homology(f) # h: ker(f) -> ker(f) / im(f)
sage: h.codomain() # This is the homology module.
Finitely presented left module on 1 generator and 4 relations
over mod 2 Steenrod algebra, milnor basis
>>> from sage.all import *
>>> f * f == Integer(0) # Does the kernel of f contain the image of f ?
True
>>> K = f.kernel_inclusion() # k: ker(f) -> Hko
>>> h = f.homology(f) # h: ker(f) -> ker(f) / im(f)
>>> h.codomain() # This is the homology module.
Finitely presented left module on 1 generator and 4 relations
over mod 2 Steenrod algebra, milnor basis
f * f == 0 # Does the kernel of f contain the image of f ? K = f.kernel_inclusion() # k: ker(f) -> Hko h = f.homology(f) # h: ker(f) -> ker(f) / im(f) h.codomain() # This is the homology module.
Free resolutions¶
Finally, free resolutions can be computed. These calculations usually take some time to complete, so it is usually a good idea to raise the verbose flag to output progress information.
The following examples are taken from Michael Catanzaro’s thesis where the first version of this software appeared:
sage: res = Hko.resolution(6, verbose=True)
Computing f_1 (1/6)
Computing f_2 (2/6)
Computing using the profile:
(2, 1)
Resolving the kernel in the range of dimensions [1, 8]: 1 2 3 4 5 6 7 8.
Computing f_3 (3/6)
Computing using the profile:
(2, 1)
Resolving the kernel in the range of dimensions [2, 10]: 2 3 4 5 6 7 8 9 10.
Computing f_4 (4/6)
Computing using the profile:
(2, 1)
Resolving the kernel in the range of dimensions [3, 13]: 3 4 5 6 7 8 9 10 11 12 13.
Computing f_5 (5/6)
Computing using the profile:
(2, 1)
Resolving the kernel in the range of dimensions [4, 18]: 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18.
Computing f_6 (6/6)
Computing using the profile:
(2, 1)
Resolving the kernel in the range of dimensions [5, 20]: 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20.
>>> from sage.all import *
>>> res = Hko.resolution(Integer(6), verbose=True)
Computing f_1 (1/6)
Computing f_2 (2/6)
Computing using the profile:
(2, 1)
Resolving the kernel in the range of dimensions [1, 8]: 1 2 3 4 5 6 7 8.
Computing f_3 (3/6)
Computing using the profile:
(2, 1)
Resolving the kernel in the range of dimensions [2, 10]: 2 3 4 5 6 7 8 9 10.
Computing f_4 (4/6)
Computing using the profile:
(2, 1)
Resolving the kernel in the range of dimensions [3, 13]: 3 4 5 6 7 8 9 10 11 12 13.
Computing f_5 (5/6)
Computing using the profile:
(2, 1)
Resolving the kernel in the range of dimensions [4, 18]: 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18.
Computing f_6 (6/6)
Computing using the profile:
(2, 1)
Resolving the kernel in the range of dimensions [5, 20]: 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20.
res = Hko.resolution(6, verbose=True)
The result of the calculation is a list of all the maps in the resolution:
sage: [f.domain() for f in res]
[Free graded left module on 1 generator over mod 2 Steenrod algebra, milnor basis,
Free graded left module on 2 generators over mod 2 Steenrod algebra, milnor basis,
Free graded left module on 2 generators over mod 2 Steenrod algebra, milnor basis,
Free graded left module on 2 generators over mod 2 Steenrod algebra, milnor basis,
Free graded left module on 3 generators over mod 2 Steenrod algebra, milnor basis,
Free graded left module on 4 generators over mod 2 Steenrod algebra, milnor basis,
Free graded left module on 4 generators over mod 2 Steenrod algebra, milnor basis]
sage: def is_complex(res):
....: for i in range(len(res)-1):
....: f = (res[i]*res[i+1])
....: if not f.is_zero():
....: return False
....: return True
....:
sage: is_complex(res)
True
sage: def is_exact(res):
....: for i in range(len(res)-1):
....: h = res[i].homology(res[i+1])
....: if not h.codomain().is_trivial():
....: return False
....: return True
sage: is_exact(res)
True
sage: [r.codomain().generator_degrees() for r in res]
[(0,), (0,), (2, 1), (2, 4), (3, 7), (4, 8, 12), (5, 9, 13, 14)]
sage: [r.values() for r in res]
[(g[0],),
(Sq(2)*g[0], Sq(1)*g[0]),
(Sq(1)*g[1], Sq(3)*g[1] + Sq(2)*g[2]),
(Sq(1)*g[2], Sq(2,1)*g[2] + Sq(3)*g[4]),
(Sq(1)*g[3], Sq(2,1)*g[3] + Sq(1)*g[7], Sq(2,1)*g[7]),
(Sq(1)*g[4],
Sq(2,1)*g[4] + Sq(1)*g[8],
Sq(2,1)*g[8] + Sq(1)*g[12],
Sq(2)*g[12]),
(Sq(1)*g[5],
Sq(2,1)*g[5] + Sq(1)*g[9],
Sq(2,1)*g[9] + Sq(1)*g[13],
Sq(0,1)*g[13] + Sq(2)*g[14])]
>>> from sage.all import *
>>> [f.domain() for f in res]
[Free graded left module on 1 generator over mod 2 Steenrod algebra, milnor basis,
Free graded left module on 2 generators over mod 2 Steenrod algebra, milnor basis,
Free graded left module on 2 generators over mod 2 Steenrod algebra, milnor basis,
Free graded left module on 2 generators over mod 2 Steenrod algebra, milnor basis,
Free graded left module on 3 generators over mod 2 Steenrod algebra, milnor basis,
Free graded left module on 4 generators over mod 2 Steenrod algebra, milnor basis,
Free graded left module on 4 generators over mod 2 Steenrod algebra, milnor basis]
>>> def is_complex(res):
... for i in range(len(res)-Integer(1)):
... f = (res[i]*res[i+Integer(1)])
... if not f.is_zero():
... return False
... return True
....:
>>> is_complex(res)
True
>>> def is_exact(res):
... for i in range(len(res)-Integer(1)):
... h = res[i].homology(res[i+Integer(1)])
... if not h.codomain().is_trivial():
... return False
... return True
>>> is_exact(res)
True
>>> [r.codomain().generator_degrees() for r in res]
[(0,), (0,), (2, 1), (2, 4), (3, 7), (4, 8, 12), (5, 9, 13, 14)]
>>> [r.values() for r in res]
[(g[0],),
(Sq(2)*g[0], Sq(1)*g[0]),
(Sq(1)*g[1], Sq(3)*g[1] + Sq(2)*g[2]),
(Sq(1)*g[2], Sq(2,1)*g[2] + Sq(3)*g[4]),
(Sq(1)*g[3], Sq(2,1)*g[3] + Sq(1)*g[7], Sq(2,1)*g[7]),
(Sq(1)*g[4],
Sq(2,1)*g[4] + Sq(1)*g[8],
Sq(2,1)*g[8] + Sq(1)*g[12],
Sq(2)*g[12]),
(Sq(1)*g[5],
Sq(2,1)*g[5] + Sq(1)*g[9],
Sq(2,1)*g[9] + Sq(1)*g[13],
Sq(0,1)*g[13] + Sq(2)*g[14])]
[f.domain() for f in res]
def is_complex(res):
for i in range(len(res)-1):
f = (res[i]*res[i+1])
if not f.is_zero():
return False
return True
is_complex(res)
def is_exact(res):
for i in range(len(res)-1):
h = res[i].homology(res[i+1])
if not h.codomain().is_trivial():
return False
return True
is_exact(res)
[r.codomain().generator_degrees() for r in res]
[r.values() for r in res]
Example: Counting lifts¶
In this more elaborate example we show how to find all possible lifts of a particular homomorphism. We will do this in two ways, and as a check of validity, we will compare the results in the end.
We will work with the following modules over the mod 2 Steenrod algebra \(A\):
\begin{align}
H\ZZ &= A/A\cdot \operatorname{Sq}(1)\\
Hko &= A/A\cdot \{\operatorname{Sq}(2), \operatorname{Sq}(1)\} \,.
\end{align}
There is a natural projection \(q: H\ZZ\to Hko\), and a non-trivial endomorphism of degree 28, represented as a degree zero map \(f: \Sigma^{28}Hko\to Hko\) that we define below.
The problem we will solve is to find all possible homomorphisms \(f': \Sigma^{28}Hko\to H\ZZ\), making the following diagram into a commuting triangle:
\[H\ZZ\xrightarrow{q} Hko \xleftarrow{f} \Sigma^{28} Hko.\]
We begin by defining the modules and the homomorphisms \(f\) and \(q\). In the following, we let \(L = \Sigma^{28}Hko\):
sage: from sage.modules.fp_graded.steenrod.module import SteenrodFPModule
sage: A = SteenrodAlgebra(2)
sage: Hko = SteenrodFPModule(A, [0], [[Sq(2)],[Sq(1)]])
sage: HZ = SteenrodFPModule(A, [0], [[Sq(1)]])
sage: L = Hko.suspension(28)
>>> from sage.all import *
>>> from sage.modules.fp_graded.steenrod.module import SteenrodFPModule
>>> A = SteenrodAlgebra(Integer(2))
>>> Hko = SteenrodFPModule(A, [Integer(0)], [[Sq(Integer(2))],[Sq(Integer(1))]])
>>> HZ = SteenrodFPModule(A, [Integer(0)], [[Sq(Integer(1))]])
>>> L = Hko.suspension(Integer(28))
from sage.modules.fp_graded.steenrod.module import SteenrodFPModule A = SteenrodAlgebra(2) Hko = SteenrodFPModule(A, [0], [[Sq(2)],[Sq(1)]]) HZ = SteenrodFPModule(A, [0], [[Sq(1)]]) L = Hko.suspension(28)
The projection:
sage: q = Hom(HZ, Hko)([Hko.generator(0)])
sage: q
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 1 generator and 2 relations
over mod 2 Steenrod algebra, milnor basis
Defn: g[0] |--> g[0]
>>> from sage.all import *
>>> q = Hom(HZ, Hko)([Hko.generator(Integer(0))])
>>> q
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 1 generator and 2 relations
over mod 2 Steenrod algebra, milnor basis
Defn: g[0] |--> g[0]
q = Hom(HZ, Hko)([Hko.generator(0)]) q
The map to lift over \(q\):
sage: f = Hom(L, Hko)([Sq(0,2,1,1)*Hko.generator(0)])
sage: f
Module morphism:
From: Finitely presented left module on 1 generator and 2 relations
over mod 2 Steenrod algebra, milnor basis
To: Finitely presented left module on 1 generator and 2 relations
over mod 2 Steenrod algebra, milnor basis
Defn: g[28] |--> Sq(0,2,1,1)*g[0]
sage: f.is_zero() # f is non-trivial.
False
>>> from sage.all import *
>>> f = Hom(L, Hko)([Sq(Integer(0),Integer(2),Integer(1),Integer(1))*Hko.generator(Integer(0))])
>>> f
Module morphism:
From: Finitely presented left module on 1 generator and 2 relations
over mod 2 Steenrod algebra, milnor basis
To: Finitely presented left module on 1 generator and 2 relations
over mod 2 Steenrod algebra, milnor basis
Defn: g[28] |--> Sq(0,2,1,1)*g[0]
>>> f.is_zero() # f is non-trivial.
False
f = Hom(L, Hko)([Sq(0,2,1,1)*Hko.generator(0)]) f f.is_zero() # f is non-trivial.
We will count the number of different lifts in two ways. First, we will simply compute the vector space of all possible maps \(L \to H\ZZ\), and then check which of those become \(f\) when composed with \(q\):
sage: basis = Hom(L, HZ).basis_elements(0) # The basis for the vector space of degree 0 maps L -> HZ
sage: from itertools import product
sage: def from_coords(c):
....: '''
....: Create a linear combination of the three basis homomorphisms.
....: '''
....: return c[0]*basis[0] + c[1]*basis[1] + c[2]*basis[2]
sage: for coords in product([0,1], repeat=3):
....: print('%s: %s' % (coords, q*from_coords(coords) == f))
(0, 0, 0): False
(0, 0, 1): False
(0, 1, 0): True
(0, 1, 1): True
(1, 0, 0): True
(1, 0, 1): True
(1, 1, 0): False
(1, 1, 1): False
>>> from sage.all import *
>>> basis = Hom(L, HZ).basis_elements(Integer(0)) # The basis for the vector space of degree 0 maps L -> HZ
>>> from itertools import product
>>> def from_coords(c):
... '''
... Create a linear combination of the three basis homomorphisms.
... '''
... return c[Integer(0)]*basis[Integer(0)] + c[Integer(1)]*basis[Integer(1)] + c[Integer(2)]*basis[Integer(2)]
>>> for coords in product([Integer(0),Integer(1)], repeat=Integer(3)):
... print('%s: %s' % (coords, q*from_coords(coords) == f))
(0, 0, 0): False
(0, 0, 1): False
(0, 1, 0): True
(0, 1, 1): True
(1, 0, 0): True
(1, 0, 1): True
(1, 1, 0): False
(1, 1, 1): False
basis = Hom(L, HZ).basis_elements(0) # The basis for the vector space of degree 0 maps L -> HZ
from itertools import product
def from_coords(c):
'''
Create a linear combination of the three basis homomorphisms.
'''
return c[0]*basis[0] + c[1]*basis[1] + c[2]*basis[2]
for coords in product([0,1], repeat=3):
print('%s: %s' % (coords, q*from_coords(coords) == f))
From this we conclude that four out of eight different homomorphisms \(L \to H\ZZ\) are lifts of \(f\):
sage: lifts = [from_coords((0,1,0)),
....: from_coords((0,1,1)),
....: from_coords((1,0,0)),
....: from_coords((1,0,1))]
sage: lifts
[Module morphism:
From: Finitely presented left module on 1 generator and 2 relations over mod 2 Steenrod algebra, milnor basis
To: Finitely presented left module on 1 generator and 1 relation over mod 2 Steenrod algebra, milnor basis
Defn: g[28] |--> Sq(6,5,1)*g[0],
Module morphism:
From: Finitely presented left module on 1 generator and 2 relations over mod 2 Steenrod algebra, milnor basis
To: Finitely presented left module on 1 generator and 1 relation over mod 2 Steenrod algebra, milnor basis
Defn: g[28] |--> (Sq(6,5,1)+Sq(18,1,1))*g[0],
Module morphism:
From: Finitely presented left module on 1 generator and 2 relations over mod 2 Steenrod algebra, milnor basis
To: Finitely presented left module on 1 generator and 1 relation over mod 2 Steenrod algebra, milnor basis
Defn: g[28] |--> Sq(10,1,0,1)*g[0],
Module morphism:
From: Finitely presented left module on 1 generator and 2 relations over mod 2 Steenrod algebra, milnor basis
To: Finitely presented left module on 1 generator and 1 relation over mod 2 Steenrod algebra, milnor basis
Defn: g[28] |--> (Sq(10,1,0,1)+Sq(18,1,1))*g[0]]
>>> from sage.all import *
>>> lifts = [from_coords((Integer(0),Integer(1),Integer(0))),
... from_coords((Integer(0),Integer(1),Integer(1))),
... from_coords((Integer(1),Integer(0),Integer(0))),
... from_coords((Integer(1),Integer(0),Integer(1)))]
>>> lifts
[Module morphism:
From: Finitely presented left module on 1 generator and 2 relations over mod 2 Steenrod algebra, milnor basis
To: Finitely presented left module on 1 generator and 1 relation over mod 2 Steenrod algebra, milnor basis
Defn: g[28] |--> Sq(6,5,1)*g[0],
Module morphism:
From: Finitely presented left module on 1 generator and 2 relations over mod 2 Steenrod algebra, milnor basis
To: Finitely presented left module on 1 generator and 1 relation over mod 2 Steenrod algebra, milnor basis
Defn: g[28] |--> (Sq(6,5,1)+Sq(18,1,1))*g[0],
Module morphism:
From: Finitely presented left module on 1 generator and 2 relations over mod 2 Steenrod algebra, milnor basis
To: Finitely presented left module on 1 generator and 1 relation over mod 2 Steenrod algebra, milnor basis
Defn: g[28] |--> Sq(10,1,0,1)*g[0],
Module morphism:
From: Finitely presented left module on 1 generator and 2 relations over mod 2 Steenrod algebra, milnor basis
To: Finitely presented left module on 1 generator and 1 relation over mod 2 Steenrod algebra, milnor basis
Defn: g[28] |--> (Sq(10,1,0,1)+Sq(18,1,1))*g[0]]
lifts = [from_coords((0,1,0)),
from_coords((0,1,1)),
from_coords((1,0,0)),
from_coords((1,0,1))]
lifts
Alternatively we can use left-exactness of the functor \(\operatorname{Hom}_A(L, -)\) to enumerate all possible lifts of \(f\). Start by finding a single lift of \(f\) over the projection \(q\):
sage: fl = f.lift(q); fl
Module morphism:
From: Finitely presented left module on 1 generator and 2 relations
over mod 2 Steenrod algebra, milnor basis
To: Finitely presented left module on 1 generator and 1 relation
over mod 2 Steenrod algebra, milnor basis
Defn: g[28] |--> (Sq(4,3,0,1)+Sq(6,0,1,1)+Sq(7,2,0,1)+Sq(10,1,0,1))*g[0]
>>> from sage.all import *
>>> fl = f.lift(q); fl
Module morphism:
From: Finitely presented left module on 1 generator and 2 relations
over mod 2 Steenrod algebra, milnor basis
To: Finitely presented left module on 1 generator and 1 relation
over mod 2 Steenrod algebra, milnor basis
Defn: g[28] |--> (Sq(4,3,0,1)+Sq(6,0,1,1)+Sq(7,2,0,1)+Sq(10,1,0,1))*g[0]
fl = f.lift(q); fl
We verify that fl is indeed a lift:
sage: q * fl == f
True
>>> from sage.all import *
>>> q * fl == f
True
q * fl == f
There is an exact sequence
\[0 \to \operatorname{Hom}_A(L, \ker(q)) \xrightarrow{iK_*}
\operatorname{Hom}_A(L, H\ZZ) \xrightarrow{q_*} \operatorname{Hom}_A(L, Hko),\]
which means that the indeterminacy of choosing a lift for \(f \in \operatorname{Hom}_A(L, Hko)\) is represented by an element in \(\operatorname{Hom}_A(L,\ker(q))\). Therefore, we can proceed to count the number of lifts by computing this vector space of homomorphisms:
sage: iK = q.kernel_inclusion()
sage: K = iK.domain()
sage: K.generator_degrees()
(2,)
sage: K.relations()
(Sq(2)*g[2],)
sage: ind = Hom(L, K).basis_elements(0); len(ind)
2
>>> from sage.all import *
>>> iK = q.kernel_inclusion()
>>> K = iK.domain()
>>> K.generator_degrees()
(2,)
>>> K.relations()
(Sq(2)*g[2],)
>>> ind = Hom(L, K).basis_elements(Integer(0)); len(ind)
2
iK = q.kernel_inclusion() K = iK.domain() K.generator_degrees() K.relations() ind = Hom(L, K).basis_elements(0); len(ind)
So now we know that the vector space of indeterminacies is 2-dimensional over the field of two elements. This means that there are four distinct lifts of \(f\) over \(q\), and we can construct these by taking the one lift we already found, and add to it all the different elements in the image of \(iK_*\):
sage: flift = [fl,
....: fl + iK * ind[0],
....: fl + iK * ind[1],
....: fl + iK * (ind[0] + ind[1])]
sage: flift
[Module morphism:
From: Finitely presented left module on 1 generator and 2 relations over mod 2 Steenrod algebra, milnor basis
To: Finitely presented left module on 1 generator and 1 relation over mod 2 Steenrod algebra, milnor basis
Defn: g[28] |--> (Sq(4,3,0,1)+Sq(6,0,1,1)+Sq(7,2,0,1)+Sq(10,1,0,1))*g[0],
Module morphism:
From: Finitely presented left module on 1 generator and 2 relations over mod 2 Steenrod algebra, milnor basis
To: Finitely presented left module on 1 generator and 1 relation over mod 2 Steenrod algebra, milnor basis
Defn: g[28] |--> (Sq(0,7,1)+Sq(3,6,1)+Sq(4,1,3)+Sq(6,0,1,1)+Sq(6,5,1)+Sq(7,0,3))*g[0],
Module morphism:
From: Finitely presented left module on 1 generator and 2 relations over mod 2 Steenrod algebra, milnor basis
To: Finitely presented left module on 1 generator and 1 relation over mod 2 Steenrod algebra, milnor basis
Defn: g[28] |--> (Sq(4,3,0,1)+Sq(6,0,1,1)+Sq(7,2,0,1)+Sq(10,1,0,1)+Sq(12,3,1)+Sq(15,2,1)+Sq(18,1,1))*g[0],
Module morphism:
From: Finitely presented left module on 1 generator and 2 relations over mod 2 Steenrod algebra, milnor basis
To: Finitely presented left module on 1 generator and 1 relation over mod 2 Steenrod algebra, milnor basis
Defn: g[28] |--> (Sq(0,7,1)+Sq(3,6,1)+Sq(4,1,3)+Sq(6,0,1,1)+Sq(6,5,1)+Sq(7,0,3)+Sq(12,3,1)+Sq(15,2,1)+Sq(18,1,1))*g[0]]
>>> from sage.all import *
>>> flift = [fl,
... fl + iK * ind[Integer(0)],
... fl + iK * ind[Integer(1)],
... fl + iK * (ind[Integer(0)] + ind[Integer(1)])]
>>> flift
[Module morphism:
From: Finitely presented left module on 1 generator and 2 relations over mod 2 Steenrod algebra, milnor basis
To: Finitely presented left module on 1 generator and 1 relation over mod 2 Steenrod algebra, milnor basis
Defn: g[28] |--> (Sq(4,3,0,1)+Sq(6,0,1,1)+Sq(7,2,0,1)+Sq(10,1,0,1))*g[0],
Module morphism:
From: Finitely presented left module on 1 generator and 2 relations over mod 2 Steenrod algebra, milnor basis
To: Finitely presented left module on 1 generator and 1 relation over mod 2 Steenrod algebra, milnor basis
Defn: g[28] |--> (Sq(0,7,1)+Sq(3,6,1)+Sq(4,1,3)+Sq(6,0,1,1)+Sq(6,5,1)+Sq(7,0,3))*g[0],
Module morphism:
From: Finitely presented left module on 1 generator and 2 relations over mod 2 Steenrod algebra, milnor basis
To: Finitely presented left module on 1 generator and 1 relation over mod 2 Steenrod algebra, milnor basis
Defn: g[28] |--> (Sq(4,3,0,1)+Sq(6,0,1,1)+Sq(7,2,0,1)+Sq(10,1,0,1)+Sq(12,3,1)+Sq(15,2,1)+Sq(18,1,1))*g[0],
Module morphism:
From: Finitely presented left module on 1 generator and 2 relations over mod 2 Steenrod algebra, milnor basis
To: Finitely presented left module on 1 generator and 1 relation over mod 2 Steenrod algebra, milnor basis
Defn: g[28] |--> (Sq(0,7,1)+Sq(3,6,1)+Sq(4,1,3)+Sq(6,0,1,1)+Sq(6,5,1)+Sq(7,0,3)+Sq(12,3,1)+Sq(15,2,1)+Sq(18,1,1))*g[0]]
flift = [fl,
fl + iK * ind[0],
fl + iK * ind[1],
fl + iK * (ind[0] + ind[1])]
flift
As a test of correctness, we now compare the two sets of lifts. As they stand,
it is not obvious that the lists flift and lifts are the same (up to a
re-ordering of list elements), so the following comparison is reassuring:
sage: flift[0] == lifts[2]
True
sage: flift[1] == lifts[0]
True
sage: flift[2] == lifts[3]
True
sage: flift[3] == lifts[1]
True
>>> from sage.all import *
>>> flift[Integer(0)] == lifts[Integer(2)]
True
>>> flift[Integer(1)] == lifts[Integer(0)]
True
>>> flift[Integer(2)] == lifts[Integer(3)]
True
>>> flift[Integer(3)] == lifts[Integer(1)]
True
flift[0] == lifts[2] flift[1] == lifts[0] flift[2] == lifts[3] flift[3] == lifts[1]