Morphisms between Ore modules

Let $R$ be a commutative ring, $\theta :R\to R$ by a ring endomorphism and $\partial :R\to R$ be a $\theta$-derivation. Let also $\mathcal{S}=R[X;\theta ,\partial ]$ denote the associated Ore polynomial ring.

By definition, a Ore module is a module over $\mathcal{S}$. In SageMath, there are rather represented as modules over $R$ equipped with the map giving the action of the Ore variable $X$. We refer to sage.modules.ore_module for more details.

A morphism of Ore modules is a $R$-linear morphism commuting with the Ore action, or equivalenty a $\mathcal{S}$-linear map.

Construction of morphisms

There are several ways for creating Ore modules morphisms in SageMath. First of all, one can use the method sage.modules.ore_module.OreModule.hom(), passing to it the matrix (in the canonical bases) of the morphism we want to build:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: M.<e0,e1> = S.quotient_module(X^2 + X + z)

sage: mat = matrix(2, 2, [z,     3*z^2 + z + 2,
....:                     z + 1,       4*z + 4])
sage: f = M.hom(mat)
sage: f
Ore module endomorphism of Ore module <e0, e1> over Finite Field in z of size 5^3 twisted by z |--> z^5

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module(X**Integer(2) + X + z, names=('e0', 'e1',)); (e0, e1,) = M._first_ngens(2)

>>> mat = matrix(Integer(2), Integer(2), [z,     Integer(3)*z**Integer(2) + z + Integer(2),
...                     z + Integer(1),       Integer(4)*z + Integer(4)])
>>> f = M.hom(mat)
>>> f
Ore module endomorphism of Ore module <e0, e1> over Finite Field in z of size 5^3 twisted by z |--> z^5

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
M.<e0,e1> = S.quotient_module(X^2 + X + z)
mat = matrix(2, 2, [z,     3*z^2 + z + 2,
                    z + 1,       4*z + 4])
f = M.hom(mat)
f

Clearly, this method is not optimal: typing all the entries of the defining matrix is long and a potential source of errors.

Instead, one can use a dictionary encoding the values taken by the morphism on a set of generators; the morphism is then automatically prolonged by $\mathcal{S}$-linearity. Actually here, $f$ was just the multiplication by $X^3$ on $M$. We can then redefine it simply as follows:

Sage

sage: g = M.hom({e0: X^3*e0})
sage: g
Ore module endomorphism of Ore module <e0, e1> over Finite Field in z of size 5^3 twisted by z |--> z^5

Python

>>> from sage.all import *
>>> g = M.hom({e0: X**Integer(3)*e0})
>>> g
Ore module endomorphism of Ore module <e0, e1> over Finite Field in z of size 5^3 twisted by z |--> z^5

Sage Live

g = M.hom({e0: X^3*e0})
g

One can then recover the matrix by using the method sage.modules.ore_module_morphism.OreModuleMorphism.matrix():

Sage

sage: g.matrix()
[            z 3*z^2 + z + 2]
[        z + 1       4*z + 4]

Python

>>> from sage.all import *
>>> g.matrix()
[            z 3*z^2 + z + 2]
[        z + 1       4*z + 4]

Sage Live

g.matrix()

Alternatively, one can use the method sage.modules.ore_module.OreModule.multiplication_map():

Sage

sage: h = M.multiplication_map(X^3)
sage: h
Ore module endomorphism of Ore module <e0, e1> over Finite Field in z of size 5^3 twisted by z |--> z^5
sage: g == h
True

Python

>>> from sage.all import *
>>> h = M.multiplication_map(X**Integer(3))
>>> h
Ore module endomorphism of Ore module <e0, e1> over Finite Field in z of size 5^3 twisted by z |--> z^5
>>> g == h
True

Sage Live

h = M.multiplication_map(X^3)
h
g == h

Of course, this method also accepts values in the base ring:

Sage

sage: h = M.multiplication_map(2)
sage: h
Ore module endomorphism of Ore module <e0, e1> over Finite Field in z of size 5^3 twisted by z |--> z^5
sage: h.matrix()
[2 0]
[0 2]

Python

>>> from sage.all import *
>>> h = M.multiplication_map(Integer(2))
>>> h
Ore module endomorphism of Ore module <e0, e1> over Finite Field in z of size 5^3 twisted by z |--> z^5
>>> h.matrix()
[2 0]
[0 2]

Sage Live

h = M.multiplication_map(2)
h
h.matrix()

Be careful that scalar multiplications do not always properly define a morphism of Ore modules:

Sage

sage: M.multiplication_map(z)
Traceback (most recent call last):
...
ValueError: does not define a morphism of Ore modules

Python

>>> from sage.all import *
>>> M.multiplication_map(z)
Traceback (most recent call last):
...
ValueError: does not define a morphism of Ore modules

Sage Live

M.multiplication_map(z)

SageMath provides methods to compute kernels, cokernels, images and coimages. In order to illustrate this, we will build the sequence

$$0\to \mathcal{S}/\mathcal{S}P\to \mathcal{S}/\mathcal{S}PQ\to \mathcal{S}/\mathcal{S}Q\to 0$$

and check that it is exact. We first build the Ore modules:

Sage

sage: P = X^2 + z*X + 1
sage: Q = X^3 + z^2*X^2 + X + z
sage: U = S.quotient_module(P, names='u')
sage: U.inject_variables()
Defining u0, u1
sage: V = S.quotient_module(P*Q, names='v')
sage: V.inject_variables()
Defining v0, v1, v2, v3, v4
sage: W = S.quotient_module(Q, names='w')
sage: W.inject_variables()
Defining w0, w1, w2

Python

>>> from sage.all import *
>>> P = X**Integer(2) + z*X + Integer(1)
>>> Q = X**Integer(3) + z**Integer(2)*X**Integer(2) + X + z
>>> U = S.quotient_module(P, names='u')
>>> U.inject_variables()
Defining u0, u1
>>> V = S.quotient_module(P*Q, names='v')
>>> V.inject_variables()
Defining v0, v1, v2, v3, v4
>>> W = S.quotient_module(Q, names='w')
>>> W.inject_variables()
Defining w0, w1, w2

Sage Live

P = X^2 + z*X + 1
Q = X^3 + z^2*X^2 + X + z
U = S.quotient_module(P, names='u')
U.inject_variables()
V = S.quotient_module(P*Q, names='v')
V.inject_variables()
W = S.quotient_module(Q, names='w')
W.inject_variables()

Next, we build the morphisms:

Sage

sage: f = U.hom({u0: Q*v0})
sage: g = V.hom({v0: w0})

Python

>>> from sage.all import *
>>> f = U.hom({u0: Q*v0})
>>> g = V.hom({v0: w0})

Sage Live

f = U.hom({u0: Q*v0})
g = V.hom({v0: w0})

We can now check that $f$ is injective by computing its kernel:

Sage

sage: f.kernel()
Ore module of rank 0 over Finite Field in z of size 5^3 twisted by z |--> z^5

Python

>>> from sage.all import *
>>> f.kernel()
Ore module of rank 0 over Finite Field in z of size 5^3 twisted by z |--> z^5

Sage Live

f.kernel()

We see on the output that it has dimension $0$; so it vanishes. Instead of reading the output, one can check programmatically the vanishing of a Ore module using the method sage.modules.ore_module.OreModule.is_zero():

Sage

sage: f.kernel().is_zero()
True

Python

>>> from sage.all import *
>>> f.kernel().is_zero()
True

Sage Live

f.kernel().is_zero()

Actually, in our use case, one can, more simply, use the method sage.modules.ore_module_morphism.OreModuleMorphism.is_injective():

Sage

sage: f.is_injective()
True

Python

>>> from sage.all import *
>>> f.is_injective()
True

Sage Live

f.is_injective()

Similarly, one checks that $g$ is surjective:

Sage

sage: g.is_surjective()
True

Python

>>> from sage.all import *
>>> g.is_surjective()
True

Sage Live

g.is_surjective()

or equivalently:

Sage

sage: g.cokernel().is_zero()
True

Python

>>> from sage.all import *
>>> g.cokernel().is_zero()
True

Sage Live

g.cokernel().is_zero()

Now, we need to check that the kernel of $g$ equals the image of $f$. For this, we compute both and compare the results:

Sage

sage: ker = g.kernel()
sage: im = f.image()
sage: ker == im
True

Python

>>> from sage.all import *
>>> ker = g.kernel()
>>> im = f.image()
>>> ker == im
True

Sage Live

ker = g.kernel()
im = f.image()
ker == im

As a sanity check, one can also verity that the composite $g\circ f$ vanishes:

Sage

sage: h = g * f
sage: h
Ore module morphism:
  From: Ore module <u0, u1> over Finite Field in z of size 5^3 twisted by z |--> z^5
  To:   Ore module <w0, w1, w2> over Finite Field in z of size 5^3 twisted by z |--> z^5
sage: h.is_zero()
True

Python

>>> from sage.all import *
>>> h = g * f
>>> h
Ore module morphism:
  From: Ore module <u0, u1> over Finite Field in z of size 5^3 twisted by z |--> z^5
  To:   Ore module <w0, w1, w2> over Finite Field in z of size 5^3 twisted by z |--> z^5
>>> h.is_zero()
True

Sage Live

h = g * f
h
h.is_zero()

Let us now consider another morphism $f$ and build the canonical isomorphism $\text{coim }f\to \text{im }f$ that it induces. We start by defining $f$:

Sage

sage: A = X + z
sage: B = X + z + 1
sage: P = X^2 + X + z
sage: U = S.quotient_module(B*P, names='u')
sage: U.inject_variables()
Defining u0, u1, u2
sage: V = S.quotient_module(P*A, names='v')
sage: V.inject_variables()
Defining v0, v1, v2

sage: f = U.hom({u0: A*v0})
sage: f
Ore module morphism:
  From: Ore module <u0, u1, u2> over Finite Field in z of size 5^3 twisted by z |--> z^5
  To:   Ore module <v0, v1, v2> over Finite Field in z of size 5^3 twisted by z |--> z^5

Python

>>> from sage.all import *
>>> A = X + z
>>> B = X + z + Integer(1)
>>> P = X**Integer(2) + X + z
>>> U = S.quotient_module(B*P, names='u')
>>> U.inject_variables()
Defining u0, u1, u2
>>> V = S.quotient_module(P*A, names='v')
>>> V.inject_variables()
Defining v0, v1, v2

>>> f = U.hom({u0: A*v0})
>>> f
Ore module morphism:
  From: Ore module <u0, u1, u2> over Finite Field in z of size 5^3 twisted by z |--> z^5
  To:   Ore module <v0, v1, v2> over Finite Field in z of size 5^3 twisted by z |--> z^5

Sage Live

A = X + z
B = X + z + 1
P = X^2 + X + z
U = S.quotient_module(B*P, names='u')
U.inject_variables()
V = S.quotient_module(P*A, names='v')
V.inject_variables()
f = U.hom({u0: A*v0})
f

Now we compute the image and the coimage:

Sage

sage: I = f.image(names='im')
sage: I
Ore module <im0, im1> over Finite Field in z of size 5^3 twisted by z |--> z^5

sage: C = f.coimage(names='co')
sage: C
Ore module <co0, co1> over Finite Field in z of size 5^3 twisted by z |--> z^5

Python

>>> from sage.all import *
>>> I = f.image(names='im')
>>> I
Ore module <im0, im1> over Finite Field in z of size 5^3 twisted by z |--> z^5

>>> C = f.coimage(names='co')
>>> C
Ore module <co0, co1> over Finite Field in z of size 5^3 twisted by z |--> z^5

Sage Live

I = f.image(names='im')
I
C = f.coimage(names='co')
C

We can already check that the image and the coimage have the same rank. We now want to construct the isomorphism between them. For this, we first need to corestrict $f$ to its image. This is achieved via the method sage.modules.ore_module.OreSubmodule.morphism_corestriction() of the Ore module:

Sage

sage: g = I.morphism_corestriction(f)
sage: g
Ore module morphism:
  From: Ore module <u0, u1, u2> over Finite Field in z of size 5^3 twisted by z |--> z^5
  To:   Ore module <im0, im1> over Finite Field in z of size 5^3 twisted by z |--> z^5

Python

>>> from sage.all import *
>>> g = I.morphism_corestriction(f)
>>> g
Ore module morphism:
  From: Ore module <u0, u1, u2> over Finite Field in z of size 5^3 twisted by z |--> z^5
  To:   Ore module <im0, im1> over Finite Field in z of size 5^3 twisted by z |--> z^5

Sage Live

g = I.morphism_corestriction(f)
g

Next, we want to factor $g$ by the coimage. We proceed as follows:

Sage

sage: h = C.morphism_quotient(g)
sage: h
Ore module morphism:
  From: Ore module <co0, co1> over Finite Field in z of size 5^3 twisted by z |--> z^5
  To:   Ore module <im0, im1> over Finite Field in z of size 5^3 twisted by z |--> z^5

Python

>>> from sage.all import *
>>> h = C.morphism_quotient(g)
>>> h
Ore module morphism:
  From: Ore module <co0, co1> over Finite Field in z of size 5^3 twisted by z |--> z^5
  To:   Ore module <im0, im1> over Finite Field in z of size 5^3 twisted by z |--> z^5

Sage Live

h = C.morphism_quotient(g)
h

We have found the morphism we were looking for: it is $h$. We can now check that it is an isomorphism:

Sage

sage: h.is_isomorphism()
True

Python

>>> from sage.all import *
>>> h.is_isomorphism()
True

Sage Live

h.is_isomorphism()

As a shortcut, we can use explicit conversions as follows:

Sage

sage: H = Hom(C, I)  # the hom space
sage: h2 = H(f)
sage: h2
Ore module morphism:
  From: Ore module <co0, co1> over Finite Field in z of size 5^3 twisted by z |--> z^5
  To:   Ore module <im0, im1> over Finite Field in z of size 5^3 twisted by z |--> z^5
sage: h == h2
True

Python

>>> from sage.all import *
>>> H = Hom(C, I)  # the hom space
>>> h2 = H(f)
>>> h2
Ore module morphism:
  From: Ore module <co0, co1> over Finite Field in z of size 5^3 twisted by z |--> z^5
  To:   Ore module <im0, im1> over Finite Field in z of size 5^3 twisted by z |--> z^5
>>> h == h2
True

Sage Live

H = Hom(C, I)  # the hom space
h2 = H(f)
h2
h == h2

For endomorphisms, one can compute classical invariants as determinants and characteristic polynomials. To illustrate this, we check on an example that the characteristic polynomial of the multiplication by $X^3$ on the quotient $\mathcal{S}/\mathcal{S}P$ is the reduced norm of $P$:

Sage

sage: P = X^5 + z*X^4 + z^2*X^2 + z + 1
sage: M = S.quotient_module(P)
sage: f = M.multiplication_map(X^3)
sage: f.charpoly()
x^5 + x^4 + x^3 + x^2 + 1

sage: P.reduced_norm('x')
x^5 + x^4 + x^3 + x^2 + 1

Python

>>> from sage.all import *
>>> P = X**Integer(5) + z*X**Integer(4) + z**Integer(2)*X**Integer(2) + z + Integer(1)
>>> M = S.quotient_module(P)
>>> f = M.multiplication_map(X**Integer(3))
>>> f.charpoly()
x^5 + x^4 + x^3 + x^2 + 1

>>> P.reduced_norm('x')
x^5 + x^4 + x^3 + x^2 + 1

Sage Live

P = X^5 + z*X^4 + z^2*X^2 + z + 1
M = S.quotient_module(P)
f = M.multiplication_map(X^3)
f.charpoly()
P.reduced_norm('x')

AUTHOR:

• Xavier Caruso (2024-10)

class sage.modules.ore_module_morphism.OreModuleMorphism(parent, im_gens, check=True)

Bases: Morphism

Generic class for morphism between Ore modules.

characteristic_polynomial(var='x')

Return the determinant of this endomorphism.

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: P = X^3 + z*X^2 + (z^2 + 3)*X + z^5
sage: M = S.quotient_module(P)
sage: f = M.multiplication_map(X^3)
sage: f.characteristic_polynomial()
x^3 + x^2 + 2*x + 2

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> P = X**Integer(3) + z*X**Integer(2) + (z**Integer(2) + Integer(3))*X + z**Integer(5)
>>> M = S.quotient_module(P)
>>> f = M.multiplication_map(X**Integer(3))
>>> f.characteristic_polynomial()
x^3 + x^2 + 2*x + 2

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
P = X^3 + z*X^2 + (z^2 + 3)*X + z^5
M = S.quotient_module(P)
f = M.multiplication_map(X^3)
f.characteristic_polynomial()

We check that the latter is equal to the reduced norm of $P$:

Sage

sage: P.reduced_norm('x')
x^3 + x^2 + 2*x + 2

Python

>>> from sage.all import *
>>> P.reduced_norm('x')
x^3 + x^2 + 2*x + 2

Sage Live

P.reduced_norm('x')

charpoly(var='x')

Return the determinant of this endomorphism.

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: P = X^3 + z*X^2 + (z^2 + 3)*X + z^5
sage: M = S.quotient_module(P)
sage: f = M.multiplication_map(X^3)
sage: f.characteristic_polynomial()
x^3 + x^2 + 2*x + 2

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> P = X**Integer(3) + z*X**Integer(2) + (z**Integer(2) + Integer(3))*X + z**Integer(5)
>>> M = S.quotient_module(P)
>>> f = M.multiplication_map(X**Integer(3))
>>> f.characteristic_polynomial()
x^3 + x^2 + 2*x + 2

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
P = X^3 + z*X^2 + (z^2 + 3)*X + z^5
M = S.quotient_module(P)
f = M.multiplication_map(X^3)
f.characteristic_polynomial()

We check that the latter is equal to the reduced norm of $P$:

Sage

sage: P.reduced_norm('x')
x^3 + x^2 + 2*x + 2

Python

>>> from sage.all import *
>>> P.reduced_norm('x')
x^3 + x^2 + 2*x + 2

Sage Live

P.reduced_norm('x')

coimage(names=None)

Return True if this morphism is injective.

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: P = X^3 + z*X^2 + z^2
sage: M = S.quotient_module(P^2, names='m')
sage: M.inject_variables()
Defining m0, m1, m2, m3, m4, m5
sage: N = S.quotient_module(P, names='n')
sage: N.inject_variables()
Defining n0, n1, n2

sage: f = M.hom({m0: n0})
sage: coim = f.coimage()
sage: coim
Ore module of rank 3 over Finite Field in z of size 5^3 twisted by z |--> z^5
sage: coim.basis()
[m3, m4, m5]

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> P = X**Integer(3) + z*X**Integer(2) + z**Integer(2)
>>> M = S.quotient_module(P**Integer(2), names='m')
>>> M.inject_variables()
Defining m0, m1, m2, m3, m4, m5
>>> N = S.quotient_module(P, names='n')
>>> N.inject_variables()
Defining n0, n1, n2

>>> f = M.hom({m0: n0})
>>> coim = f.coimage()
>>> coim
Ore module of rank 3 over Finite Field in z of size 5^3 twisted by z |--> z^5
>>> coim.basis()
[m3, m4, m5]

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
P = X^3 + z*X^2 + z^2
M = S.quotient_module(P^2, names='m')
M.inject_variables()
N = S.quotient_module(P, names='n')
N.inject_variables()
f = M.hom({m0: n0})
coim = f.coimage()
coim
coim.basis()

cokernel(names=None)

Return True if this morphism is injective.

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: P = X^3 + z*X^2 + z^2
sage: M = S.quotient_module(P^2, names='m')
sage: M.inject_variables()
Defining m0, m1, m2, m3, m4, m5
sage: N = S.quotient_module(P, names='n')
sage: N.inject_variables()
Defining n0, n1, n2

sage: f = N.hom({n0: P*m0})
sage: coker = f.cokernel()
sage: coker
Ore module of rank 3 over Finite Field in z of size 5^3 twisted by z |--> z^5
sage: coker.basis()
[m3, m4, m5]

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> P = X**Integer(3) + z*X**Integer(2) + z**Integer(2)
>>> M = S.quotient_module(P**Integer(2), names='m')
>>> M.inject_variables()
Defining m0, m1, m2, m3, m4, m5
>>> N = S.quotient_module(P, names='n')
>>> N.inject_variables()
Defining n0, n1, n2

>>> f = N.hom({n0: P*m0})
>>> coker = f.cokernel()
>>> coker
Ore module of rank 3 over Finite Field in z of size 5^3 twisted by z |--> z^5
>>> coker.basis()
[m3, m4, m5]

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
P = X^3 + z*X^2 + z^2
M = S.quotient_module(P^2, names='m')
M.inject_variables()
N = S.quotient_module(P, names='n')
N.inject_variables()
f = N.hom({n0: P*m0})
coker = f.cokernel()
coker
coker.basis()

det()

Return the determinant of this endomorphism.

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: M.<m0,m1> = S.quotient_module(X^2 + z)
sage: f = M.multiplication_map(X^3)
sage: f.determinant()
2

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module(X**Integer(2) + z, names=('m0', 'm1',)); (m0, m1,) = M._first_ngens(2)
>>> f = M.multiplication_map(X**Integer(3))
>>> f.determinant()
2

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
M.<m0,m1> = S.quotient_module(X^2 + z)
f = M.multiplication_map(X^3)
f.determinant()

If the domain differs from the codomain (even if they have the same rank), an error is raised:

Sage

sage: N.<n0,n1> = S.quotient_module(X^2 + z^25)
sage: g = M.hom({z*m0: n0})
sage: g.determinant()
Traceback (most recent call last):
...
ValueError: determinants are only defined for endomorphisms

Python

>>> from sage.all import *
>>> N = S.quotient_module(X**Integer(2) + z**Integer(25), names=('n0', 'n1',)); (n0, n1,) = N._first_ngens(2)
>>> g = M.hom({z*m0: n0})
>>> g.determinant()
Traceback (most recent call last):
...
ValueError: determinants are only defined for endomorphisms

Sage Live

N.<n0,n1> = S.quotient_module(X^2 + z^25)
g = M.hom({z*m0: n0})
g.determinant()

determinant()

Return the determinant of this endomorphism.

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: M.<m0,m1> = S.quotient_module(X^2 + z)
sage: f = M.multiplication_map(X^3)
sage: f.determinant()
2

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module(X**Integer(2) + z, names=('m0', 'm1',)); (m0, m1,) = M._first_ngens(2)
>>> f = M.multiplication_map(X**Integer(3))
>>> f.determinant()
2

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
M.<m0,m1> = S.quotient_module(X^2 + z)
f = M.multiplication_map(X^3)
f.determinant()

If the domain differs from the codomain (even if they have the same rank), an error is raised:

Sage

sage: N.<n0,n1> = S.quotient_module(X^2 + z^25)
sage: g = M.hom({z*m0: n0})
sage: g.determinant()
Traceback (most recent call last):
...
ValueError: determinants are only defined for endomorphisms

Python

>>> from sage.all import *
>>> N = S.quotient_module(X**Integer(2) + z**Integer(25), names=('n0', 'n1',)); (n0, n1,) = N._first_ngens(2)
>>> g = M.hom({z*m0: n0})
>>> g.determinant()
Traceback (most recent call last):
...
ValueError: determinants are only defined for endomorphisms

Sage Live

N.<n0,n1> = S.quotient_module(X^2 + z^25)
g = M.hom({z*m0: n0})
g.determinant()

image(names=None)

Return True if this morphism is injective.

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: P = X^3 + z*X^2 + z^2
sage: M = S.quotient_module(P^2, names='m')
sage: M.inject_variables()
Defining m0, m1, m2, m3, m4, m5
sage: N = S.quotient_module(P, names='n')
sage: N.inject_variables()
Defining n0, n1, n2

sage: f = N.hom({n0: P*m0})
sage: im = f.image()
sage: im
Ore module of rank 3 over Finite Field in z of size 5^3 twisted by z |--> z^5
sage: im.basis()
[m0 + (2*z^2+3*z+1)*m3 + (4*z^2+3*z+3)*m4 + (2*z^2+3*z)*m5,
 m1 + (z+3)*m3 + (z^2+z+4)*m4,
 m2 + (2*z^2+4*z+2)*m4 + (2*z^2+z+1)*m5]

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> P = X**Integer(3) + z*X**Integer(2) + z**Integer(2)
>>> M = S.quotient_module(P**Integer(2), names='m')
>>> M.inject_variables()
Defining m0, m1, m2, m3, m4, m5
>>> N = S.quotient_module(P, names='n')
>>> N.inject_variables()
Defining n0, n1, n2

>>> f = N.hom({n0: P*m0})
>>> im = f.image()
>>> im
Ore module of rank 3 over Finite Field in z of size 5^3 twisted by z |--> z^5
>>> im.basis()
[m0 + (2*z^2+3*z+1)*m3 + (4*z^2+3*z+3)*m4 + (2*z^2+3*z)*m5,
 m1 + (z+3)*m3 + (z^2+z+4)*m4,
 m2 + (2*z^2+4*z+2)*m4 + (2*z^2+z+1)*m5]

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
P = X^3 + z*X^2 + z^2
M = S.quotient_module(P^2, names='m')
M.inject_variables()
N = S.quotient_module(P, names='n')
N.inject_variables()
f = N.hom({n0: P*m0})
im = f.image()
im
im.basis()

inverse()

Return the inverse of this morphism.

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: P = X^2 + z
sage: M.<e0,e1,e2,e3> = S.quotient_module(P^2)

sage: f = M.multiplication_map(X^3)
sage: g = f.inverse()
sage: (f*g).is_identity()
True
sage: (g*f).is_identity()
True

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> P = X**Integer(2) + z
>>> M = S.quotient_module(P**Integer(2), names=('e0', 'e1', 'e2', 'e3',)); (e0, e1, e2, e3,) = M._first_ngens(4)

>>> f = M.multiplication_map(X**Integer(3))
>>> g = f.inverse()
>>> (f*g).is_identity()
True
>>> (g*f).is_identity()
True

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
P = X^2 + z
M.<e0,e1,e2,e3> = S.quotient_module(P^2)
f = M.multiplication_map(X^3)
g = f.inverse()
(f*g).is_identity()
(g*f).is_identity()

If the morphism is not invertible, an error is raised:

Sage

sage: h = M.hom({e0: P*e0})
sage: h.inverse()
Traceback (most recent call last):
...
ValueError: this morphism is not invertible

Python

>>> from sage.all import *
>>> h = M.hom({e0: P*e0})
>>> h.inverse()
Traceback (most recent call last):
...
ValueError: this morphism is not invertible

Sage Live

h = M.hom({e0: P*e0})
h.inverse()

is_bijective()

Return True if this morphism is bijective.

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: M = S.quotient_module(X^2 + z)
sage: f = M.multiplication_map(X^3)
sage: f.is_bijective()
True

sage: N = S.quotient_module(X^2)
sage: g = N.multiplication_map(X^3)
sage: g.is_bijective()
False

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module(X**Integer(2) + z)
>>> f = M.multiplication_map(X**Integer(3))
>>> f.is_bijective()
True

>>> N = S.quotient_module(X**Integer(2))
>>> g = N.multiplication_map(X**Integer(3))
>>> g.is_bijective()
False

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
M = S.quotient_module(X^2 + z)
f = M.multiplication_map(X^3)
f.is_bijective()
N = S.quotient_module(X^2)
g = N.multiplication_map(X^3)
g.is_bijective()

is_identity()

Return True if this morphism is the identity.

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: M.<v,w> = S.quotient_module(X^2 + z)
sage: f = M.hom({v: v})
sage: f.is_identity()
True

sage: f = M.hom({v: 2*v})
sage: f.is_identity()
False

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module(X**Integer(2) + z, names=('v', 'w',)); (v, w,) = M._first_ngens(2)
>>> f = M.hom({v: v})
>>> f.is_identity()
True

>>> f = M.hom({v: Integer(2)*v})
>>> f.is_identity()
False

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
M.<v,w> = S.quotient_module(X^2 + z)
f = M.hom({v: v})
f.is_identity()
f = M.hom({v: 2*v})
f.is_identity()

is_injective()

Return True if this morphism is injective.

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: P = X^3 + z*X^2 + z^2
sage: M = S.quotient_module(P^2, names='m')
sage: M.inject_variables()
Defining m0, m1, m2, m3, m4, m5
sage: N = S.quotient_module(P, names='n')
sage: N.inject_variables()
Defining n0, n1, n2

sage: f = N.hom({n0: P*m0})
sage: f.is_injective()
True

sage: g = M.hom({m0: n0})
sage: g.is_injective()
False

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> P = X**Integer(3) + z*X**Integer(2) + z**Integer(2)
>>> M = S.quotient_module(P**Integer(2), names='m')
>>> M.inject_variables()
Defining m0, m1, m2, m3, m4, m5
>>> N = S.quotient_module(P, names='n')
>>> N.inject_variables()
Defining n0, n1, n2

>>> f = N.hom({n0: P*m0})
>>> f.is_injective()
True

>>> g = M.hom({m0: n0})
>>> g.is_injective()
False

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
P = X^3 + z*X^2 + z^2
M = S.quotient_module(P^2, names='m')
M.inject_variables()
N = S.quotient_module(P, names='n')
N.inject_variables()
f = N.hom({n0: P*m0})
f.is_injective()
g = M.hom({m0: n0})
g.is_injective()

is_isomorphism()

Return True if this morphism is an isomorphism.

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: M = S.quotient_module(X^2 + z)
sage: f = M.multiplication_map(X^3)
sage: f.is_isomorphism()
True

sage: N = S.quotient_module(X^2)
sage: g = N.multiplication_map(X^3)
sage: g.is_isomorphism()
False

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module(X**Integer(2) + z)
>>> f = M.multiplication_map(X**Integer(3))
>>> f.is_isomorphism()
True

>>> N = S.quotient_module(X**Integer(2))
>>> g = N.multiplication_map(X**Integer(3))
>>> g.is_isomorphism()
False

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
M = S.quotient_module(X^2 + z)
f = M.multiplication_map(X^3)
f.is_isomorphism()
N = S.quotient_module(X^2)
g = N.multiplication_map(X^3)
g.is_isomorphism()

is_surjective()

Return True if this morphism is surjective.

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: P = X^3 + z*X^2 + z^2
sage: M = S.quotient_module(P^2, names='m')
sage: M.inject_variables()
Defining m0, m1, m2, m3, m4, m5
sage: N = S.quotient_module(P, names='n')
sage: N.inject_variables()
Defining n0, n1, n2

sage: f = N.hom({n0: P*m0})
sage: f.is_surjective()
False

sage: g = M.hom({m0: n0})
sage: g.is_surjective()
True

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> P = X**Integer(3) + z*X**Integer(2) + z**Integer(2)
>>> M = S.quotient_module(P**Integer(2), names='m')
>>> M.inject_variables()
Defining m0, m1, m2, m3, m4, m5
>>> N = S.quotient_module(P, names='n')
>>> N.inject_variables()
Defining n0, n1, n2

>>> f = N.hom({n0: P*m0})
>>> f.is_surjective()
False

>>> g = M.hom({m0: n0})
>>> g.is_surjective()
True

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
P = X^3 + z*X^2 + z^2
M = S.quotient_module(P^2, names='m')
M.inject_variables()
N = S.quotient_module(P, names='n')
N.inject_variables()
f = N.hom({n0: P*m0})
f.is_surjective()
g = M.hom({m0: n0})
g.is_surjective()

is_zero()

Return True if this morphism is zero.

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: P = X^3 + z*X^2 + z^2 + 1
sage: M = S.quotient_module(P^2, names='e')
sage: M.inject_variables()
Defining e0, e1, e2, e3, e4, e5

sage: f = M.hom({e0: P*e0})
sage: f.is_zero()
False
sage: (f*f).is_zero()
True

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> P = X**Integer(3) + z*X**Integer(2) + z**Integer(2) + Integer(1)
>>> M = S.quotient_module(P**Integer(2), names='e')
>>> M.inject_variables()
Defining e0, e1, e2, e3, e4, e5

>>> f = M.hom({e0: P*e0})
>>> f.is_zero()
False
>>> (f*f).is_zero()
True

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
P = X^3 + z*X^2 + z^2 + 1
M = S.quotient_module(P^2, names='e')
M.inject_variables()
f = M.hom({e0: P*e0})
f.is_zero()
(f*f).is_zero()

kernel(names=None)

Return True if this morphism is injective.

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: P = X^3 + z*X^2 + z^2
sage: M = S.quotient_module(P^2, names='m')
sage: M.inject_variables()
Defining m0, m1, m2, m3, m4, m5
sage: N = S.quotient_module(P, names='n')
sage: N.inject_variables()
Defining n0, n1, n2

sage: f = M.hom({m0: n0})
sage: ker = f.kernel()
sage: ker
Ore module of rank 3 over Finite Field in z of size 5^3 twisted by z |--> z^5
sage: ker.basis()
[m0 + (2*z^2+3*z+1)*m3 + (4*z^2+3*z+3)*m4 + (2*z^2+3*z)*m5,
 m1 + (z+3)*m3 + (z^2+z+4)*m4,
 m2 + (2*z^2+4*z+2)*m4 + (2*z^2+z+1)*m5]

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> P = X**Integer(3) + z*X**Integer(2) + z**Integer(2)
>>> M = S.quotient_module(P**Integer(2), names='m')
>>> M.inject_variables()
Defining m0, m1, m2, m3, m4, m5
>>> N = S.quotient_module(P, names='n')
>>> N.inject_variables()
Defining n0, n1, n2

>>> f = M.hom({m0: n0})
>>> ker = f.kernel()
>>> ker
Ore module of rank 3 over Finite Field in z of size 5^3 twisted by z |--> z^5
>>> ker.basis()
[m0 + (2*z^2+3*z+1)*m3 + (4*z^2+3*z+3)*m4 + (2*z^2+3*z)*m5,
 m1 + (z+3)*m3 + (z^2+z+4)*m4,
 m2 + (2*z^2+4*z+2)*m4 + (2*z^2+z+1)*m5]

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
P = X^3 + z*X^2 + z^2
M = S.quotient_module(P^2, names='m')
M.inject_variables()
N = S.quotient_module(P, names='n')
N.inject_variables()
f = M.hom({m0: n0})
ker = f.kernel()
ker
ker.basis()

matrix()

Return the matrix defining this morphism.

EXAMPLES:

Sage

sage: K.<z> = GF(5^3)
sage: S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: M = S.quotient_module(X^2 + z)
sage: f = M.multiplication_map(2)
sage: f.matrix()
[2 0]
[0 2]

sage: g = M.multiplication_map(X^3)
sage: g.matrix()
[            0 3*z^2 + z + 1]
[      2*z + 1             0]

Python

>>> from sage.all import *
>>> K = GF(Integer(5)**Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> S = OrePolynomialRing(K, K.frobenius_endomorphism(), names=('X',)); (X,) = S._first_ngens(1)
>>> M = S.quotient_module(X**Integer(2) + z)
>>> f = M.multiplication_map(Integer(2))
>>> f.matrix()
[2 0]
[0 2]

>>> g = M.multiplication_map(X**Integer(3))
>>> g.matrix()
[            0 3*z^2 + z + 1]
[      2*z + 1             0]

Sage Live

K.<z> = GF(5^3)
S.<X> = OrePolynomialRing(K, K.frobenius_endomorphism())
M = S.quotient_module(X^2 + z)
f = M.multiplication_map(2)
f.matrix()
g = M.multiplication_map(X^3)
g.matrix()

class sage.modules.ore_module_morphism.OreModuleRetraction

Bases: Map

Conversion (partially defined) map from an ambient module to one of its submodule.

class sage.modules.ore_module_morphism.OreModuleSection

Bases: Map

Section map of the projection onto a quotient. It is not necessarily compatible with the Ore action.

Make live