homalg · FiniteGroupCohomologyElementaryAbelianRestriction

5 accepted declarations. Python and Rust forms, contracts, evidence, failure boundaries, and checked usages are kept together below.

Worked example

The complete checked program Exact affine systems and canonical submodules over Z and Z/nZ uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Exact affine systems and canonical submodules over Z and Z/nZ."""

from __future__ import annotations

from noether import IntegerRing, Z
from noether.commalg import ResidueRing
from noether.homalg import FreeModule, ModuleElement, ModuleHomomorphism, Submodule
from noether.linalg import Matrix, Vector

operator = Matrix.from_rows(Z, [[2, 4, 0], [0, 0, 3]])
right_hand_side = Vector.from_values(Z, [6, 9])
integer_result = operator.solve_exact(right_hand_side)
assert integer_result.is_consistent
integer_solutions = integer_result.solutions
assert integer_solutions is not None
assert operator @ integer_solutions.particular_solution == right_hand_side
assert operator @ integer_solutions.homogeneous_generators == Matrix.zeros(Z, (2, 1))
integer_result.verify()

# Empty systems retain a replayable divisibility obstruction.
impossible = Matrix.from_rows(Z, [[6]]).solve_exact(Vector.from_values(Z, [9]))
assert not impossible.is_consistent
assert impossible.obstruction is not None
assert impossible.obstruction.left_gcd == 6
assert abs(impossible.obstruction.combined_rhs) == 9
impossible.obstruction.verify()

# Composite moduli are not treated as fields.  The result is an affine module,
# including all homogeneous torsion directions.
z12 = ResidueRing(12)
congruence = Matrix.from_rows(z12, [[8]]).solve_exact(Vector.from_values(z12, [4]))
assert congruence.is_consistent
congruence_solutions = congruence.solutions
assert congruence_solutions is not None
assert {
    (int(congruence_solutions.particular_solution[0]) + 3 * offset) % 12 for offset in range(4)
} == {
    2,
    5,
    8,
    11,
}
congruence.verify()

# Submodules retain a canonical normal form, so equality and lattice operations
# compare mathematics rather than the supplied generating lists.
ambient = FreeModule(Z, 2)


def element(x: int, y: int) -> ModuleElement[IntegerRing]:
    return ModuleElement(ambient, Vector.from_values(Z, [x, y]))


lattice = Submodule.from_generators(ambient, [element(2, 0), element(0, 3)])
assert lattice.finite_index() == 6
witness = lattice.membership_witness(element(6, 9))
assert witness is not None
witness.verify()
assert lattice.saturation().finite_index() == 1

homomorphism = ModuleHomomorphism(
    FreeModule(Z, 3),
    ambient,
    operator,
)
kernel_result = homomorphism.kernel()
image_result = homomorphism.image()
kernel = kernel_result.submodule
image = image_result.submodule
assert kernel.generator_count == 1
assert image.contains(element(2, 0))
kernel_result.verify()
image_result.verify()

print(
    "exact systems: integer affine rank 1; four Z/12Z solutions; "
    "index-6 lattice saturated and verified"
)

Run it with .venv/bin/python examples/38_exact_linear_systems.py.

homalg.FiniteGroupCohomologyElementaryAbelianRestriction

One maximal elementary abelian restriction in a Symonds certificate.

PythonRust

class FiniteGroupCohomologyElementaryAbelianRestriction

noether::homalg::FiniteGroupCohomologyElementaryAbelianRestriction · Direct Rust declaration

pub struct FiniteGroupCohomologyElementaryAbelianRestriction { /* private fields */ }

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core, groebner-kernel, m4rst, m4rste, module-core, resolution-kernel, sparla
GIL	attached / bounded

Executable usage: validation/unit/homalg/test_finite_group_cohomology.py:15 · validation/unit/homalg/test_finite_group_cohomology.py:219

homalg.FiniteGroupCohomologyElementaryAbelianRestriction.quotient_generator_power_bounds

Return the quotient generator power bounds recorded by this FiniteGroupCohomologyElementaryAbelianRestriction.

Detailed contract

A pure power of each non-exterior presentation generator killed by the restricted parameter ideal. Exterior generators record the intrinsic square-zero bound 2.

PythonRust

(self) -> list[int]

noether::homalg::FiniteGroupCohomologyElementaryAbelianRestriction::quotient_generator_power_bounds · Direct Rust declaration

pub fn quotient_generator_power_bounds(&self) -> &[u32]

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core, groebner-kernel, m4rst, m4rste, module-core, resolution-kernel, sparla
GIL	attached / bounded

Executable usage: validation/unit/homalg/test_finite_group_cohomology.py:223

homalg.FiniteGroupCohomologyElementaryAbelianRestriction.rank

Return the rank recorded by this FiniteGroupCohomologyElementaryAbelianRestriction.

Detailed contract

Elementary abelian rank.

PythonRust

(self) -> int

noether::homalg::FiniteGroupCohomologyElementaryAbelianRestriction::rank · Direct Rust declaration

pub const fn rank(&self) -> usize

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core, groebner-kernel, m4rst, m4rste, module-core, resolution-kernel, sparla
GIL	attached / bounded

Executable usage: examples/38_exact_linear_systems.py:78 · examples/distributed/f2_linear_algebra.py:34 · examples/distributed/f2_linear_algebra.py:54 · examples/distributed/f2_linear_algebra.py:55 · examples/distributed/f2_linear_algebra.py:60 · validation/unit/combin/test_acceptance_workload.py:74

homalg.FiniteGroupCohomologyElementaryAbelianRestriction.restricted_parameters

Return the restricted parameters recorded by this FiniteGroupCohomologyElementaryAbelianRestriction.

Detailed contract

Restrictions of the selected parameters in a deterministic complete presentation of this subgroup’s cohomology.

PythonRust

(self) -> list[GradedCommutativePolynomial]

noether::homalg::FiniteGroupCohomologyElementaryAbelianRestriction::restricted_parameters · Direct Rust declaration

pub fn restricted_parameters(&self) -> &[GradedCommutativePolynomial]

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core, groebner-kernel, m4rst, m4rste, module-core, resolution-kernel, sparla
GIL	attached / bounded

Executable usage: validation/unit/homalg/test_finite_group_cohomology.py:222

homalg.FiniteGroupCohomologyElementaryAbelianRestriction.subgroup

Return the subgroup recorded by this FiniteGroupCohomologyElementaryAbelianRestriction.

Detailed contract

Literal maximal elementary abelian subgroup of the retained Sylow subgroup.

PythonRust

(self) -> PermutationGroup

noether::homalg::FiniteGroupCohomologyElementaryAbelianRestriction::subgroup · Direct Rust declaration

pub const fn subgroup(&self) -> &PermutationGroup

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core, groebner-kernel, m4rst, m4rste, module-core, resolution-kernel, sparla
GIL	attached / bounded

Executable usage: validation/unit/homalg/test_finite_group_cohomology.py:221 · validation/unit/infra/test_magma_release_protocol.py:192 · validation/unit/infra/test_magma_release_protocol.py:193 · validation/parity/m1_bsgs.py:50 · validation/bench/magma_release.py:1580 · validation/bench/magma_release.py:1581