Skip to content

topology · EilenbergMacLaneIntegralCohomologyPresentation

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

Worked example

The complete checked program Decompose a polynomial by a nonstandard positive grading uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Decompose a polynomial by a nonstandard positive grading."""

from noether import Z
from noether.commalg import PolynomialRing, WeightedGrading


def main() -> None:
    ring = PolynomialRing(Z, ["x", "y", "z"], grading=WeightedGrading([1, 2, 3]))
    x, y, z = ring.generators()
    sparse = ring.from_terms([([0, 0, 0], 1), ([2, 0, 0], 1), ([0, 1, 0], 1)])
    polynomial = sparse + z
    assert sparse * (x + y) == (1 + x**2 + y) * (x + y)
    components = polynomial.homogeneous_components()
    assert [degree for degree, _ in components] == [0, 2, 3]
    assert x**2 + y == components[1][1]
    print(f"weighted homogeneous degrees: {[degree for degree, _ in components]}")


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/14_weighted_polynomials.py.

topology.EilenbergMacLaneIntegralCohomologyPresentation

Complete integral cohomology-ring presentation of a symbolic K(A,n).

class EilenbergMacLaneIntegralCohomologyPresentation

noether::topology::EilenbergMacLaneIntegralCohomologyPresentation · Direct Rust declaration

pub struct EilenbergMacLaneIntegralCohomologyPresentation { /* private fields */ }
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, geom-core, m4rst, m4rste
GIL attached / bounded
Semantic ID noether/topology/EilenbergMacLaneIntegralCohomologyPresentation@1

Executable usage: validation/unit/topology/test_eilenberg_maclane.py:104 · validation/unit/topology/test_eilenberg_maclane.py:14 · validation/unit/topology/test_eilenberg_maclane.py:86

topology.EilenbergMacLaneIntegralCohomologyPresentation.additive_group

Compute additive group for this EilenbergMacLaneIntegralCohomologyPresentation.

Detailed contract

Additive integral cohomology in one degree, derived from the same closed-form family and checked by the bounded homology/UCT lane.

(self, degree: int) -> FinitelyGeneratedAbelianGroup

noether::topology::EilenbergMacLaneIntegralCohomologyPresentation::additive_group · Direct Rust declaration

pub fn additive_group( &self, degree: usize, ) -> Result<FinitelyGeneratedAbelianGroup, EilenbergMacLaneError>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, geom-core, m4rst, m4rste
GIL attached / bounded
Semantic ID noether/topology/EilenbergMacLaneIntegralCohomologyPresentation@1

Executable usage: validation/unit/homalg/test_finite_group_cohomology.py:119 · validation/unit/homalg/test_finite_group_cohomology.py:61 · validation/unit/homalg/test_finite_group_cohomology.py:82 · validation/unit/homalg/test_finite_group_cohomology.py:83 · validation/unit/topology/test_eilenberg_maclane.py:101 · validation/unit/topology/test_eilenberg_maclane.py:114

topology.EilenbergMacLaneIntegralCohomologyPresentation.coefficient_ring

Return the coefficient ring recorded by this EilenbergMacLaneIntegralCohomologyPresentation.

Detailed contract

Coefficient-ring display name.

(self) -> Literal['Z']

noether::topology::EilenbergMacLaneIntegralCohomologyPresentation::coefficient_ring · Direct Rust declaration

pub fn coefficient_ring(&self) -> &'static str
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, geom-core, m4rst, m4rste
GIL attached / bounded
Semantic ID noether/topology/EilenbergMacLaneIntegralCohomologyPresentation@1

Executable usage: validation/unit/commalg/test_affine_normalization_facade.py:115 · validation/unit/commalg/test_affine_normalization_facade.py:117 · validation/unit/commalg/test_affine_normalization_facade.py:119 · validation/unit/commalg/test_base_change.py:31 · validation/unit/commalg/test_base_change.py:53 · validation/unit/commalg/test_construction_algebras.py:19

topology.EilenbergMacLaneIntegralCohomologyPresentation.from_json

Deserialize and revalidate a EilenbergMacLaneIntegralCohomologyPresentation from typed JSON.

Detailed contract

Deserialize and exactly replay a complete integral presentation.

(cls, value: str) -> EilenbergMacLaneIntegralCohomologyPresentation

noether::topology::EilenbergMacLaneIntegralCohomologyPresentation::from_json · Direct Rust declaration

pub fn from_json(value: &str) -> Result<Self, NoetherError>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, geom-core, m4rst, m4rste
GIL attached / bounded
Semantic ID noether/topology/EilenbergMacLaneIntegralCohomologyPresentation@1

Executable usage: validation/unit/category/test_category.py:14 · validation/unit/combin/test_acceptance_workload.py:65 · validation/unit/combin/test_acceptance_workload.py:86 · validation/unit/combin/test_graph_facade.py:106 · validation/unit/combin/test_graph_facade.py:25 · validation/unit/combinatorics/test_kashiwara_crystal.py:32

topology.EilenbergMacLaneIntegralCohomologyPresentation.generators

Return the generators recorded by this EilenbergMacLaneIntegralCohomologyPresentation.

Detailed contract

Ordered homogeneous algebra generators.

(self) -> list[EilenbergMacLaneIntegralCohomologyGenerator]

noether::topology::EilenbergMacLaneIntegralCohomologyPresentation::generators · Direct Rust declaration

pub fn generators(&self) -> &[EilenbergMacLaneIntegralCohomologyGenerator]
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, geom-core, m4rst, m4rste
GIL attached / bounded
Semantic ID noether/topology/EilenbergMacLaneIntegralCohomologyPresentation@1

Executable usage: examples/14_weighted_polynomials.py:11 · examples/15_groebner_twisted_cubic.py:11 · examples/15_groebner_twisted_cubic.py:19 · examples/17_weighted_power_series.py:15 · examples/18_finite_field_schemes.py:12 · examples/30_modules_and_resolutions.py:20

topology.EilenbergMacLaneIntegralCohomologyPresentation.graded_commutative

Return the graded commutative recorded by this EilenbergMacLaneIntegralCohomologyPresentation.

(self) -> Literal[True]

noether::topology::EilenbergMacLaneIntegralCohomologyPresentation · Python projection

Python projection convenience over this Rust mathematical type
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, geom-core, m4rst, m4rste
GIL attached / bounded
Semantic ID noether/topology/EilenbergMacLaneIntegralCohomologyPresentation@1

Executable usage: validation/unit/topology/test_eilenberg_maclane.py:89

topology.EilenbergMacLaneIntegralCohomologyPresentation.relations

Return the relations recorded by this EilenbergMacLaneIntegralCohomologyPresentation.

Detailed contract

Homogeneous defining relations.

(self) -> list[EilenbergMacLaneIntegralCohomologyRelation]

noether::topology::EilenbergMacLaneIntegralCohomologyPresentation::relations · Direct Rust declaration

pub fn relations(&self) -> &[EilenbergMacLaneIntegralCohomologyRelation]
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, geom-core, m4rst, m4rste
GIL attached / bounded
Semantic ID noether/topology/EilenbergMacLaneIntegralCohomologyPresentation@1

Executable usage: validation/unit/combinatorics/test_partition_types.py:63 · validation/unit/combinatorics/test_partition_types.py:68 · validation/unit/combinatorics/test_partition_types.py:72 · validation/unit/homalg/test_exterior_algebra.py:263 · validation/unit/homalg/test_exterior_algebra.py:267 · validation/unit/homalg/test_exterior_algebra.py:456

topology.EilenbergMacLaneIntegralCohomologyPresentation.space

Return the space recorded by this EilenbergMacLaneIntegralCohomologyPresentation.

Detailed contract

Symbolic source space.

(self) -> EilenbergMacLaneSpace

noether::topology::EilenbergMacLaneIntegralCohomologyPresentation::space · Direct Rust declaration

pub fn space(&self) -> &EilenbergMacLaneSpace
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, geom-core, m4rst, m4rste
GIL attached / bounded
Semantic ID noether/topology/EilenbergMacLaneIntegralCohomologyPresentation@1

Executable usage: examples/03_exact_linear_systems.py:3 · validation/unit/commalg/test_curve_facade.py:116 · validation/unit/commalg/test_curve_facade.py:123 · validation/unit/commalg/test_curve_facade.py:124 · validation/unit/commalg/test_curve_facade.py:126 · validation/unit/commalg/test_curve_facade.py:73

topology.EilenbergMacLaneIntegralCohomologyPresentation.strategy

Return the strategy recorded by this EilenbergMacLaneIntegralCohomologyPresentation.

Detailed contract

Closed-form completeness strategy.

(self) -> Literal['trivial_space', 'free_and_cyclic_classifying_space', 'free_and_rank_two_abelian_classifying_space', 'free_abelian_second_delooping']

noether::topology::EilenbergMacLaneIntegralCohomologyPresentation::strategy · Direct Rust declaration

pub fn strategy(&self) -> EilenbergMacLaneIntegralCohomologyPresentationStrategy
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, geom-core, m4rst, m4rste
GIL attached / bounded
Semantic ID noether/topology/EilenbergMacLaneIntegralCohomologyPresentation@1

Executable usage: validation/unit/linalg/test_exact_linear_algebra_facade.py:319 · validation/unit/linalg/test_exact_linear_algebra_facade.py:330 · validation/unit/linalg/test_exact_linear_algebra_facade.py:428 · validation/unit/linalg/test_exact_linear_algebra_facade.py:439 · validation/unit/topology/test_eilenberg_maclane.py:112 · validation/unit/topology/test_eilenberg_maclane.py:119

topology.EilenbergMacLaneIntegralCohomologyPresentation.to_json

Serialize this EilenbergMacLaneIntegralCohomologyPresentation to its canonical typed JSON representation.

Detailed contract

Serialize the complete integral presentation as typed JSON.

(self) -> str

noether::topology::EilenbergMacLaneIntegralCohomologyPresentation::to_json · Direct Rust declaration

pub fn to_json(&self) -> Result<String, NoetherError>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, geom-core, m4rst, m4rste
GIL attached / bounded
Semantic ID noether/topology/EilenbergMacLaneIntegralCohomologyPresentation@1

Executable usage: validation/unit/category/test_category.py:14 · validation/unit/category/test_category.py:15 · validation/unit/combin/test_acceptance_workload.py:30 · validation/unit/combin/test_acceptance_workload.py:65 · validation/unit/combin/test_acceptance_workload.py:86 · validation/unit/combin/test_graph_facade.py:106

topology.EilenbergMacLaneIntegralCohomologyPresentation.verify

Replay and verify the mathematical evidence carried by this EilenbergMacLaneIntegralCohomologyPresentation.

Detailed contract

Reconstruct the entire presentation from the retained symbolic space.

(self) -> None

noether::topology::EilenbergMacLaneIntegralCohomologyPresentation::verify · Direct Rust declaration

pub fn verify(&self) -> Result<(), VerificationError>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, geom-core, m4rst, m4rste
GIL attached / bounded
Semantic ID noether/topology/EilenbergMacLaneIntegralCohomologyPresentation@1

Executable usage: examples/01_bezout_certificates.py:10 · examples/01_bezout_certificates.py:3 · examples/03_exact_linear_systems.py:13 · examples/03_exact_linear_systems.py:3 · examples/04_smith_normal_form.py:12 · examples/06_shortest_lattice_vectors.py:12