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homalg · ResolutionMinimality

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

Worked example

The complete checked program Reduce a module element and certify a minimal free resolution uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Reduce a module element and certify a minimal free resolution."""

from noether import Q
from noether.commalg import Degree, MonomialOrder, PolynomialRing
from noether.homalg import (
    FreeModule,
    ModuleElement,
    ModuleHomomorphism,
    ModuleMonomialOrder,
    PresentedModule,
    Submodule,
)
from noether.linalg import Matrix, Vector


def main() -> None:
    ring = PolynomialRing(Q, ["x", "y", "z", "w"])
    x, y, z, w = ring.generators()
    order = ModuleMonomialOrder.term_over_position(MonomialOrder.GREVLEX)

    ambient = FreeModule(ring, 2)
    first = ModuleElement(ambient, Vector.from_values(ring, [x, y]))
    second = ModuleElement(ambient, Vector.from_values(ring, [y, 1]))
    submodule = Submodule.from_generators(ambient, [first, second])
    basis = submodule.groebner_basis(order)
    reduction = basis.reduce(first + second * x)
    basis.verify()
    reduction.verify()
    assert reduction.is_member()

    target = FreeModule(ring, 1, grading_rank=1, degree_shifts=[Degree([0])])
    source = FreeModule(ring, 3, grading_rank=1, degree_shifts=[Degree([-2])] * 3)
    twisted_cubic = PresentedModule(
        ModuleHomomorphism(
            source,
            target,
            Matrix.from_rows(ring, [[x * z - y**2, x * w - y * z, y * w - z**2]]),
        )
    )
    resolution = twisted_cubic.minimal_free_resolution(order)
    resolution.verify()
    betti = resolution.graded_betti_table()
    betti.verify()
    assert betti.entries == [
        (0, Degree([0]), 1),
        (1, Degree([2]), 3),
        (2, Degree([3]), 2),
    ]
    print(f"module member certified; minimal Betti entries={betti.entries}")


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/30_modules_and_resolutions.py.

homalg.ResolutionMinimality

Exact minimality claim of a graded free resolution.

class ResolutionMinimality

noether::homalg::ResolutionMinimality · Direct Rust declaration

pub enum ResolutionMinimality { NotClaimed, Minimal, }
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
Semantic ID noether/homalg/ResolutionMinimality@1

Executable usage: validation/unit/homalg/test_module_facade.py:41 · validation/unit/homalg/test_module_facade.py:833 · validation/unit/homalg/test_module_facade.py:835 · validation/unit/homalg/test_module_facade.py:836 · validation/unit/homalg/test_module_facade.py:847 · validation/bench/macaulay2_core.py:1493

homalg.ResolutionMinimality.from_json

Deserialize and revalidate a ResolutionMinimality from typed JSON.

Detailed contract

Deserialize an exact minimality claim.

(cls, value: str) -> ResolutionMinimality

noether::homalg::ResolutionMinimality::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, flint-core, groebner-kernel, m4rst, m4rste, module-core, resolution-kernel, sparla
GIL attached / bounded
Semantic ID noether/homalg/ResolutionMinimality@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

homalg.ResolutionMinimality.is_minimal

Test whether this ResolutionMinimality is minimal.

Detailed contract

Whether this value carries a verified minimality claim.

(self) -> bool

noether::homalg::ResolutionMinimality::is_minimal · Direct Rust declaration

pub const fn is_minimal(&self) -> bool
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
Semantic ID noether/homalg/ResolutionMinimality@1

Executable usage: validation/unit/homalg/test_module_facade.py:500 · validation/unit/homalg/test_module_facade.py:815 · validation/unit/homalg/test_module_facade.py:830 · validation/unit/homalg/test_module_facade.py:832 · validation/unit/homalg/test_module_facade.py:836 · validation/unit/homalg/test_module_facade.py:873

homalg.ResolutionMinimality.kind

Return the kind recorded by this ResolutionMinimality.

Detailed contract

Stable minimality kind.

(self) -> Literal['not_claimed', 'minimal']

noether::homalg::ResolutionMinimality::kind · Direct Rust declaration

pub const fn kind(&self) -> &'static str
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
Semantic ID noether/homalg/ResolutionMinimality@1

Executable usage: validation/unit/combinatorics/test_poset.py:95 · validation/unit/commalg/test_curve_facade.py:137 · validation/unit/commalg/test_curve_facade.py:43 · validation/unit/commalg/test_curve_facade.py:44 · validation/unit/commalg/test_curve_facade.py:45 · validation/unit/commalg/test_curve_facade.py:80

homalg.ResolutionMinimality.minimal

Construct a ResolutionMinimality using minimal.

(cls) -> ResolutionMinimality

noether::homalg::ResolutionMinimality · 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, flint-core, groebner-kernel, m4rst, m4rste, module-core, resolution-kernel, sparla
GIL attached / bounded
Semantic ID noether/homalg/ResolutionMinimality@1

Executable usage: examples/30_modules_and_resolutions.py:3 · examples/30_modules_and_resolutions.py:51 · validation/unit/commalg/test_polynomial_ideal_facade.py:1150 · validation/unit/commalg/test_polynomial_ideal_facade.py:676 · validation/unit/commalg/test_polynomial_ideal_facade.py:679 · validation/unit/commalg/test_polynomial_ideal_facade.py:684

homalg.ResolutionMinimality.not_claimed

Construct a ResolutionMinimality using not claimed.

(cls) -> ResolutionMinimality

noether::homalg::ResolutionMinimality · 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, flint-core, groebner-kernel, m4rst, m4rste, module-core, resolution-kernel, sparla
GIL attached / bounded
Semantic ID noether/homalg/ResolutionMinimality@1

Executable usage: validation/unit/homalg/test_module_facade.py:814 · validation/unit/homalg/test_module_facade.py:836

homalg.ResolutionMinimality.to_json

Serialize this ResolutionMinimality to its canonical typed JSON representation.

Detailed contract

Serialize the exact minimality claim as typed JSON.

(self) -> str

noether::homalg::ResolutionMinimality::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, flint-core, groebner-kernel, m4rst, m4rste, module-core, resolution-kernel, sparla
GIL attached / bounded
Semantic ID noether/homalg/ResolutionMinimality@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