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commalg · WeightedProjectiveScheme

8 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.

commalg.WeightedProjectiveScheme

A weighted-homogeneous closed subscheme.

class WeightedProjectiveScheme

noether::commalg::WeightedProjectiveScheme · Direct Rust declaration

pub struct WeightedProjectiveScheme<R: CommutativeRing> { /* private fields */ }
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, module-core, poly-core, resolution-kernel
GIL attached / bounded
Semantic ID noether/commalg/WeightedProjectiveScheme@1

commalg.WeightedProjectiveScheme.ambient

Return the ambient recorded by this WeightedProjectiveScheme.

Detailed contract

Ambient weighted projective space.

(self) -> WeightedProjectiveSpace[C]

noether::commalg::WeightedProjectiveScheme::ambient · Direct Rust declaration

pub fn ambient(&self) -> &WeightedProjectiveSpace<R>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, module-core, poly-core, resolution-kernel
GIL attached / bounded
Semantic ID noether/commalg/WeightedProjectiveScheme@1

Executable usage: examples/30_modules_and_resolutions.py:23 · examples/30_modules_and_resolutions.py:24 · examples/30_modules_and_resolutions.py:25 · examples/30_modules_and_resolutions.py:26 · examples/38_exact_linear_systems.py:49 · examples/38_exact_linear_systems.py:53

commalg.WeightedProjectiveScheme.chart

Compute chart for this WeightedProjectiveScheme.

Detailed contract

Pull this presentation back to one affine cover of a quotient chart.

(self, index: int) -> WeightedProjectiveSchemeChart[C]

noether::commalg::WeightedProjectiveScheme::chart · Direct Rust declaration

pub fn chart( &self, index: usize, ) -> Result<WeightedProjectiveSchemeChart<R>, WeightedProjectiveError>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, module-core, poly-core, resolution-kernel
GIL attached / bounded
Semantic ID noether/commalg/WeightedProjectiveScheme@1

Executable usage: validation/unit/commalg/test_acceptance_workload.py:105 · validation/unit/commalg/test_acceptance_workload.py:106 · validation/unit/commalg/test_acceptance_workload.py:107 · validation/unit/commalg/test_acceptance_workload.py:110 · validation/unit/commalg/test_acceptance_workload.py:117 · validation/unit/commalg/test_acceptance_workload.py:118

commalg.WeightedProjectiveScheme.coordinate_ring

Return the coordinate ring recorded by this WeightedProjectiveScheme.

Detailed contract

Weighted homogeneous coordinate ring.

(self) -> PolynomialRing[C]

noether::commalg::WeightedProjectiveScheme::coordinate_ring · Direct Rust declaration

pub fn coordinate_ring(&self) -> &PolynomialRing<R>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, module-core, poly-core, resolution-kernel
GIL attached / bounded
Semantic ID noether/commalg/WeightedProjectiveScheme@1

Executable usage: examples/18_finite_field_schemes.py:12 · examples/18_finite_field_schemes.py:13 · validation/unit/commalg/test_acceptance_workload.py:106 · validation/unit/commalg/test_acceptance_workload.py:107 · validation/unit/commalg/test_acceptance_workload.py:113 · validation/unit/commalg/test_acceptance_workload.py:114

commalg.WeightedProjectiveScheme.defining_ideal

Return the defining ideal recorded by this WeightedProjectiveScheme.

Detailed contract

Canonical weighted-homogeneous defining ideal.

(self) -> Ideal[PolynomialRing[C]]

noether::commalg::WeightedProjectiveScheme::defining_ideal · Direct Rust declaration

pub fn defining_ideal(&self) -> &Ideal<PolynomialRing<R>>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, module-core, poly-core, resolution-kernel
GIL attached / bounded
Semantic ID noether/commalg/WeightedProjectiveScheme@1

Executable usage: validation/unit/commalg/test_acceptance_workload.py:107 · validation/unit/commalg/test_acceptance_workload.py:114 · validation/unit/commalg/test_affine_normalization_facade.py:108 · validation/unit/commalg/test_affine_normalization_facade.py:163 · validation/unit/commalg/test_affine_normalization_facade.py:182 · validation/unit/commalg/test_affine_normalization_facade.py:89

commalg.WeightedProjectiveScheme.dimension

Compute dimension for this WeightedProjectiveScheme.

Detailed contract

Dimension of this weighted projective presentation.

(self) -> int

noether::commalg::WeightedProjectiveScheme::dimension · Direct Rust declaration

pub fn dimension(&self) -> Result<i64, WeightedProjectiveError>
Contract Value
Guarantee checked deterministic result
Determinism deterministic
Availability default CPU build
Domain implementation crates noether, flint-core, groebner-kernel, module-core, poly-core, resolution-kernel
GIL attached / bounded
Semantic ID noether/commalg/WeightedProjectiveScheme@1

Executable usage: examples/03_exact_linear_systems.py:15 · examples/03_exact_linear_systems.py:16 · examples/15_groebner_twisted_cubic.py:14 · examples/15_groebner_twisted_cubic.py:28 · examples/27_weight_characters.py:25 · examples/34_finite_group_cohomology.py:13

commalg.WeightedProjectiveScheme.from_json

Deserialize and revalidate a WeightedProjectiveScheme from typed JSON.

Detailed contract

Restore and revalidate this weighted-projective scheme.

(cls, value: str) -> WeightedProjectiveScheme[SchemeBase]

noether::commalg::WeightedProjectiveScheme::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, module-core, poly-core, resolution-kernel
GIL attached / bounded
Semantic ID noether/commalg/WeightedProjectiveScheme@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

commalg.WeightedProjectiveScheme.to_json

Serialize this WeightedProjectiveScheme to its canonical typed JSON representation.

Detailed contract

Serialize this weighted-projective scheme.

(self) -> str

noether::commalg::WeightedProjectiveScheme::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, module-core, poly-core, resolution-kernel
GIL attached / bounded
Semantic ID noether/commalg/WeightedProjectiveScheme@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