Skip to content

commalg · CotangentComplexHypothesis

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

Worked example

The complete checked program Differentiate exactly and distinguish a cusp's smooth and singular points uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Differentiate exactly and distinguish a cusp's smooth and singular points."""

from noether import Q
from noether.commalg import KahlerDifferentials, PolynomialRing


def main() -> None:
    ring = PolynomialRing(Q, ["x", "y"])
    x, y = ring.generators()
    relation = y**2 - x**3
    cusp = ring.quotient(ring.ideal([relation]))
    omega = cusp.kahler_differentials()

    assert isinstance(omega, KahlerDifferentials)
    dx = omega.differentiate(cusp(x))
    dy = omega.differentiate(cusp(y))
    assert omega.differentiate(cusp(x * y)) == dx * cusp(y) + dy * cusp(x)
    assert omega.differentiate(cusp(relation)).is_zero()

    singular = cusp.cotangent_space_at([0, 0])
    smooth = cusp.cotangent_space_at([1, 1])
    assert singular.embedding_dimension == 2
    assert smooth.embedding_dimension == 1
    assert singular.presentation_matrix == singular.evaluated_jacobian.transpose()

    complex = cusp.cotangent_complex()
    assert complex.codimension == 1
    assert (complex.amplitude.lowest_degree, complex.amplitude.highest_degree) == (-1, 0)
    assert complex.hypothesis.equation_count == 1
    complex.verify()

    print(
        "cusp cotangent dimensions: "
        f"singular={singular.embedding_dimension}, smooth={smooth.embedding_dimension}; "
        "complete-intersection amplitude=[-1,0]"
    )


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/39_kahler_differentials.py.

commalg.CotangentComplexHypothesis

Sealed exact hypothesis issued only after replaying a CI proof.

class CotangentComplexHypothesis

noether::commalg::CotangentComplexHypothesis · Direct Rust declaration

pub enum CotangentComplexHypothesis { #[non_exhaustive] ProvedAffineCompleteIntersection { ambient_dimension: usize, quotient_dimension: usize, codimension: usize, equation_count: usize, }, }
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

Executable usage: validation/unit/commalg/test_kahler_differentials.py:159 · validation/unit/commalg/test_kahler_differentials.py:16 · validation/unit/commalg/test_kahler_differentials.py:95

commalg.CotangentComplexHypothesis.ambient_dimension

Krull dimension of the ambient polynomial algebra.

Detailed contract

Dimension of the ambient polynomial algebra used in the proof.

(self) -> int

noether::commalg::CotangentComplexHypothesis::ambient_dimension · Direct Rust declaration

pub const fn ambient_dimension(self) -> usize
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

Executable usage: validation/unit/commalg/test_kahler_differentials.py:98 · validation/unit/geometry/test_geometry_facade.py:23 · validation/unit/geometry/test_geometry_facade.py:46 · validation/unit/geometry/test_polytope_facade.py:30 · validation/unit/linalg/test_exact_linear_algebra_facade.py:152 · validation/unit/linalg/test_exact_linear_algebra_facade.py:153

commalg.CotangentComplexHypothesis.codimension

Exact difference between ambient and quotient dimensions.

Detailed contract

Codimension computed as ambient dimension minus quotient dimension.

(self) -> int

noether::commalg::CotangentComplexHypothesis::codimension · Direct Rust declaration

pub const fn codimension(self) -> usize
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

Executable usage: examples/39_kahler_differentials.py:29 · validation/unit/commalg/test_intersection_theory.py:31 · validation/unit/commalg/test_intersection_theory.py:32 · validation/unit/commalg/test_intersection_theory.py:50 · validation/unit/commalg/test_kahler_differentials.py:100 · validation/unit/commalg/test_kahler_differentials.py:121

commalg.CotangentComplexHypothesis.equation_count

Number of equations in the verified regular-sequence presentation.

Detailed contract

Number of chosen equations compared exactly with the computed codimension.

(self) -> int

noether::commalg::CotangentComplexHypothesis::equation_count · Direct Rust declaration

pub const fn equation_count(self) -> usize
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

Executable usage: examples/39_kahler_differentials.py:31 · validation/unit/commalg/test_kahler_differentials.py:101 · validation/unit/commalg/test_kahler_differentials.py:152

commalg.CotangentComplexHypothesis.quotient_dimension

Proven Krull dimension of the quotient algebra.

Detailed contract

Quotient dimension computed exactly from the defining ideal’s Gröbner leading ideal.

(self) -> int

noether::commalg::CotangentComplexHypothesis::quotient_dimension · Direct Rust declaration

pub const fn quotient_dimension(self) -> usize
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

Executable usage: validation/unit/commalg/test_curve_facade.py:34 · validation/unit/commalg/test_kahler_differentials.py:99 · validation/unit/commalg/test_polynomial_ideal_facade.py:101 · validation/unit/commalg/test_polynomial_ideal_facade.py:113 · validation/unit/commalg/test_polynomial_ideal_facade.py:1178 · validation/unit/commalg/test_polynomial_ideal_facade.py:454