ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The associated graded ring of a regular local ring and of a cusp local ring can be computed explicitly

Example

Let $R_{1} : = k [x, y]_{(x, y)}$ with maximal ideal $m = (x, y)$ . Then ${gr}_{m} (R_{1}) ≅ k [X, Y],$ because $m^{n} / m^{n + 1}$ has basis given by degree- $n$ monomials in the initial classes of $x$ and $y$ .

For the cusp local ring $R_{2} : = k [x, y]_{(x, y)} / (y^{2} - x^{3}),$ the initial form of the relation has degree $2$ , so ${gr}_{m} (R_{2}) ≅ k [X, Y] / (Y^{2}) .$

Facts & Assumptions

Given: A field $k$ , the local rings $R_{1}$ and $R_{2}$ above, and the maximal-ideal filtrations.

[L1]

The associated graded ring is ${gr}_{m} (R) = ⨁_{n \geq 0} m^{n} / m^{n + 1}$ (The associated graded ring and associated graded module of an ideal-adic filtration).

Verification

technique · direct

1.1L1givenalgebra

In $R_{1}$ , the classes of $x$ and $y$ in $m / m^{2}$ generate every graded piece: the images of the degree- $n$ monomials $x^{n - i} y^{i}$ form a basis of $m^{n} / m^{n + 1}$ . Therefore the map $k [X, Y] \to {gr}_{m} (R_{1})$ sending $X, Y$ to the initial classes of $x, y$ is a graded isomorphism.

1.2L1algebra

In $R_{2}$ , the relation $y^{2} - x^{3}$ lies in $m^{2}$ and its lowest-degree term is $y^{2}$ . Hence the only initial relation in degree $2$ is $Y^{2} = 0$ . As in the remaining monomials $X^{n}$ and $X^{n - 1} Y$ survive and span the graded pieces, so ${gr}_{m} (R_{2}) ≅ k [X, Y] / (Y^{2}) .$

2.1algebra∎

These explicit computations exhibit the regular local and cusp cases.

Depends on

The associated graded ring and associated graded module of an ideal-adic filtration

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Stacks Project, Section 10.57: Graded modules (standard reference, not scraped)
Craig Huneke and Irena Swanson, Integral Closure, Chapter 1 (standard reference, not scraped)