LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-09-24 (gpt-6-sol)

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.

regular local regular quotient ideal is parameter generated

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $(R, m, k)$ be regular local of dimension $d$ and $I \subseteq m$ . The following are equivalent: $R / I$ is regular; $I$ is generated by an initial part of a regular system of parameters; and $\dim_{k} ((I + m^{2}) / m^{2}) = d - \dim (R / I)$ .

Facts & Assumptions

Given: The objects and hypotheses in the statement. We work with the Axiom of Choice; cited dependent-choice and resolution-existence hypotheses are retained.

[F1]

regular local quotient by parameter is regular: Let $(R, m, k)$ be regular local of dimension $d$ , and let $x \in m ∖ m^{2}$ . Then $R / (x)$ is regular local, of dimension and embedding dimension $d - 1$ .

[F2]

regular local domain induction: Every regular local ring is an integral domain.

[F3]

regular system of parameters equivalent basis: Let $(R, m, k)$ be a nonzero Noetherian local ring of dimension $d$ , and let $x = (x_{1}, \dots, x_{d}) \in m^{d}$ . Then $x$ is a regular system of parameters if and only if its classes form a $k$ -basis of $m / m^{2}$ . In particular every lift of a cotangent basis in a regular local ring generates $m$ and is a system of parameters.

Proof

1.1givenalgebra

Put $c = \dim_{k} ((I + m^{2}) / m^{2})$ . The cotangent space of $R / I$ is $m / (I + m^{2})$ and has dimension $d - c$ . Consequently the numerical equality is precisely the definition of regularity of $R / I$ .

2.1F3F1step 1.1

If $R / I$ is regular, choose $x_{1}, \dots, x_{c} \in I$ lifting a basis of that subspace and extend their classes to a cotangent basis of $R$ . Put $S = R / (x_{1}, \dots, x_{c})$ . Repeated parameter reduction makes $S$ regular of dimension $d - c$ , and the extended tuple is a regular system.

3.1F2F1step 2.1algebra∎

The ring $S$ is a domain. If the kernel $J$ of $S \to R / I$ were nonzero, any prime chain in $S / J$ would lift to a chain of nonzero primes of $S$ , to which $(0)$ can be prepended. Hence $\dim (S / J) \leq \dim S - 1$ , contradicting equality of dimensions. Thus $I = (x_{1}, \dots, x_{c})$ . Conversely, repeated parameter reduction makes every such quotient regular. This includes $c = 0$ , when $I = 0$ , and $c = d$ , when $I = m$ and the quotient is $k$ .

Depends on

Dependency tree · 2 levels

Sources

Proposition 12.13, pp.116–117