LemmaStatement: AI-generatedProof: AI-generated

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.

Dimension and purity of an affine polynomial cylinder

Statement

Assume AC. Let $R$ be a nonzero noetherian commutative ring. Then $\dim Spec R = \dim R$ , where ring dimension is Krull dimension and scheme dimension is the supremum of lengths of strict chains of nonempty irreducible closed subsets. If $\dim R = d < \infty$ , then $\dim Spec R [t] = d + 1$ . If $Spec R$ has pure dimension $d$ , meaning that every irreducible component has dimension $d$ , then $Spec R [t]$ has pure dimension $d + 1$ for every finite $d$ .

Facts & Assumptions

Given: AC and the ring $R$ in the statement. All prime ideals are proper; an irreducible component is a maximal irreducible closed subset. A minimal prime is a prime with no strictly smaller prime. All spectra carry the Zariski topology and all subsets carry the subspace topology.

[F0]

Commutative rings, ideals and prime ideals have their usual unital-ring meanings. Noetherian means that every ideal is finitely generated. Polynomials are finite coefficient sequences with coefficientwise addition and convolution multiplication; $Spec R$ is the set of prime ideals, and $V (I)$ is the set of primes containing $I$ . The sets $V (I)$ are precisely its Zariski-closed sets. Irreducible means nonempty and not the union of two proper closed subsets, using the subspace topology for subsets. Commutative ring, Left, right and two-sided ideals, Prime ideals and maximal ideals in a commutative ring, Noetherian commutative rings and modules, The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution, The prime spectrum and vanishing sets, The vanishing sets define the Zariski topology on the prime spectrum, Irreducible topological spaces and irreducible subsets in the subspace topology

[F1]

Ring dimension counts strict prime chains, whereas dimension of a noetherian space counts strict chains of nonempty irreducible closed subsets. Krull dimension of a nonzero ring, Chain dimension and the empty-space convention

[F2]

Under AC, the nonempty irreducible closed subsets of a spectrum are exactly the sets $V (p)$ for prime ideals $p$ , each with unique radical defining ideal $p$ . Thus inclusion is reversed, and irreducible components correspond to minimal prime ideals. A Zariski-closed subset is irreducible exactly when its radical defining ideal is prime, and then it has a unique generic point Primes of $R / I$ correspond, preserving strict inclusions, to primes of $R$ containing $I$ . Prime ideals of a quotient ring are exactly the prime ideals containing the ideal

[F3]

A finite-type algebra over a noetherian ring is noetherian, quotients of noetherian rings are noetherian, and under AC the spectrum of a noetherian ring is a noetherian space. Every algebra of finite type over a Noetherian ring is a Noetherian ring, Every quotient and every localisation of a Noetherian ring is Noetherian, The spectrum of a Noetherian ring is a Noetherian topological space

[F4]

A polynomial ring over a domain is a domain. Under AC a noetherian ring of finite Krull dimension gains exactly one dimension on adjoining a variable. A polynomial ring over an integral domain is an integral domain, A Noetherian polynomial ring has dimension one larger

[F5]

AC is assumed through the spectrum and polynomial-dimension results; the coefficient calculations below make no additional infinite choices. The Axiom of Choice

[F6]

A unital homomorphism of commutative rings contracts prime ideals to prime ideals. For an ideal $I$ of a commutative ring $A$ , the quotient $A / I$ is a domain exactly when $I$ is prime. A ring homomorphism $f$ induces $A / \ker f ≅ im f$ . A ring map induces a contraction map on prime spectra, R/P is an integral domain if and only if P is a prime ideal, First isomorphism theorem for rings: R/ker f congimf

[F7]

A quotient ring has additive cosets and multiplication $(a + I) (b + I) = a b + I$ ; its canonical projection is a surjective unital homomorphism with kernel $I$ . Polynomial convolution makes $A [t]$ a commutative ring with injective constant homomorphism $A \to A [t]$ . Ring homomorphisms preserve addition, multiplication and identity; a bijective such map is an isomorphism. A domain is a nonzero commutative ring without zero divisors. The quotient ring R/I with (r+I)(s+I)=rs+I, The canonical projection R→ R/I is a surjective ring homomorphism with kernel I, Polynomial convolution makes R[x] a commutative ring containing R as its constant subring, Ring homomorphism: additive, multiplicative, and required to send 1 to 1, Zero divisor, and integral domain: a commutative ring with 1 ≠ 0 and no zero divisors

[F8]

Noetherian topological spaces satisfy the descending chain condition on closed subsets. Closed subsets of a subspace are intersections with ambient closed subsets, and subspace topologies are transitive under inclusions. Noetherian topological spaces via ACC on opens or DCC on closed subsets, Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace

[F9]

A commutative $R$ -algebra is of finite type when finitely many elements generate it as an $R$ -algebra. In particular every polynomial is a finite sum of coefficient multiples of powers of $t$ . Subalgebra generated by a subset, algebras of finite type, and module-finite algebras, The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution

Proof

Given: The ring and AC hypothesis in the Statement.

Goal: Establish all three dimension and purity assertions.

1.1F0F1F2F3F4F5F7F8F9given

The constant map of [F7] makes $R [t]$ an $R$ -algebra, generated by $t$ by [F9], so [F3] makes both spectra noetherian. For primes $p, q$ , the definition of $V$ gives $p \subseteq q$ if and only if $V (q) \subseteq V (p)$ : the reverse implication follows by testing the point $q \in V (q)$ . Thus the bijection of [F2] reverses each finite prime chain without changing its length; [F1] gives $\dim Spec R = \dim R$ and $\dim Spec R [t] = \dim R [t]$ without a finiteness assumption on dimension. If $\dim R = d < \infty$ , [F4] makes the latter dimension $d + 1$ . More generally, for a proper ideal $I$ of either ring, $V (I)$ is closed by [F0]. By [F8], its closed subsets are closed in the ambient spectrum, so it is noetherian and its irreducible closed subsets are exactly the ambient ones contained in $V (I)$ . By [F2] these correspond to primes containing $I$ , and the quotient-prime bijection preserves their strict inclusions. Hence [F1] gives $\dim V (I) = \dim (R / I)$ .

2.1F0F1F2F3F4F5F6F7step 1.1constructalgebra∎

For any ideal $I$ of $R$ , write $I [t]$ for the polynomials all of whose coefficients lie in $I$ . It is an ideal: addition and negatives preserve this condition, and each coefficient of a product with an arbitrary polynomial is a finite sum of multiples of elements of $I$ . Coefficient reduction defines $ρ_{I} : R [t] \to (R / I) [t]$ by $\sum_{i} a_{i} t^{i} \mapsto \sum_{i} (a_{i} + I) t^{i}$ . This is well-defined on finite coefficient sequences, preserves addition and identity, and preserves multiplication because its degree- $n$ coefficient is $\sum_{i + j = n} (a_{i} + I) (b_{j} + I) = (\sum_{i + j = n} a_{i} b_{j}) + I$ . It is surjective by lifting the finitely many coefficients of a target polynomial, and its kernel is exactly $I [t]$ by [F7]. Thus [F6] gives $R [t] / I [t] ≅ (R / I) [t]$ . If $I$ is prime, [F6] makes $R / I$ a domain and [F4] makes $(R / I) [t]$ a domain; the isomorphism and [F6] therefore show that $I [t]$ is prime. Now suppose $Spec R$ is pure of finite dimension $d$ . For a minimal prime $Q$ of $R [t]$ , its contraction $p = Q \cap R$ is prime by [F6] applied to the injective constant map in [F7]. Every coefficient in $p$ lies in $Q$ , so $p [t] \subseteq Q$ ; the primeness just proved and minimality give $Q = p [t]$ . Any prime $q ⊊ p$ would give $q [t] ⊊ p [t]$ , with strictness witnessed by a constant in $p ∖ q$ , contradicting minimality of $Q$ . Conversely, for minimal $p$ and prime $Q^{'} \subseteq p [t]$ , contraction is prime and contained in $p$ (the constant coefficients of $p [t]$ are exactly $p$ ), hence equals $p$ . The ideal $Q^{'}$ then contains $p [t]$ , so equality follows. These are exactly the minimal primes of $R [t]$ , and [F2] identifies their vanishing sets with its components. For each minimal $p$ of $R$ , purity and step 1.1 give $\dim (R / p) = \dim V (p) = d$ ; [F3] makes this nonzero quotient noetherian. By [F4], $\dim ((R / p) [t]) = d + 1$ . The coefficient isomorphism gives the same dimension for $R [t] / p [t]$ : contracting primes along a ring isomorphism and its inverse gives inverse inclusion-preserving prime bijections by [F6]. Step 1.1 now gives $\dim V (p [t]) = d + 1$ for every component, proving purity.

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