To prove that for a Noetherian commutative ring ( R ), the Krull dimension of the polynomial ring ( R[T] ) satisfies ( \dim(R[T]) = \dim(R) + 1 ) using the Coquand-Lombardi characterization, we proceed as follows:

Key Background: Coquand-Lombardi Characterization

The Krull dimension of a ring ( R ) is defined via a first-order condition: ( \dim(R) \leq k ) if and only if for every s...

Yes. In fact, the Coquand–Lombardi characterization is tailor‑made for this. One proves directly, without mentioning prime ideals, that for every polynomial f ∈ R[T] the “analytic” localization Bf := (R[T]){f} = localization of R[T] at fℕ · (1 + f·R[T]) has Krull dimension (in the Coquand–Lombardi sense) at most that of R. Then the identity dim(R[T]) = supf∈R[T](dim((R[T]){f}) + 1) from Coquand–Lo...

Proving Using the Coquand-Lombardi Characterization

This is an interesting question about using the Coquand-Lombardi first-order characterization of Krull dimension to prove a fundamental result in dimension theory without resorting to prime ideals.

Let me work through a proof based on the Coquand-Lombardi characterization.

Background

For a commutative ring ,...

Throughout the proof we use only the Coquand–Lombardi characterisation

     dim A ≤ k  ⇔  (∀x0 ,…, xk ∈ A) (∃a0 ,…, ak , m0 ,…, mk )
                   x0m0 ( …( xkmk ( 1+ak xk)+ …)+ a0 x0)=0                 (CLk)

and the elementary facts which were established in their paper:

(i) localisation does not increase the dimension;
(ii) if S is integral over A then dim S = dim A ;
(i...

We want to prove that for a Noetherian commutative ring R, dim(R[T]) = dim(R) + 1, using only this characterization and without using prime ideals.

Since R is Noetheria...

Below is a sketch of how one shows

dim R[T] = dim R + 1

purely in the Coquand–Lombardi (C–L) language (no prime‐ideals anywhere), under the hypothesis that R is Noetherian. Recall that in C–L one sets

dim A = sup_{x∈A} ( dim A_{ {x} } +1 )  (∗)

and one shows equivalently that “dim A≤k” if and only if for every k+1 elements x₀,…,x_k of A there are aᵢ∈A and mᵢ∈ℕ...

This is an interesting question about finding a proof of the dimension formula for Noetherian rings using the Coquand-Lombardi characterization.

Let me outline the key challenges and potential approaches:

The Challenge

The Coquand-Lombardi characterization states that:

where is the localization at $x^{...

Below I give a completely “prime–free’’ proof – i.e. a proof that only uses the Coquand–Lombardi first–order description of Krull dimension – of

    dim ( R[T] ) = dim ( R ) + 1        (∗)

for every Noetherian commutative ring R. Nothing that follows mentions prime ideals, maximal ideals, chains of prime ideals, Krull’s intersection theorem …; everything is done inside the language

  ...

Excellent question. You are asking for a proof of for a commutative Noetherian ring that uses the Coquand-Lombardi (C-L) characterization of Krull dimension and avoids the language of prime ideals.

You've correctly identified the main steps and the core of the problem:

1. The inequality is elementary. Given a chain of primes $\mathfrak{p}_0...