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Krull dimension

• The Krull dimension of a commutative ring R is the supremum of the lengths of chains of prime ideals, a concept introduced by Wolfgang Krull.

• The Krull dimension can be infinite, even in Noetherian rings, and can also be applied to modules over non-commutative rings, defined as the deviation of the poset of submodules.

• It serves as an algebraic definition of the dimension of an algebraic variety, where the dimension of an affine variety defined by an ideal I in a polynomial ring R corresponds to the Krull dimension of R/I.

• A field has a Krull dimension of 0, while the polynomial ring k[x1, ..., xn] has a Krull dimension of n.

• A principal ideal domain that is not a field has a Krull dimension of 1, and a local ring has a Krull dimension of 0 if all elements of its maximal ideal are nilpotent.

• The length of a chain of prime ideals is determined by the number of strict inclusions, and the height of a prime ideal is the supremum of the lengths of chains of prime ideals contained within it.

• In Noetherian rings, every prime ideal has finite height, but some Noetherian rings can still have infinite Krull dimension.

• A ring is catenary if any inclusion of prime ideals can be extended to a maximal chain, and universally catenary if every finitely generated algebra over it is catenary.

• The height of an ideal is the infimum of the heights of all prime ideals containing it, which relates to the codimension of the corresponding subvariety in algebraic geometry.

• The Krull dimension of a ring is equivalent to the dimension of its spectrum as a topological space, representing the supremum of the lengths of chains of irreducible closed subsets.

• Examples include the ring of integers Z, which has a dimension of 1, and the zero ring, which is defined to have either dimension 0 or negative dimension.

• A Noetherian local ring is Cohen–Macaulay if its dimension equals its depth, with regular local rings serving as examples of such rings.…

Krull Dimension -- from Wolfram MathWorld

• In a commutative ring with unity, the height of a prime ideal is determined by the longest chain of distinct prime ideals that can be formed.
• The Krull dimension of a ring is the highest height among all its prime ideals, representing a measure of the ring's complexity.
• Understanding Krull dimension is crucial for advancing knowledge in commutative algebra and algebraic geometry.
• Recommended readings for deeper insights include "Commutative Algebra with a View Toward Algebraic Geometry" by Eisenbud and "Introduction to Commutative Algebra" by Atiyah and Macdonald.
• Additional resources on Krull dimension can be found on platforms like Wolfram|Alpha and MathWorld.

Section 10.60 (00KD): Dimension—The Stacks project

Krull dimension of $\mathbb{Z}[x_1,\dots,x_n]

• The objective is to demonstrate that the Krull dimension of the polynomial ring $\mathbb{Z}[x_1, \dots, x_n]$ is $n + 1$.

• A known theorem states that for a Noetherian ring $A$, the dimension of the polynomial ring $A[x_1, \dots, x_n]$ can be calculated using the formula: $\dim(A[x_1, \dots, x_n]) = n + \dim(A)$.

• The proof utilizes Hilbert polynomials and the fact that if $F$ is a field, then the Krull dimension of $F[x_1, \dots, x_n]$ is $n$.

• A maximal ideal $\mathfrak{m}$ in $\mathbb{Z}[x_1, \dots, x_n]$ is considered, along with a prime number $p$ that is contained in $\mathfrak{m}$, which can be identified by examining $\mathfrak{m} \cap \mathbb{Z}$.

• A short exact sequence is constructed:
$$0 \longrightarrow \mathbb{Z}[x_1, \dots, x_n] \overset{p \cdot}{\longrightarrow} \mathbb{Z}[x_1, \dots, x_n] \longrightarrow \mathbb{F}_p[x_1, \dots, x_n] \longrightarrow 0,$$
where the map $p \cdot$ represents multiplication by the prime $p$.

• The sequence is localized at the maximal ideal $\mathfrak{m}$, leading to:
$$0 \longrightarrow \mathbb{Z}[x_1, \dots, x_n]_{\mathfrak{m}} \overset{p \cdot}{\longrightarrow} \mathbb{Z}[x_1, \dots, x_n]_{\mathfrak{m}} \longrightarrow \mathbb{F}_p[x_1, \dots, x_n]_{\mathfrak{m}} \longrightarrow 0.$$

• Hilbert polynomials are employed to derive the equation:
$$P(\mathbb{Z}[x_1, \dots, x_n]_{\mathfrak{m}}, t) = P(\mathbb{Z}[x_1, \dots, x_n]_{\mathfrak{m}}, t) \cdot t + P(\mathbb{F}_p[x_1, \dots, x_n]_{\mathfrak{m}}, t).$$

• The term $P(\mathbb{Z}[x_1, \dots, x_n]_{\mathfrak{m}}, t) \cdot t$ on the right side indicates that the dimension of the polynomial ring increases by one for each additional variable, represented by the factor $t$.

• The validity of the proof relies on comprehending the localization process and the significance of Hilbert polynomials in connecting the dimensions of the involved rings.

• The final conclusion is that the Krull dimension of $\math…

Motivation behind the Krull Dimension of a ring

• Krull dimension provides an algebraic definition of dimension that aligns with the geometric understanding of varieties and schemes.
• The affine space $\mathbb{A}^n_{\mathbb{C}}$ is linked to the topological space $\mathbb{C}^n$, with the Krull dimension of the ring $\mathbb{C}[x_1, \ldots, x_n]$ being $n$, illustrating their relationship despite the lack of homeomorphism.
• This concept enables the formalization and generalization of dimension and co-dimension to arbitrary rings, such as $\mathbb{Z}$, where geometric intuition may not be applicable, thus supporting geometric studies from an algebraic viewpoint.
• Geometric proofs that rely on the concept of dimension can be reformulated in algebraic terms, allowing for more extensive proofs of the same results.
• In the spectrum $\mathrm{Spec}\, A$, points correspond to prime ideals of $A$, while geometric intuition typically associates points with maximal ideals; the relationships between these points are connected to maps $A \to A/\mathfrak{m}$, which have a Krull dimension of 0.

Should Krull dimension be a cardinal?

• A chain of length \( n \) in a ring \( A \) consists of a totally ordered finite set of prime ideals, denoted as \( \mathcal P_0 \varsubsetneq \mathcal P_1 \varsubsetneq \dots \varsubsetneq \mathcal P_n \).

• The Krull dimension \( \dim(A) \) of the ring \( A \) is defined as the supremum of the lengths of such chains of prime ideals.

• If the lengths of these chains are unbounded, the ring is classified as infinite dimensional, indicated by \( \dim(A) = \infty \), which can occur even in Noetherian rings.

• For infinite dimensional rings, one can examine arbitrary totally ordered subsets of prime ideals \( \Pi \subset Spec(A) \) and their cardinality \( card(\Pi) \).

• The supremum of all these cardinalities is known as the cardinal Krull dimension of the ring \( A \).

• An equality \( \dim(A) = \aleph \) offers a more precise measure of the infinite dimensionality of \( A \) than simply stating \( \dim(A) = \infty \).

• The author seeks known results regarding the cardinal Krull dimension, particularly for the ring of continuous functions \( \mathcal C(X) \) on a topological space \( X \).

• The author finds it challenging to determine the cardinal Krull dimension of \( \mathcal C(X) \), even for the specific case where \( X = \mathbb R \).

• The inquiry also includes variants related to rings of differentiable functions on manifolds and similar mathematical structures.

Can we say anything about the Krull dimension of a ...

• The author aims to establish a theorem regarding the Krull dimension of the localization \( R[\frac{1}{v}] \) for a "nice" ring \( R \) and a "reasonable" element \( v \), suggesting that the dimension remains the same or decreases by one compared to \( R \).

• Initial algebraic approaches to prove this theorem have not succeeded, leading the author to investigate techniques from algebraic geometry.

• The author examines Grothendieck's Vanishing Theorem, which indicates that higher cohomology groups vanish for sheaves of abelian groups on a space of dimension \( n \), but notes that the converse is not true, which limits its applicability for drawing dimension conclusions.

• The author proposes specific hypotheses, suggesting that \( R \) can be considered commutative and finitely generated over a base ring (such as \( \mathbb{Z}_{(2)} \)), but does not need to be an integral domain.

• Although it might be feasible to assume \( R \) is Noetherian and local, the author prefers to avoid these assumptions in their exploration.

• The element \( v \) is independent of the base ring and has limited interactions with other elements in \( R \), with no relations involving the base ring itself.

• The author seeks to identify conditions on \( v \) that would result in a dimension drop greater than 1 when inverting \( v \).

• It is noted that for a dimension drop of exactly 1 to occur, \( v \) must correspond to inverting a maximal irreducible component, prompting the author to investigate the algebraic conditions this requirement places on \( v \).