Krull Dimension in Algebra and Geometry

• Krull dimension is a numerical invariant defined as the supremum of lengths of chains of prime ideals in a ring, variety, or scheme.
• It extends to various contexts—including differential, graded, and tropical settings—revealing rich links between algebraic structure and geometric intuition.
• Its applications span computational, constructive, and homological methods, providing practical tools for analyzing algebraic complexity.

Krull dimension is a foundational invariant in commutative algebra and algebraic geometry, capturing the maximal length of chains of prime ideals, and thus providing a quantitative measure of the “size,” complexity, or geometric dimension of algebraic structures such as rings, varieties, and schemes. Across classical, differential, graded, combinatorial, and tropical settings, Krull dimension acquires significant variations and extensions, underlying deep connections between algebraic, geometric, and homological properties.

1. Core Definition and Generalizations

The classical Krull dimension of a commutative ring $R$ is defined as the supremum of the lengths $n$ of strictly increasing chains of prime ideals

$$\mathfrak{p}_0 \subsetneq \mathfrak{p}_1 \subsetneq \cdots \subsetneq \mathfrak{p}_n$$

such that each inclusion is proper. Its geometric importance is seen in the coordinate ring of an affine variety: $\dim R$ agrees with the topological dimension of the associated variety, as in $\dim F[t_1,\dotsc,t_n]=n$ for a field $F$ (Nathanson, 2016). In advanced settings, this notion admits generalizations:

• Cardinal Krull dimension: Allows arbitrary infinite chains with dimension defined as the supremum cardinality of such chains (Loper et al., 2017).
• Krull dimension for semirings and tropical objects: Replaces prime ideals with prime congruences, adapting the concept to the idempotent or characteristic-one context (Joó et al., 2015, Song et al., 5 Aug 2024).
• Differential Krull dimension: Considers chains of differential prime ideals, important in differential algebra (Smirnov, 2011).
• Graded, DG, and super-dimensions: Incorporate gradings and additional algebraic structure, such as Z$_2$ gradings in super-commutative super-rings (Masuoka et al., 2019, Beck et al., 2012).
• q-Krull and w-Krull dimensions: Generalize the dimension theory with respect to closure operations or filtrations relevant in non-Noetherian settings (Zhang, 15 Apr 2024).

2. Chains, Regular Sequences, and Constructive Characterizations

Krull dimension is intimately related to the existence of chains of prime ideals or, equivalently, to algebraic sequences that “detect” dimension—a perspective that is both classical and constructively robust:

• A ring $A$ has Krull dimension at least $\ell$ if and only if it admits a pseudo-regular sequence of length $\ell$, i.e., sequences $x_1, ..., x_\ell$ for which certain algebraic identities do not collapse under arbitrary algebraic substitutions (Lombardi, 2023, Coquand et al., 2017, Coquand et al., 2017).
• The elementary, constructive approach reformulates the existence (or obstruction) of chains of primes as the (non-)vanishing of explicit algebraic expressions:

$$x_1^{m_1}\bigl(x_2^{m_2}( \cdots (x_\ell^{m_\ell}(1 + a_\ell x_\ell) + a_{\ell-1} x_{\ell-1}) + \cdots ) + a_1 x_1\bigr) \neq 0$$

for all choices of $a_i\in A$, $m_i\in\mathbb{N}$, for a pseudo-regular sequence of length $\ell$ (Lombardi, 2023, Coquand et al., 2017).

• Constructive treatments, often invoking “idealistic chains” or collapse certificates, provide algorithmic or combinatorial interpretations of Krull dimension, connecting classical algebra, distributive lattice theory, and proof theory (Coquand et al., 2017, Coquand et al., 2017, Lombardi, 2023).

3. Krull Dimension in Polynomial, Differential, and DG Extensions

Krull dimension exhibits additive and structural behavior under extensions:

• For polynomial rings over a commutative ring or semiring with Krull dimension $d$, the dimension often increases by the number of added indeterminates in rings:

$\dim R[t_1,\dotsc,t_n]=\dim R + n$

• In the context of additively idempotent semirings (e.g., tropical), the analogous result holds: $\dim A[x]=\dim A+1$ (Joó et al., 2015).
• In differential algebra, the differential Krull dimension of the ring $R\{z_1,\ldots,z_n\}$ satisfies strong additivity properties under standardness and finiteness conditions:

$$\operatorname{ht}^\mathfrak{p} = \operatorname{ht}^{\mathfrak{q}\{z_1,\ldots, z_n\}} + \operatorname{ht}^{\mathfrak{p}/\mathfrak{q}\{z_1,\ldots, z_n\}}$$

where $\mathfrak{p}$ is a differential prime in $R\{z_1,\ldots, z_n\}$ and $\mathfrak{q}=\mathfrak{p}\cap R$ (Smirnov, 2011). This is the differential Special Chain Theorem.

• For differential graded (DG) algebras, several definitions via anchor primes, systems of parameters, or DG prime ideals coincide, particularly in the case of DG algebras generated in odd degrees or bounded, yielding

$$\operatorname{DGdim}(A) = \operatorname{dim}_{A_0}(A) = \operatorname{dim}(H_0(A))$$

(Beck et al., 2012).

4. Geometric, Tropical, and Combinatorial Interpretations

Krull dimension serves as a geometric invariant, tightly linked with the dimension of topological, polyhedral, or combinatorial structures:

• For rings of semialgebraic functions $\mathcal{S}(M)$, Krull dimension agrees with the topological dimension:

$\dim \mathcal{S}(M) = \dim M$

and at a point $p\in M$, the height of the maximal ideal equals the local (topological) dimension at $p$. The semialgebraic depth of a prime ideal corresponds to the transcendence degree of the associated function field (Fernando et al., 2013).

• In tropical geometry, the Krull dimension of a $\boldsymbol{T}$-algebra (e.g., tropical polynomial semiring or quotient by a congruence) is determined by the dimension of the associated congruence variety as a polyhedral complex. For a proper congruence $C$ on $\boldsymbol{T}[X_1^{\pm1},..., X_n^{\pm1}]$ (with a finite congruence tropical basis), the key formula is

$$\dim \boldsymbol{T}[X_1^{\pm1},..., X_n^{\pm1}]/C = \max\{ \dim \boldsymbol{V}(C) + 1, \dim \boldsymbol{V}(C_{\boldsymbol{B}}) \}$$

where $\boldsymbol{V}(C)$ and $\boldsymbol{V}(C_{\boldsymbol{B}})$ are polyhedral sets associated to $C$ and a “flattened” version, respectively (Song et al., 5 Aug 2024).

• For function semifields of nondegenerate tropical curves, the Krull dimension is always two, reflecting a structural feature analogous to classical function fields of algebraic curves (Song et al., 5 Aug 2024).
• Krull dimension can be formulated in terms of independent sequences (analytic or monomial), with the supremum of independence length equalling the dimension (Kemper et al., 2013).

5. Krull Dimension in Infinite, Noncommutative, and Exotic Contexts

Krull dimension extends beyond Noetherian or commutative cases, as well as to infinitary settings:

• Cardinal-valued Krull dimension admits rings of prescribed cardinality $K$ and dimension $L$ for a wide range of $(K, L)$, utilizing valuation rings, polynomial rings in a large family of indeterminates, and Leavitt path algebras (Loper et al., 2017). There are constructions with uncountable Krull dimension, as in the ring $W_{\mathcal{O}_E}(R)$ (Witt vectors over a non-discrete perfect valuation ring), where the dimension is at least continuum (Du, 2020).
• In noncommutative settings, e.g., quantum tori, the Krull dimension is expressed via the supremum of the ranks of commutative subgroups embedded in a free abelian group, with the dimension coinciding with the global dimension and exhibiting super-additivity under tensor products (Gupta, 2014).
• Semirings and tropical objects require the use of prime congruences rather than ideals. The passage to polynomial or Laurent polynomial semirings increases dimension by one, echoing the ring-theoretic case but with adjustments for idempotency and lack of additive inverses (Joó et al., 2015, Song et al., 5 Aug 2024).

6. Homological, Derived, and Categorical Connections

Krull dimension manifests in, or is reflected by, a range of categorical and homological invariants:

• In the bounded derived category of coherent sheaves or perfect complexes, the Rouquier dimension is a central invariant measuring categorical “complexity.” For regular rings and for normal toric varieties, the Rouquier dimension exactly equals the Krull dimension (Favero et al., 2023, Letz, 30 Jun 2025). For any commutative noetherian regular ring $A$,

$\mathrm{Rdim}(\mathrm{Perf}(A)) = \dim(A)$

For perfect complexes, the lower bound of the Rouquier dimension is governed by the maximal length of a regular sequence in $A$, tightly linking homological and Krull dimensions (Letz, 30 Jun 2025).

• In the context of complexes of modules over Noetherian rings, the Krull dimension can be calculated using Fitting ideals of the matrices representing differentials, linking module presentations, regular sequences, and homological dimensions (Christensen et al., 2020).
• In constructive treatments of dimension theory, collapsing of idealistic chains or pseudo-regular sequences serves as a computational witness to finiteness or limitations of Krull dimension, highlighting connections with logical, lattice-theoretic, and algorithmic perspectives (Coquand et al., 2017, Coquand et al., 2017, Lombardi, 2023).

7. Specialized Dimensions: q-Krull, Super, and Differential Extensions

Specialized versions of Krull dimension accommodate the structure of rings, modules, filtrations, and additional algebraic operations:

• q-Krull dimension is defined via maximal q-ideals and relates via precise inequalities and structural equivalences to regularity, total quotient rings, and properties of $w$-dimensional rings (Zhang, 15 Apr 2024).
• Krull super-dimension for super-commutative super-rings captures both the “even” (classical Krull dimension of $R_0$) and “odd” (maximal system of odd parameters) structure, playing a decisive role in regularity, Kähler superdifferentials, and the geometry of superschemes (Masuoka et al., 2019).
• Differential Krull dimension quantifies the size of differential spectra, dictating the behavior of rings under differential polynomial extensions and supporting a systematic theory of dimension in differential algebraic geometry (with absence of anomaly in J-rings) (Smirnov, 2011).

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Krull dimension, in all its forms and under various algebraic, categorical, and geometric generalizations, remains a fundamental invariant: it encapsulates both the combinatorial and topological complexity of algebraic objects, bridges algebraic and geometric perspectives, enables computational and homological techniques, and adapts robustly to novel algebraic frameworks and categorical settings. Its modern treatments extend well beyond the classical paradigm, affirming its centrality in both structure theory and practice across commutative algebra, algebraic geometry, and their applications.