On Krull's Dimension Theorem for Certain Graded Rings and Its Applications

Abstract: This paper explores the dimension theory of non-Noetherian graded rings by introducing the class of Hilbert-Serre rings. We generalize Krull's dimension theorem and Smoke's dimension theorem by establishing the fundamental inequalities $\dim(R) \le \operatorname{GKdim}_k(R) \le d(R)$ for any Hilbert-Serre ring $R$, where $d(R)$ is the pole order of its Poincaré series at $t=1$. Furthermore, we apply these results to initial algebras, proving that all these dimensions, including the transcendence degree, coincide for monomial algebras. Finally, we provide explicit examples demonstrating that these inequalities can be strict in general, even for integral domains.

• The paper introduces Hilbert-Serre rings, extending the framework of Noetherian graded rings with strict finiteness and analytic conditions for controlled non-Noetherian examples.
• It establishes key inequality relations among invariants by proving that for these rings, dim(R) ≤ GKdimₖ(R) ≤ d(R) and, in domains, dim(R) ≤ tr.degₖ(R) = GKdimₖ(R) ≤ d(R).
• By applying these results to monomial and initial algebras, the work offers practical computational methodologies and explicit counterexamples to demonstrate the optimality of the bounds.

Dimension Theory in Non-Noetherian Graded Rings: The Hilbert-Serre Class and Generalized Dimension Inequalities

Introduction and Context

The dimension theory of Noetherian rings is a foundational subject in commutative algebra, crystallized by Krull’s dimension theorem, which equates Krull dimension, analytic spread, and certain growth invariants for Noetherian local rings. For Noetherian graded rings over a field, Smoke’s theorem similarly relates Krull dimension with the pole order of the Poincaré series. However, removing the Noetherian condition leads to a collapse in these equivalences, with classical dimension-theoretic invariants potentially diverging. The paper "On Krull's Dimension Theorem for Certain Graded Rings and Its Applications" (2605.00463) rigorously analyzes this divergence, introducing the class of Hilbert-Serre rings and systematically elucidating the relationships and inequalities among key dimension invariants in this broader, less restrictive setting.

Hilbert-Serre Rings and Fundamental Dimension Invariants

A Hilbert-Serre ring is defined as an $N$-graded $k$-algebra $R = \bigoplus_{n \ge 0} R_n$ with $R_0 = k$ a field, subject to strict finiteness conditions on the graded components and the existence of a pole of finite order of the Poincaré series at $t = 1$. This class properly expands the familiarly tractable Noetherian $N$-graded rings, admitting many non-Noetherian but controllable examples.

For such rings, the paper investigates five invariants:

• Krull dimension $\dim(R)$,
• Graded Krull dimension $(R)$,
• Transcendence degree $tr.deg_k(R)$,
• Gelfand-Kirillov dimension $GKdim_k(R)$,
• Poincaré pole order $k$0 (the Hilbert-Serre dimension).

A major technical achievement is the proof that for any Hilbert-Serre ring,

$k$1

and, when $k$2 is a domain,

$k$3

These inequalities generalize the exact equalities of the Noetherian case, with $k$4 playing a crucial interpolating role when transcendence degree is not defined.

Structural Results and Technical Insights

The structural properties of Hilbert-Serre rings are systematically developed. The paper establishes that Hilbert-Serre rings necessarily have finite Krull dimension, and that the Hilbert-Serre dimension controls chains of homogeneous prime ideals. Homogenization and localization methods are employed to relate general prime chains to homogeneous ones, ensuring that the Hilbert-Serre condition is robust under passage to quotients and certain subalgebras.

The Gelfand-Kirillov dimension is shown to agree with the transcendence degree for integral domains, in line with the classical theory for finitely generated algebras. In the overall framework, $k$5 provides the appropriate substitute for transcendence degree in non-domain contexts, facilitating the main inequality for arbitrary Hilbert-Serre rings.

The paper constructs explicit non-Noetherian Hilbert-Serre rings where the dimension inequalities are strict, demonstrating the optimality and limitations of these results.

Monomial Algebras, Initial Algebras, and SAGBI Bases

A major application addressed is the dimension theory of initial algebras and their connection to SAGBI bases. The initial algebra, even of a Noetherian subalgebra, is not necessarily Noetherian; nevertheless, its Poincaré series matches that of the original algebra. For the class of monomial algebras—equivalently, $k$6-subalgebras of a polynomial ring generated by monomials, and encompassing all initial algebras—the paper proves

$k$7

thus extending Smoke's theorem to this infinitely generated setting. This follows via an analysis using the dimension theory of commutative monoid rings and explicit growth estimates, rendering dimension theory for monomial algebras fully computable.

For initial algebras of finitely generated homogeneous subalgebras, all these dimensions are shown to coincide with those of the original subalgebra, demonstrating invariance under the initial algebra operation in this technically important setting.

Counterexamples and Optimality

The final sections provide explicit constructions where the inequalities of the main theorem are strict, both in the case of domains and in more degenerate examples, such as idealizations and rings with rapidly growing graded pieces. These demonstrate that none of the inequalities can be improved in general. Moreover, the examples are carefully constructed to separate the influence of the various conditions in the definition of Hilbert-Serre rings, clarifying the necessity of each.

Open Problems

The paper concludes with a question on the possible equality of Krull dimension and graded Krull dimension for Hilbert-Serre domains, while providing counterexamples when the Hilbert-Serre property is relaxed. This points toward further investigation into the subtle structure of prime ideals and homogeneity in the context of dimension theory for non-Noetherian graded rings.

Conclusion

This work provides a definitive treatment of dimension theory beyond the Noetherian context for a significant class of graded rings. By leveraging analytic, algebraic, and combinatorial tools, the paper establishes the precise relationships among dimension invariants, highlights classes (such as monomial algebras and initial algebras) where these invariants coincide, and carefully documents the possible strictness of all dimension inequalities when generalizing Krull's and Smoke’s theorems. These results have immediate implications both for the structure theory of graded rings and the effective computation of invariants in commutative algebra and algebraic combinatorics. The introduction of Hilbert-Serre rings offers a natural setting for future developments in graded ring theory, broadening the understanding of dimension in infinite generation scenarios.