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Krull Dimension and the Nullstellensatz: From Classical Foundations to Constructive Mathematics

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Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2025, Director: Carlos D’Andrea

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This work presents a comprehensive study of fundamental concepts in commutative algebra through both classical and constructive approaches. The first chapter focuses on affine algebraic varieties, the Zariski topology, and Hilbert’s Nullstellensatz. This chapter analyzes foundational notions in detail, explaining why they are defined as such and exploring their deep interrelations. Special emphasis is placed on the geometric intuition behind these concepts (for instance, studying the the $T_2$ separability of the Zariski topology depending on the underlying ring), and on linking algebraic definitions to their geometric counterparts.

The second chapter develops the notion of Krull dimension. The effort is placed on deeply understanding how this dimension works, especially in relation to prime ideals. Each theorem is studied carefully, examining the ideas and tools used in the proofs. A completely geometric study of different ideals and their associated varieties is carried out, where the Krull dimension plays a central role in determining the geometric dimension of these varieties. Concepts like symbolic powers and primary decomposition are explored not just as technical tools, but as meaningful objects that help understand the structure of algebraic spaces.

The third chapter is the core and most extensive part of the work, centered on the constructive approach to commutative algebra developed by H. Lombardi, T. Coquand, and collaborators. This methodology seeks to avoid classical non-constructive principles like the law of excluded middle or the axiom of choice, instead providing explicit, algorithmic formulations of concepts and proofs. The chapter highlights the notion of ’collapse’ in dynamic algebraic structures, such as potential primes, which impose more computationally verifiable conditions on classical prime ideals. Applying this approach implies that classical results such as the Nullstellensatz and Krull dimension are redefined in a fully constructive framework. The effort here lies not only in understanding these new definitions and their formal development but also in filling gaps in the literature by proving many propositions that are stated without proof in foundational texts. This chapter illustrates how constructive algebra enables explicit computations and verifications that classical algebra approaches without fully specifying the elements involved.

Finally, the fourth chapter offers a historical overview of the development of the Nullstellensatz theorem, summarizing key contributions from figures such as Kronecker and Noether. This retrospective contextualizes the earlier technical chapters by showing the evolution of ideas and terminology leading up to modern formulations.

In conclusion, this work bridges classical theory and modern constructive methods, providing detailed conceptual, geometric, and algebraic insight into the core structures of commutative algebra.

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cc-by-nc-nd (c) Saad Boulaich Marso, 2025

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Treballs Finals de Grau (TFG) - Matemàtiques

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Krull Dimension and the Nullstellensatz: From Classical Foundations to Constructive Mathematics