Commutative algebra 57: Krull versus Hilbert

Item: Commutative algebra 57: Krull versus Hilbert
Rating: 5
Author: PostSingularity Institute

by Richard E Borcherds

Formal & Physical Sciences Mathematics PBMathematics PBFAlgebra

🎙 Richard E Borcherds 👥 82K 📅 December 13, 2020 ⏱ 12 min 👁 2K 📄 lecture 🧭 2026-08-17

Available in: English (current) Français

Keywords

Krull dimensionHilbert dimensionNoetherian local ringHilbert polynomialexact sequence

Summary

This lecture is part of an online course on commutative algebra, following Eisenbud’s book. The goal is to prove that three definitions of dimension for Noetherian local rings coincide: Krull dimension, Hilbert dimension (via Hilbert polynomials), and dimension via systems of parameters. The previous lecture proved one inequality; this lecture proves that Krull dimension is at most Hilbert dimension. The proof begins with a lemma: if x is in the maximal ideal and not a zero divisor, then the Hilbert dimension of R/xR is at most the Hilbert dimension of R minus one. The lemma is proven using an exact sequence and the Artin-Rees lemma to show that the Hilbert polynomials have the same degree and leading coefficient. Then, assuming Krull dimension at least n, a chain of prime ideals is used to find a nonzero divisor x in p1, and induction on n completes the proof. The next lecture will prove the remaining inequality.

155 words

Critical Evaluation

Value of the Information & Strength of the Argument

The lecture provides a clear and rigorous proof of a fundamental result in commutative algebra. The argumentation is solid: it builds on previous material, introduces a necessary lemma, and proves it carefully. The use of exact sequences and the Artin-Rees lemma is appropriate and well-motivated. The example illustrating the necessity of the non-zero-divisor condition is helpful. The proof is logically structured and easy to follow for an audience familiar with the prerequisites.

Scientific Rigor, Source Quality, Title Accuracy

The lecture is based on a standard textbook (Eisenbud) and is delivered by an expert. The mathematical content is rigorous, with no apparent errors. The title accurately describes the content. No external sources are cited beyond the textbook, but this is appropriate for a lecture. The description mentions the textbook and the course context, which adds credibility.

144 words

Title / Content Match

The title accurately reflects the content, which focuses on comparing Krull dimension and Hilbert dimension.

Quality & Reliability

9/10

The lecture is part of a formal course by a renowned mathematician, following a standard textbook. The proof is rigorous and well-explained, with clear logical steps and appropriate examples.

Key Moments

Markers derived by PSI from the transcript: the creator did not define chapters.

Cited Sources

Commutative algebra with a view toward algebraic geometry — The course follows this book by David Eisenbud.

Concurring Sources

Commutative algebra with a view toward algebraic geometry — The lecture follows this standard textbook.

Contribution & Novelties

This lecture provides a clear and rigorous proof of the inequality between Krull dimension and Hilbert dimension, which is a key step in showing the equivalence of three dimension notions. The presentation is didactic, with careful attention to technical details such as the use of the Artin-Rees lemma. It is particularly valuable for students learning commutative algebra.

Pour aller plus loin :

Krull dimension — Provides background on Krull dimension.
Hilbert polynomial — Background on Hilbert polynomials.
Artin-Rees lemma — The lemma used in the proof.

86 words

Radar Profile

The radar profile shows high scores across all dimensions, indicating a well-balanced and reliable lecture. The quantity of information is substantial, the quality is high, the technical level is advanced, and the overall reliability is excellent.

Reliability 9/10