Schemes 22: Valuation rings

Item: Schemes 22: Valuation rings
Rating: 5
Author: PostSingularity Institute

by Richard E Borcherds

Formal & Physical Sciences Mathematics PBMathematics PBFAlgebra

🎙 Richard E Borcherds 👥 82K 📅 July 16, 2020 ⏱ 24 min 👁 4K 📄 lecture 🧭 2026-08-17

Available in: English (current) Français

Keywords

valuation ringdiscrete valuation ringrankZariski–Riemann surfacescheme

Summary

This lecture is part of an online algebraic geometry course on schemes, based on chapter II of Hartshorne’s ‘Algebraic Geometry’. The speaker, Richard Borcherds, reviews valuation rings, which are integral domains with the property that for any nonzero element in the quotient field, either it or its inverse lies in the ring. He gives examples such as formal power series over a field and the localization of integers at a prime. He explains that valuation rings are local rings and introduces the valuation group, which is totally ordered. When the valuation group is isomorphic to the integers, the ring is a discrete valuation ring (DVR), which is a principal ideal domain. The spectrum of a DVR consists of a closed point and a generic point. The lecture then discusses non-discrete valuation rings, which have a complicated history. Examples include the Puiseux series, which has rational valuations and is rank 1 but not discrete. Higher rank valuation rings can be constructed using formal power series with well-ordered support over any totally ordered group. These rings appear in algebraic geometry through infinite sequences of blow-ups, leading to Zariski–Riemann surfaces, which are sets of valuation rings of a function field. Zariski used these to attempt resolution of singularities, but Grothendieck later eliminated non-discrete valuations from algebraic geometry, preferring discrete ones. The lecture concludes by mentioning that valuation rings will be used in criteria for separated and proper morphisms.

235 words

Critical Evaluation

Value of the Information & Strength of the Argument

The lecture provides a clear and rigorous exposition of valuation rings, starting from the definition and building up to more advanced concepts like rank and Zariski–Riemann surfaces. The argumentation is solid, with each concept motivated by examples and connections to algebraic geometry. The historical context, including Grothendieck’s negative view, adds depth. The speaker’s expertise ensures the accuracy and relevance of the content.

Scientific Rigor, Source Quality, Title Accuracy

The lecture is based on standard material from Hartshorne’s textbook, ensuring scientific rigor. The speaker mentions specific references, such as Hartshorne Chapter I Section 6 for the correspondence between curves and function fields, and a quote from a letter by Grothendieck. The title accurately reflects the content, which is a focused review of valuation rings within a schemes course. No comments were provided for analysis.

142 words

Title / Content Match

The title accurately reflects the content, which is a focused review of valuation rings within a schemes course.

Quality & Reliability

9/10

Lecture by a renowned mathematician (Fields medalist) based on a standard textbook (Hartshorne). Content is mathematically rigorous, with clear definitions and examples. Historical context is provided with a primary source quote (letter by Grothendieck).

Key Moments

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

Cited Sources

Algebraic Geometry (book) — Main reference for the course, specifically Chapter II on schemes and Chapter I Section 6 for the correspondence between curves and function fields.

Concurring Sources

Valuation ring (Wikipedia) — Confirms the definition and properties of valuation rings.
Discrete valuation ring (Wikipedia) — Confirms the characterization of DVRs as principal ideal domains.

Contribution & Novelties

The lecture provides a concise yet comprehensive review of valuation rings, bridging classical algebraic geometry and modern scheme theory. It highlights the historical development and the role of non-discrete valuation rings, which are often omitted in standard treatments. The discussion of Zariski–Riemann surfaces as precursors to schemes offers valuable insight.

Pour aller plus loin :

Valuation ring (Wikipedia) — General overview and properties.
Discrete valuation ring (Wikipedia) — Detailed treatment of DVRs.
Zariski–Riemann space (nLab) — Modern perspective on Zariski–Riemann spaces.
Hartshorne’s Algebraic Geometry — Standard reference for the course.

90 words

Radar Profile

The radar profile shows high scores in quality of information, technical level, and reliability, with a slightly lower score for quantity of information due to the concise nature of the lecture. This indicates a dense, expert-level presentation with strong scientific foundation.

Reliability 9/10