Introductory Commutative Algebra Reading Group

This group systematically studies Miles Reid's "Undergraduate Commutative Algebra" (LMS Student Texts 29). The text focuses on the "bridge" between commutative rings and geometric spaces, teaching participants to view rings as functions and ideals as subvarieties. What Will Happen: We will work through the book chapter-by-chapter. Sessions involve presenting core theorems, discussing Reid's geometric illustrations, and reviewing the exercises at the end of each section. Topic Progression: 1. Basics of Spec A: Defining prime and maximal ideals and the geometric interpretation of the spectrum. 2. Modules and Finiteness: Developing linear algebra over rings, exact sequences, Nakayama's Lemma, and Noetherian rings. 3. The Nullstellensatz: Establishing the core link between varieties and coordinate rings. 4. Localisation and Primary Decomposition: Learning to study rings "locally" and factorizing ideals into primary components. 5. DVRs and Normalization: Studying Discrete Valuation Rings and the resolution of singularities in algebraic curves. 6. Pathological Examples: Examining advanced cases where standard conditions fail, such as examples by Akizuki and Nagata. Expectations and Methodology: The group emphasizes intuition alongside rigor. We will spend significant time on the end-of-chapter exercises to ensure a practical grasp of the tools. Reid's "opinionated" commentary on the history of algebra will also serve as a basis for discussion regarding the subject's development. Prerequisites Participants should have a standard undergraduate background in linear algebra (bases, determinants) and ring theory (ideals, quotients, UFDs). Basic knowledge of set theory (Zorn's Lemma) is required.