Section outline

  • Collapse Expand

    General

    Collapse all Expand all

    Commutative Algebra fall 2026


    Meetings:     Mondays at 13:00 in room Cramér

    Teachers: Luís Duarte and Samuel Lundqvist

    Prerequisites: Knowledge of basic concepts in Abstract algebra such as ideals, rings, and modules. 

    Content: The goal of this course is to provide an introduction to some key topics in Commutative Algebra, with focus on homological methods.

    We will begin by covering some basic fundamental concepts such as localization, Noetherian rings, Artinian rings and associated primes and we will arrive at the concepts of height and dimension. We will also cover some aspects of graded rings.  

    We will then move on to the study of projective and injective resolutions of modules, which are the basis of Homological Commutative Algebra.

    We will use homological properties to characterize regular sequences, the invariant depth and the property of a ring being regular.

    For the final stretch of the course, we will delve into the study of Cohen-Macaulay rings and Gorensteins rings, and the various ways one can characterize them.

    Evaluation: Homeworks (biweekly) and a presentation at the end.

    We will closely follow some chapters of the Matsumura's, Bruns & Herzog's and Atiyah & McDonald's books, but it is possible to follow the course just using the lecture notes.

    (1) Winfried Bruns and Jürgen Herzog, "Cohen-Macaulay rings", volume 39. Cambridge University Press, 1998.
    (2) Hideyuki Matsumura, "Commutative ring theory", volume 8, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1986.
    (3) M. Atiyah and I. G. Macdonald, "Introduction to commutative algebra", Addison-Wesley Publishing Company, 1969.
    • Course Notes File

      The lecture notes will be regularly updated and expanded (last updated: 27 September).

    • Nyheter Forum
  • Collapse Expand

    Homeworks

  • Collapse Expand

    Summaries

    Lecture 5 (05/10) - Primary ideals. Primary decompositions of ideals in Noetherian rings and the first and second uniqueness theorem for irredundant primary decompositions. Introducing graded rings, graded modules and graded submodules. (Sections 2.4 and 2.5. Start of Section 3.1)

    Lecture 4 (28/09) - Minimal primes. Associated primes. The set of zero divisors as the union of the associated primes of the module. Prime filtrations of finitely generated modules over Noetherian rings. (Sections 2.2 and 2.3)

    Lecture 3 (21/09) - The radical of an ideal as the intersection of the primes that contain the ideal. Length. Artinian rings and modules. (Sections 1.8 and 2.1)

    Lecture 2 (14/09) - Prime avoidance. Localization. Characterizing the ideals and the prime ideals in the localized ring. Noetherian rings and modules. (Sections 1.4, 1.5 and 1.7)

    Lecture 1 (07/09) - Introducing the basic concepts of commutative rings, modules and ideals. Prime and maximal ideals. Local rings and Nakayama's lemma. (Sections 1.1, 1.2 and 1.3)