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Commutative Algebra & Algebraic Geometry - MAT00108M

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  • Department: Mathematics
  • Credit value: 20 credits
  • Credit level: M
  • Academic year of delivery: 2023-24
    • See module specification for other years: 2024-25

Module summary

This module is an introduction to commutative algebra & algebraic geometry.

Related modules

Pre-requisite modules

Co-requisite modules

  • None

Prohibited combinations

  • None

Additional information

Either Topology or Metric Spaces

MSc students should have done an undergraduate course in Linear Algebra, and a module equivalent to “Groups, Rings and Fields”, covering basic properties of polynomial rings and factorizability in rings. Prior knowledge of some basic topological notions would also be an advantage, but is not essential.

Background information: Algebraic Geometry is a subject at the interface of commutative algebra and geometry and its study involves a synthesis of techniques and ideas from many Pure Modules in previous stages. This module will allow students to understand an important area of modern mathematics, giving a good preparation for research in Algebra (both in the final year project and beyond into postgraduate life). The study of Algebraic Geometry mixes algebraic tools with geometric insight: algebraic varieties are defined by the vanishing of collections of polynomials, and the geometry of these sets is tightly controlled by the structure of the associated polynomial algebra. In the treatment of the subject in this course, the emphasis will be on the commutative algebra.

Module will run

Occurrence Teaching period
A Semester 2 2023-24

Module aims

Algebraic Geometry is a subject at the interface of commutative algebra and geometry and its study involves a synthesis of techniques and ideas from many Pure Modules in previous stages. This module will allow students to understand an important area of modern mathematics, giving a good preparation for research in Algebra (both in the final year project and beyond into postgraduate life). The study of Algebraic Geometry mixes algebraic tools with geometric insight: algebraic varieties are defined by the vanishing of collections of polynomials, and the geometry of these sets is tightly controlled by the structure of the associated polynomial algebra. In the treatment of the subject in this course, the emphasis will be on the commutative algebra

Module learning outcomes

By the end of the module students will be able to demonstrate understanding of, and be able to perform computations involving:

  1. the equivalent definitions of “Noetherian” for rings and modules

  2. the Zariski topology on affine n-space and abstract affine varieties

  3. the Nullstellensatz and its applications

  4. the notion of dimension for rings and varieties

  5. the tensor product of modules and the related direct product of varieties

Module content

  • Noetherian Rings and modules.

  • Affine n-space and the Zariski topology.

  • Abstract affine varieties.

  • Hilbert's Nullstellensatz and its proof.

  • The Krull dimension of a ring and the dimension of an affine variety.

  • The tensor product of modules.

Indicative assessment

Task % of module mark
Closed/in-person Exam (Centrally scheduled) 100

Special assessment rules

None

Indicative reassessment

Task % of module mark
Closed/in-person Exam (Centrally scheduled) 100

Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.

Indicative reading

“Abstract Algebra” by Dummit and Foote, ch 15

“Undergraduate Algebraic Geometry” by Reid

“Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra” by Cox, Little, and O’Shea.


The information on this page is indicative of the module that is currently on offer. The University constantly explores ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary. In some instances it may be appropriate for the University to notify and consult with affected students about module changes in accordance with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.