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# Commutative Algebra & Algebraic Geometry - MAT00108M

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• Department: Mathematics
• Module co-ordinator: Dr. Emilie Dufresne
• Credit value: 20 credits
• Credit level: M
• Academic year of delivery: 2024-25
• See module specification for other years: 2023-24

## Module summary

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

## Related modules

• None

### Prohibited combinations

• None

It would be useful for students to have done a first course in Topology, such as the York Stage 3 module Topology (MAT00082H).

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.

## Module will run

Occurrence Teaching period
A Semester 2 2024-25

## 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.

## Assessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Closed exam : Commutative Algebra & Algebraic Geometry
3 hours 100

None

### Reassessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Closed exam : Commutative Algebra & Algebraic Geometry
3 hours 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.