MA316-7-AU-CO:
Commutative Algebra

The details
2021/22
Mathematical Sciences
Colchester Campus
Autumn
Postgraduate: Level 7
Current
Thursday 07 October 2021
Friday 17 December 2021
15
12 May 2021

 

Requisites for this module
(none)
(none)
(none)
(none)

 

(none)

Key module for

(none)

Module description

Commutative algebra is the cornerstone established by Hilbert to give a formal backing to intuitive arguments in geometry. This module will provide a solid foundation of commutative rings and module theory, as well as help developing foundational notions helpful in other areas such as number theory, algebraic geometry, and homological algebra. Examples will be key, many of them will be made 'graphic' thanks to Hilbert's Nullstellensatz.

In contrast with analytic geometry or topology, computations are very explicit. Keeping the geometry of algebraic varieties in mind, in this module, we will study modules over commutative rings as a generalization of both vector spaces and abelian groups. This will include 'good' rings, such as Principal Ideal Domains (PIDs) and Noetherian rings, and modules over them.

Module aims

1. To introduce students to basic definitions of commutative algebra and Zariski topology.
2. To develop students’ critical understanding of some main results in commutative algebra.
3. To develop understanding of how to apply such results in particular problems.

Module learning outcomes

1. Have a systemic understanding of key definitions in the theory of commutative algebra and critical awareness of how they interact and support each other.
2. Select and apply relevant theorems to examples.
3. Construct arguments to prove properties of rings.
4. Solve problems involving homomorphisms between pairs of groups.
5. Formulate counterexamples to statements.
6. Understand the notion of affine algebraic variety and its relation to the coordinate ring.
7. Recognise and work with quotient rings.
8. Being able to decide if an exact sequence of modules is exact.
9. Apply geometric techniques to obtain and illustrate algebraic properties of particular rings
10. Use properties of the coordinate ring to obtain topological information of an algebraic variety and viceversa.

Module information

Syllabus:
Recollection of abstract algebra: rings, fields, integral domains, rings of polynomials and homomorphisms.
Unique Factorisation Domains (UFDs): irreducible elements and factorisation, UFDs, polynomial ring of a UFD is a UFD.
Ideals and arithmetic of ideals: definition of ideal, intersection and sum, finitely generated ideals, principal ideals, principal ideal domains (PIDs), PIDs are UFDs.
Ideals for commutative rings: prime ideals, maximal ideals and quotient of ring by them, existence of maximal ideals using Zorn's lemma.
Prime spectrum of a ring (Spec): relation to geometry. Examples.
Local rings and localisation. Nilradical.
Modules and exact sequences.
Noetherian rings and Noetherian modules. Hilbert's basis Theorem.
Hilbert's Nullstellensatz and affine varieties.
Introduction to point-set topology. Zariski topology for affine varieties and its properties.

Learning and teaching methods

There are three lectures a week during the main teaching term and an extra lecture every fortnight. The rough equivalent of one hour a fortnight will be devoted to going over some problems. Students having issues with the module are encouraged to talk to the module supervisor during office hours. Three hours of revision will be available in the summer term.

Bibliography

  • Reid, Miles. (2010) Undergraduate commutative algebra, Cambridge: Cambridge University Press. vol. 29

The above list is indicative of the essential reading for the course. The library makes provision for all reading list items, with digital provision where possible, and these resources are shared between students. Further reading can be obtained from this module's reading list.

Assessment items, weightings and deadlines

Coursework / exam Description Deadline Weighting
Coursework   Problem Set 1  12/11/2021   
Coursework   Problem Set 2  17/12/2021   
Exam  150 minutes during Summer (Main Period) (Main) 

Overall assessment

Coursework Exam
20% 80%

Reassessment

Coursework Exam
20% 80%
Module supervisor and teaching staff
Dr Jesus Martinez-Garcia, email: jesus.martinez-garcia@essex.ac.uk.
Dr Jesus Martinez-Garcia
jesus.martinez-garcia@essex.ac.uk

 

Availability
Yes
No
Yes

External examiner

No external examiner information available for this module.
Resources
Available via Moodle
Of 1847 hours, 35 (1.9%) hours available to students:
1812 hours not recorded due to service coverage or fault;
0 hours not recorded due to opt-out by lecturer(s).

 

Further information
Mathematical Sciences

Disclaimer: The University makes every effort to ensure that this information on its Module Directory is accurate and up-to-date. Exceptionally it can be necessary to make changes, for example to programmes, modules, facilities or fees. Examples of such reasons might include a change of law or regulatory requirements, industrial action, lack of demand, departure of key personnel, change in government policy, or withdrawal/reduction of funding. Changes to modules may for example consist of variations to the content and method of delivery or assessment of modules and other services, to discontinue modules and other services and to merge or combine modules. The University will endeavour to keep such changes to a minimum, and will also keep students informed appropriately by updating our programme specifications and module directory.

The full Procedures, Rules and Regulations of the University governing how it operates are set out in the Charter, Statutes and Ordinances and in the University Regulations, Policy and Procedures.