MA316-6-AU-CO:
Commutative Algebra
2021/22
Mathematics, Statistics and Actuarial Science (School of)
Colchester Campus
Autumn
Undergraduate: Level 6
Current
Thursday 07 October 2021
Friday 17 December 2021
15
12 May 2021
Requisites for this module
MA201 and MA204
(none)
(none)
(none)
(none)
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.
1. To introduce students to basic definitions of commutative algebra.
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.
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.
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.
There are three lectures a week during the main teaching term. 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.
This module does not appear to have a published bibliography for this year.
Assessment items, weightings and deadlines
| Coursework / exam |
Description |
Deadline |
Coursework weighting |
| Coursework |
Problem Set 1 |
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| Coursework |
Problem Set 2 |
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| Exam |
Main exam: 180 minutes during Summer (Main Period)
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Exam format definitions
- Remote, open book: Your exam will take place remotely via an online learning platform. You may refer to any physical or electronic materials during the exam.
- In-person, open book: Your exam will take place on campus under invigilation. You may refer to any physical materials such as paper study notes or a textbook during the exam. Electronic devices may not be used in the exam.
- In-person, open book (restricted): The exam will take place on campus under invigilation. You may refer only to specific physical materials such as a named textbook during the exam. Permitted materials will be specified by your department. Electronic devices may not be used in the exam.
- In-person, closed book: The exam will take place on campus under invigilation. You may not refer to any physical materials or electronic devices during the exam. There may be times when a paper dictionary,
for example, may be permitted in an otherwise closed book exam. Any exceptions will be specified by your department.
Your department will provide further guidance before your exams.
Overall assessment
Reassessment
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
Yes
No
Yes
Prof Stephen Langdon
Brunel University London
Professor
Available via Moodle
Of 60 hours, 58 (96.7%) hours available to students:
2 hours not recorded due to service coverage or fault;
0 hours not recorded due to opt-out by lecturer(s), module, or event type.
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