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# 2017/18 Taught Postgraduate Module Catalogue

## MATH5253M Commutative Algebra and Algebraic Geometry

### 20 creditsClass Size: 30

Module manager: Dr Eleonore Faber
Email: E.M.Faber@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2017/18

### Pre-requisite qualifications

MATH3193 or equivalent.

This module is approved as an Elective

### Module summary

Algebraic geometry is the study of spaces defined as the solution sets of polynomials in several indeterminates. It plays an ever more important role in mathematics, for example in elliptic curve cryptography and in Wiles' proof of Fermat's Last Theorem (although these topics aren't included in the module). One of the reasons why algebraic geometry is so useful is that one can use powerful abstract results about commutative rings. At the heart of this module are two of the most famous of these results, Hilbert's Basis Theorem and Nullstellensatz (Zeros Theorem).

### Objectives

To introduce the student to the basic ideas of algebraic geometry, the associated commutative algebra, and the interplay between the ideas of algebra and geometry.

On completion of this module, students should be able to:
a) Define some of the main concepts in commutative algebra and algebraic geometry.
b) State and prove some of the basic results in commutative algebra and algebraic geometry.
c) Use Hilbert's Basis Theorem and other techniques to show that certain examples of modules and ideals are finitely generated.
d) Find the radical of a given ideal in simple cases, and hence find the ideals of varieties defined by simple equations.
e) Decompose simple examples of varieties into unions of irreducible components.

### Syllabus

Commutative rings. Prime and maximal ideals. Finitely generated modules. Hilbert's Basis Theorem. Integral extensions. The correspondence between ideals in the polynomial ring and algebraic sets in affine space. Hilbert's Nullstellensatz. Irreducible decomposition. Further topics.

1. Reminder about commutative rings, ideals and factor rings. Polynomial rings in several indeterminates over a field, and rings which are finitely generated over a field. Zorn's Lemma. Prime ideals and maximal ideals. Localisation.
2. Modules. Isomorphism Theorems. Generators of a module. Free modules. Nakayama's Lemma. Noetherian rings and modules. Hilbert's Basis Theorem. Integral extensions. Noether normalization. The weak Nullstellensatz, that a field extension which is finitely generated as an algebra is algebraic. Tensor Products.
3. Affine varieties. The Zariski topology. The ideal of an affine variety. Radical ideals. The coordinate ring of an affine variety. Hilbert's Nullstellensatz, that the ideal of the affine variety defined by an ideal is the radical of that ideal. Correspondence between the points of an affine variety and the maximal ideals of its coordinate ring. Irreducible decomposition.
4. Topic(s) chosen from: projective varieties and Grassmannians; non-singular varieties and dimension; Bezout's Theorem and the group law for elliptic curves.

### Teaching methods

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

 Delivery type Number Length hours Student hours Lecture 44 1.00 44.00 Private study hours 156.00 Total Contact hours 44.00 Total hours (100hr per 10 credits) 200.00

### Opportunities for Formative Feedback

Regular example sheets

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 3 hr 00 mins 100.00 Total percentage (Assessment Exams) 100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated