2019/20 Taught Postgraduate Module Catalogue

MATH5195M Advanced Commutative Rings and Algebraic Geometry

20 creditsClass Size: 40

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

Taught: Semester 2 View Timetable

Year running 2019/20

Pre-requisite qualifications

MATH2025 or MATH2026 or MATH2027, or equivalent.

This module is mutually exclusive with

 MATH3195 Commutative Rings and Algebraic Geometry MATH5253M Commutative Algebra and Algebraic Geometry

Module replaces

MATH5253M

This module is not approved as an Elective

Module summary

Algebraic geometry is the study of the solution sets of polynomial equations, just as linear algebra is the study of the solution sets of linear equations. Polynomial equations in several variables are fundamental in both pure and applied mathematics since they allow us to describe many shapes, such as circles, ellipses and parabolas. Examples include the trajectory of an object moving under gravity and the points that can be reached by a robot arm.Sets of solutions of polynomial equations, known as affine varieties, have beautiful properties. The role of polynomials in their description means that they can be studied via commutative rings, such as polynomial rings. There is thus a rich interplay between the geometry of varieties and the algebraic properties of commutative rings.Two lines in the plane intersect, unless they are parallel. In projective geometry, this defect is remedied by adding "points at infinity" where parallel lines intersect. The resulting space is called the projective plane, and can be constructed by taking all of the lines passing through the origin in three-dimensional space. The theory of varieties can be extended to the projective case, and many interesting geometric objects arise in this way.The module will introduce the basic ideas of algebra geometry, including projective varieties, via commutative rings. There will be an emphasis on concrete examples and explicit calculations.

Objectives

To give an introduction to the theory of algebraic geometry via affine and projective algebraic varieties.
To give an introduction to the theory and properties of the commutative rings which arise in the description of algebraic varieties.
To illustrate how commutative rings can be used to study algebraic varieties, and how the properties of each are interrelated.
In particular, to study the primary decomposition of ideals and the irreducible components of varieties and how these are related.
To introduce advanced topics in algebraic geometry.
To focus on the use of interesting examples to illustrate ideas in the theory.
To develop the skills of rigorous logical argument and problem-solving in the context of commutative rings, algebraic geometry and their interaction.

Learning outcomes
On completion of this module, students should be able to:

(a) Define and use key concepts in commutative rings and algebraic geometry;
(b) State and prove some of the basic results in commutative rings and algebraic geometry;
(c) Solve simple problems concerning varieties.
(d) Understand the relationship between ideals and varieties;
(e) Understand the basic properties of projective varieties.

Syllabus

Revision of rings and polynomials.
Commutative rings and ideals.
Varieties and an algebra-geometry dictionary.
Noetherian rings and modules.
Decomposition of ideals and irreducible components of varieties.
Brief discussion of application(s), e.g. geometric description of the points that can be reached by a robot arm.
Projective varieties, homogeneous coordinates and homogeneous ideals.
Additional topic(s) chosen from the following or similar: Groebner basis, further projective varieties, fundamental examples such as the variety of subspaces of a vector space (the Grassmannian), singular points of varieties and tangent spaces, function fields, the Zariski topology on a variety.

Teaching methods

 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

Private study

Study and revision of course material.

Completion of assignments and assessments.

Opportunities for Formative Feedback

Regular problem sheets.

Methods of assessment

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