5M: Commutative Algebra and Algebraic Geometry MATHS5045

  • Academic Session: 2019-20
  • School: School of Mathematics and Statistics
  • Credits: 20
  • Level: Level 5 (SCQF level 11)
  • Typically Offered: Either Semester 1 or Semester 2
  • Available to Visiting Students: No
  • Available to Erasmus Students: No

Short Description

The aim of this course is to provide an introduction to the subject of algebraic geometry, that is, the study of affine or projective varieties or more general geometric spaces called schemes, using algebraic methods.

Timetable

2 hours of lectures a week, over 11 weeks.

1 hour tutorial a week over 10 weeks (or equivalent)

Requirements of Entry

Mandatory Entry Requirements

 

4H Galois Theory and 4H Number Theory

 

 

Recommended Entry Requirements

Assessment

Assessment

 

100% Final Exam

 

 

Reassessment

In accordance with the University's Code of Assessment reassessments are normally set for all courses which do not contribute to the honours classifications. For non honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students, and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions are listed below in this box.

Main Assessment In: April/May

Are reassessment opportunities available for all summative assessments? No

Reassessments are normally available for all courses, except those which contribute to the Honours classification. For non Honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions for this course are described below. 

Course Aims

The aim of this course is to provide an introduction to the subject of algebraic geometry, that is, the study of affine or projective varieties or more general geometric spaces called schemes, using algebraic methods. Possible additional topics are category theory and homological algebra and applications in algebraic geometry, or the relation to complex analytic geometry.

Intended Learning Outcomes of Course

By the end of this course students will be able to:

 

 Demonstrate knowledge of the central definitions and facts of selected topics in commutative algebra and algebraic geometry and use these to solve problems of a numerical or logical nature. Four possible topics are:

1. Affine varities and schemes (irreducibility of algebraic sets, the Zariski topology, Hilbert's basis theorem, the Nullstellensatz, prime ideals)

2. Localisation (local rings, Nakayama's lemma, support of a module, associated primes).

3. Dimension and regularity (Krull, Chevally and embedded dimension, regular local rings).

4. Projective varieties (homogeneous coordinate rings, projective curves).

 

 Apply the language and methods of category theory and homological algebra to solve algebro-geometric problems (exactness of localisation, projectivity vs. local freeness, global dimension, Serre's theorem, sheaf cohomology, Riemann-Roch theorem)

 

 Apply algebro-geometric methodology to solve problems in complex analytic

geometry (complex manifolds, analytic functions, Riemannian surfaces)

Minimum Requirement for Award of Credits

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.