## MAS439 Commutative Algebra and Algebraic Geometry

Note: Information for future academic years is provisional. Timetable information and teaching staff are especially likely to change, but other details may also be altered, some courses may not run at all, and other courses may be added.

Both semesters, 2017/18 | 20 Credits | ||||

Lecturer: | Dr Paul Johnson | Home page | Timetable | Reading List | |

Aims | Outcomes | Assessment | Full Syllabus |

This module will develop both the algebraic and geometric theories of commutative rings and modules. The most basic form of interaction between these two subjects can be seen as the relationship between polynomials and their graphs. This relationship can then be extended to the relationship between ideals in polynomial rings and the corresponding vanishing loci in affine space. At a basic level, this module is about turning algebra into pictures and describing pictures using algebra. To do so, we will study many important properties of commutative rings and their modules, and then explore the geometric counterparts of these properties. Interpreted in the context of the complex numbers, this analogy between algebra and geometry reflects many of the basic intuitions one has about graphs of polynomial equations. Once the basic dictionary is set up however, it can be applied in more exotic situations, such as over finite fields.

**Prerequisites:** MAS220 (Algebra)

No other modules have this module as a prerequisite.

## Outline syllabus

Part A

- Definition of a ring and examples
- Homomorphisms and subrings
- Ideals and quotient rings
- Algebras and polynomials
- Noetherian rings
- Algebraic subsets
- Statement of Hilbert's Nullstellensatz
- Affine varieties
- Regular maps
- Localization

Part B

- Modules, isomorphism theorems
- Localization for modules
- Nakayama's Lemma
- Tensor products
- Integral extensions
- Noether normalization
- Proof of Nullstellensatz
- Dimension theory

## Aims

- To establish a basic groundwork of knowledge in commutative algebra.
- To apply that knowledge to study problems of a geometric nature.
- To develop an appropriate perspective on the techniques discussed.
- To understand the connections between algebra and geometry.

## Learning outcomes

- Understand the basic definitions concerning rings and modules;
- Understand homomorphisms and isomorphisms of rings and modules and be able to construct examples;
- Be able to describe subrings, ideals and submodules in terms of systems of generators;
- Understand the definitions of prime, maximal and radical ideals and be able to recognise them in examples;
- Understand and be able to work with quotient rings and modules;
- Understand and be able to work with localized rings and modules;
- Understand the notions of k-algebras and homomorphisms of k-algebras;
- Study Noetherian rings and modules and understand the Hilbert basis theorem;
- Understand the different versions of Hilbert's Nullstellensatz;
- Be able to compute the ring of functions on an algebraic set;
- Understand the definition of an affine variety and be able to compute the irreducible components of an algebraic set;
- Understand the relationship between morphisms of varieties and their co-ordinate rings;
- Understand the field of fractions of an integral domain and the geometric interpretation in terms of rational maps;
- Understand integral extensions;
- Understand Nakayama's Lemma and it's consequences;
- Understand tensor products of modules and algebras and be able to compute them.

40 lectures, no tutorials

## Assessment

Assessment will be via 20 problem sheets, assigned weekly through both semesters. The final mark will be the sum of the 16 best marks obtained (i.e. the worst four marks will be discarded). These problems will assess the student's knowledge of the key concepts, their ability to synthesize and generalize these concepts, and their ability to present proofs logically and coherently. Clear standards and sample solutions will be provided to students at the beginning of the module.

## Full syllabus

**Part A**

**1. Rings and Homomorphisms**

Definition of a ring; polynomial rings, residue class rings; integral domains and fields; definition of a ring homomorphism.

**2. Subrings and ideals**

Subrings and their generators; ideals and their generators; operations on ideals.

**3. Quotient rings**

Quotient rings; universal property; isomorphism theorem; prime, maximal and radical ideals; ideals in quotient rings.

**4. Algebras**

Definitions of k-algebras and k-algebra homomorphisms; examples; algebras of functions; generators for algebras; polynomial algebras and their universal property.

**5. Noetherian rings**

Definition in terms of ascending chains; finite-generation of ideals; Hilbert Basis Theorem; Artinian rings.

**6. Hilbert's Nullstellensatz**

Algebraic subsets; correspondence between ideals and subsets; statement of the Nullstellensatz.

**7. Affine varieties**

Zariski topology; irreducible components; co-ordinate rings; regular maps.

**8. Localization**

Field of fractions of an integral domain; general localization; universal property; ideals in localized rings.

**Part B****1. Modules**

Definitions; examples; submodules; quotient modules; isomorphism theorems; chain conditions and Noetherian modules.

**2. Tensor products**

Definitions; universal property; examples.

**3. Localization for modules**

Definition; basic properties; examples.

**4. Nakayama's Lemma**

Statement and proof; geometric interpretation.

**5. Integral extensions**

Definition; basic examples; geometric interpretation; Noether normalization; Proof of Hilbert's Nullstellensatz.

**6. Further topics**

Projective varieties; affine pieces; regular and rational maps; tangent spaces.

## Reading list

Type | Author(s) | Title | Library | Blackwells | Amazon |
---|---|---|---|---|---|

B |
Atiyah and Macdonald | Introduction to commutative algebra | 512.8 (A) | Blackwells | Amazon |

B |
Eisenbud | Commutative algebra with a view toward algebraic geometry | 512.8 (E) | Blackwells | Amazon |

B |
Reid | Undergraduate algebraic geometry | 513.6 (R) | Blackwells | Amazon |

B |
Reid | Undergraduate commutative algebra | 512.8 (R) | Blackwells | Amazon |

(

**A**= essential,

**B**= recommended,

**C**= background.)

Most books on reading lists should also be available from the Blackwells shop on Mappin Street.