## MAS6320 Algebra II

 Both semesters, 2021/22 20 Credits Lecturer: Dr Moty Katzman Timetable 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 (seen as elements in a ring) and their graphs. This relationship can then be extended to the relationship between certain kinds of ideals in a ring and the geometric object ("graph") such an ideal describes. At a basic level, this module can be seen as the study of 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 objects that arise from various algebraic properties. Interpreted in the context of complex numbers, this analogy between algebra and geometry reflects many of the basic intuitions one has about graphs of polynomial equations, but we will also consider the geometry that comes about in more exotic situations, such as over finite fields.

There are no prerequisites for this module.
No other modules have this module as a prerequisite.

## Outline syllabus

Part A
• Review: rings and abelian groups
• Tensor products
• Modules
• Direct sums and generators; Chinese Remainder Theorem
• Local rings
• Algebras
• Noetherian rings and modules
• Integral extensions and Noether Normalization
• Localization

Part B
• Affine algebraic varieties
• Hilbert's Nullstellensatz
• Regular functions
• Maps between varieties
• Dimension theory
• Derivations and "calculus" for algebraic varieties

## 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

By the end of the unit, a candidate will be able to demonstrate the ability to: - Classify modules over certain rings (e.g., abelian groups of given order); - Understand the theory of ideals and modules; - Make computations in commutative algebra.
40 lectures, no tutorials

## Assessment

Assessment will be via 20 assigned problems, each with an equal weighting of 5%. 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. Individual problems will be assessed via descriptor, with clear standards and sample solutions provided to students at the beginning of the module.

## Full syllabus

Part A

1. Review: rings and abelian groups
Direct sum of abelian groups; Fundamental theorem of finitely generated abelian groups; Definition of a ring; definition of a ring homomorphism; relationship between rings, abelian groups, and monoids; Z as the initial ring
2. Tensor products
Bilinear maps; universal properties; some basic properties and examples; definition of a ring revisited; Hom-tensor adjunction
3. Modules
Definitions, elementary and abstract; homomorphisms; quotients; ideals as modules; exact sequences
4. Prime ideals
Definitions of prime and maximal ideals; relationship with quotients; radicals; Nakayama's Lemma
5. Algebras
Definitions of R-algebras and R-algebra homomorphisms; examples; generators for algebras; free algebras
6. Noetherian rings and modules
Definitions; Zorn's Lemma; Chain conditions; relationship with exact sequences; Hilbert Basis Theorem
7. Integral extensions
Definitions; equivalent characterizations; algebraic independence; Noether Normalization
8. Localization
Definition of the localization of a ring; universal property; examples; definition of the localization of a module; exactness of localization

Part B

1. Affine algebraic varieties
Definitions of algebraic set, irreducibility, and affine algebraic variety; examples; basic properties; open and closed sets; quasi-affine varieties
2. Hilbert's Nullstellensatz
The ideal of vanishing functions, I(X); correspondences between ideals and varieties; various statements of Hilbert's Nullstellensatz
3. Regular functions
Functions on a variety and the coordinate ring; ring of local functions at a point; examples
4. Maps between varieties
Definition; basic examples; relationship between maps of varieties and maps of coordinate rings; different kinds of maps and their associated algebraic properties
5. Dimension theory
Possible definitions of dimension; relationships between these
6. Derivations and "calculus" on varieties
Module of derivations; tangent vectors; Kähler differentials