MATH380 (Fall 2023): Algebra.

Instructor: Prof. Ivan Loseu (email: ivan.loseu@yale.edu)

Lectures: TTh 1:00-2:15pm, location: KT 201. The first class is on 8/31, and the last class is on 12/7.

Office hours: M 2.30-4 and T 10.30-11.30 in KT 715.

This document contains an important information on the class content, homeworks, etc. Please read it!

Goals: This course serves as an introduction to Commutative algebra with a discussion of category theory and a brief discussion of connections to Algebraic geometry and Number theory. This document also contains a list of references. There is no required textbook.

Grading: Based on 6 homework sets (60%) and a take-home final exam (40%). Due dates for homeworks are TBA.

Rules: For homework sets, peer discussion is allowed but only after your own thinking; you are also required to write your solution on your own, if it (partially) comes from a discussion. Homework sets and the take-home final should be submitted via Canvas. You are encouraged to texify your work, but scans of handwriting are also acceptable. Overdue submissions will not be accepted.

Preliminary list of topics together with their descriptions is in this document as well. You can also find references there.

Assignments:

  • Homework 1, due Sept 19.
  • Homework 2, due Oct 3.
  • Homework 3, due Oct 17.
  • Homework 4, due Nov 2.
  • Homework 5, due Nov 16.
  • Homework 6, due Dec 7.
  • Final, due Dec 17.

    Index of terminology.

    Schedule: The first chapter in this class is Basics of Commutative algebra.

  • Aug 31, Lecture 1: Rings, ideals and modules, part I, notes: The goal of this part is to introduce key concepts of this class. We will also be mentioning what's going to be studied later in the class. In this particular lecture, we introduce rings together with easy examples and constructions, ring homomorphisms, ideals and, time permitting, quotient rings.
  • Sept 5, Lecture 2: Rings, ideals and modules, part II, notes: we finish our discussion of quotient rings. We'll then discuss operations with ideals. The main part is a discussion about maximal ideals. We state Nullstellensatz and have a brief discussion of the relevance of ideals in Algebraic geometry.
  • Sept 7, Lecture 3: Rings, ideals and modules, part III, notes: Prime ideals and motivations from Number theory. Modules: definitions and basic examples.
  • Sept 12, Lecture 4: Rings, ideals and modules, part IV, notes: Constructions with modules. Sub- and quotient modules. Finitely generated modules.
  • Sept 14, Lecture 5: Noetherian rings and modules, I, notes: Module overflow: free and projective modules. Noetherian rings and modules. The Hilbert basis theorem.
  • Sept 19, Lecture 6: Noetherian rings and modules, II, notes: Finitely generated algebras. Properties of Noetherian modules. Artinian modules and rings.
  • Sept 21, Lecture 7: Classification of modules over PID's, I notes: More on PID's. Classification theorem and its special cases. Proof of the classification theorem.
  • Sept 26, Lecture 8: Classification of modules over PID's, II, notes: We finish the proof of the classification theorem from Lecture 6. And start to discuss localization of rings.
  • Sept 28, Lecture 9: Localization of rings and modules, I, notes: We give further examples and properties of the localizations of rings and start to discuss the localization of modules.
  • Oct 3, Lecture 10: Localization of rings and modules, II, notes: Localizations and sub- and quotient modules and some applications to localizations of rings. Submodules in localizations of modules. Local rings.

    We proceed to the second part: introduction to Category theory and its connections to Commutative algebra.

  • Oct 5, Lecture 11: Categories, functors and functor morphisms, part I, notes: We introduce categories and start our discussion of functors.
  • Oct 10, Lecture 12: Categories, functors and functor morphisms, part II, notes (updated 10/12): We continue our discussion of functors and introduce functor morphisms. We state and prove the Yoneda lemma.
  • Oct 12, Lecture 13: Categories, functors and functor morphisms, part III, notes: We discuss the Yoneda lemma further and introduce its application: the concept of an object representing a functor. Then we discuss products in categories.
  • Oct 17, Lecture 14: Categories, functors and functor morphisms, part IV, notes: Coproducts. Adjoint functors (definition, examples and uniqueness).
  • Oct 24, Lecture 15 (with Prof. Boixeda Alvarez): Tensor products, part I, notes: Definition, existence, and examples of tensor products of modules.
  • Oct 26, Lecture 16 (with Prof. Boixeda Alvarez): Tensor products, part II, notes: Tensor product cont'd. Tensor-Hom adjunction.
  • Oct 31, Lecture 17: Tensor products, part III, notes: Tensor-Hom adjunction continued. Tensor product of algebras.

    The Evil Categorical Pumpkin!

  • Nov 2, Lecture 18: Exactness and projective modules, part I, notes: Additive functors. Exactness of functors.
  • Nov 7, Lecture 19: Exactness and projective modules, part II, notes: Projective and flat modules. Projective vs locally free.
  • Nov 9, Lecture 20: Exactness and projective modules, part III, notes: Nakayama lemma. Projective modules over local rings.

    We proceed to the third and final part: connections of Commutative algebra with Algebraic geometry and Algebraic number theory. Both require talking about integral and finite extension of rings.

  • Nov 14, Lecture 21: Integral and finite extensions of rings, I, notes: Definitions and basic results.
  • Nov 16, Lecture 22: Integral and finite extensions of rings, II, notes: Integral closures cont'd. The Noether normalization lemma.
  • Nov 28, Lecture 23: Connections to Algebraic geometry, I, notes: Nullstellensatz. Algebraic subsets and their algebras of functions.
  • Nov 30, Lecture 24: Connections to Algebraic geometry, II, notes: Prime ideals and irreducibility. Geometric singnifance of localizations.
  • Bonus lecture B1: Connections to Algebraic geometry, III, notes: Algebra homomorphisms vs polynomial maps. Category of affine varieties/schemes. More general varieties/schemes.
  • Bonus lecture B2: Connections to Algebraic geometry, IV, notes: Vector bundles on C^\infty-manifolds. Modules, algebro-geometrically.
  • Dec 5, Lecture 25: Connections to Algebraic Number theory, I, notes: Dedekind domains. Unique factorization of ideals.
  • Dec 7, Lecture 26: Connections to Algebraic Number theory, II, notes: Unique factorization of ideals cont'd. Class groups.
  • Bonus lecture B3: Connections to Algebraic geometry, V, notes: Dedekind domains and class groups in Algebraic geometry.