MATH380 (Fall 2020): Modern Algebra I.

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

Lectures: TTh 1:00-2:15pm, zoom. The first class is on 9/1, and the last class is on 12/3.

Office hours: M 10.30-11.30am, F 1-2pm, zoom.

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

Goals: This course serves an introduction to Commutative algebra with a brief discussion of category theory. Commutative algebra references include [AM],[E], and Category theory reference is [R]. There is no required text book.

Grading: Based on 5 homework sets (60%) and a take-home final exam (40%). Homeworks are due on Sept 21, Oct 5, Oct 19, Nov 2 and Dec 2.

Homeworks

Homework 1, due Sept 21, by the end of day.

Homework 2, due Oct 5, by the end of day.

Homework 3, due Oct 20, by the end of day.

Homework 4, due Nov 2, by the end of day.

Homework 5, due Dec 2, by the end of day.

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.

Final exam is here. Due on Dec 16 by the end of day.

Preliminary list of topics:

Schedule:

  • Sept 1, Lecture 1: Rings. Ring homomorphisms. Ideals and quotient rings. Updated notes (9/3).
  • Sept 3, Lecture 2: Wrap-up from Lec 1. Operations with ideals. Maximal ideals. Updated notes (9/4).
  • Sept 8, Lecture 3. Prime ideals. Modules over rings and their homomorphisms. Updated notes (9/9).
  • Sept 10, Lecture 4. Submodules and quotient modules. Finitely generated, free and projective modules. Updated notes (9/11).
  • Sept 15, Lecture 5. Noetherian rings and modules. Hilbert's basis theorem. Updated notes (9/16).
  • Sept 17, Lecture 6. Proof of Hilbert's basis theorem. Artinian and finite length modules. Updated notes (9/18).
  • Sept 22, Lecture 7. Principal ideal domains and modules over them. Updated notes (9/23).
  • Sept 24, Lecture 8. Modules over PID's continued. Localization of rings. Updated notes (9/25).
  • Sept 29, Lecture 9. Localization of rings and modules. Updated notes (10/2) (incl. a revision to Section 2.2 on Oct 1).
  • Oct 1, Lecture 10. Localization of modules, cont'd (incl. revision of Section 2.2 of Lec 9). Categories. Updated notes (10/2).
  • Oct 6, Lecture 11. Categories cont'd. Functors. Updated notes (10/7).
  • Oct 8, Lecture 12. Functor morphisms. Yoneda Lemma. Updated notes (10/13).
  • Oct 13, Lecture 13. Objects representing functors. Products and coproducts. Updated notes (10/13).
  • Oct 15, Lecture 14. Adjoint functors. Updated notes (10/16).
  • Oct 20, Lecture 15. Additive functors between categories of modules. Tensor products of modules. Updated notes (10/30).
  • Oct 22, Lecture 16. Tensor products, cont'd. Updated notes (10/27).
  • Oct 27, Lecture 17. Tensor-Hom adjunction. Tensor product of algebras. Updated notes (10/28).
  • Oct 29, Lecture 18. Tensor product of algebras. Exactness of additive functors. Updated notes (11/4).
  • Nov 3, Lecture 19. Basic properties of left/right exact functors. Localization as base change functor. Projective and flat modules. Updated notes (11/4).
  • Nov 5, Lecture 20. Integral extensions and integral closures for rings. Updated notes, now explaining how to remove the Noetherian assumption (11/10).
  • Nov 10, Lecture 21. Integral closures cont'd, incl. normalizations and normal rings, and finiteness for integral closures. The Noether normalization lemma.
    • Updated notes (11/11).
  • Nov 12, Lecture 22. Hilbert's Nullstellensatz. Algebraic subsets in F^n vs radical ideals in F[x_1,...,x_n]. Updated notes (11/13).
  • Nov 17, Lecture 23. Prime ideals and irreducibility. Irreducible components. Polynomial maps vs algebra homomorphisms. Updated notes (11/18).
  • Nov 19, Lecture 24. Affine varieties. Geometric meaning of localization. Updated notes (11/21).
  • Dec 1, Lecture 25. Modules over local rings and Nakayma lemma. Geometric meaning of modules. Updated notes (12/2).
  • Dec 3, Lecture 26 (a.k.a. the last one): Geometric meaning of projective modules. Preliminary notes.

    • References:
    [AM] M. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, 1969.
    [E] D. Eisenbud, Commutative Algebra: With a View Toward Algebraic Geometry, GTM 150, Springer-Verlag, 2004.
    [R] E. Riehl, Category theory in context. Available here.