MA802 - Module Theory (NISER, Odd Sem 2024-25)


Snake Lemma Instructor: Chitrabhanu Chaudhuri
(office M-227, email: chitrabhanu at niser dot ac dot in)

Lectures: Mon 8:30, Tue 9:30, Wed 10:30, Thu 11:30 in M4 at SMS

Office hours: By appointment.

Academic Calendar, Course Resources.
Content:
Modules, submodules, module homomorphisms, quotient modules, isomorphism theorems, Direct Sum of modules, finitely generated modules, Free modules, structure theorem of finitely generated modules over PID. Tensor product of modules.
Over commutative rings with identity: Categories and Functors, exact functors, Hom and Tensor functors, Localization of modules, Direct and Inverse Limit of modules, Projective, Injective and Flat modules, Ext, Tor. Algebras, Tensor Algebras, Symmetric Algebras, Exterior Algebras, Determinants. Length of Modules, Noetherian and Artinian modules, Hilbert Basis Theorem.

Texts:
  • Dummit, D.S.; Foote, R.M.; Abstract Algebra, Third Edition, John Wiley & Sons.
  • Rotman, J.; An Introduction to Homological Algebra.
  • Sing, Balwant; Basic Commutative Algebra, World Scientific, 2011.
References: Evaluation: There will be weekly problem sets. The points dstribution will be: Problem sets: 30 pts, Midsem 30 pts, Final 40 pts.
Exam Dates: Midsem 3rd October 2024, 10:00-12:00 hrs. Final 6th December 2024, 10:00 - 13:00 hrs.

Material covered:
  • Week 1: Definition and examples of left and right modules, submodules, module homomorphisms. (notes, PS1)
  • Week 2: Module homomorphisms, Quotient modules, Isomorphism theorems, Direct sums, Generating sets. (notes, PS2)
  • Week 3: Free modules, Presentations. (notes)
  • Week 4: Modules over PID's. (notes, PS3)
  • Week 5: Smith Normal Form and applications, Noetherian Rings, Hilbert basis theorem. (notes, PS4)
  • Week 6: Tensor product of modules. (notes, PS5)
  • Week 7: Categories and functors. (notes)
  • Week 8: Functors, Natural transformations, Universal objects in a category. (notes)
  • Week 9: Infinite direct sums and direct products, Exact sequences, Exactness properties of Hom and Tensor. (notes)
  • Week 10: Projective and Injective modules. (notes, PS6)
  • Week 11: Flat modules, Localisation. (notes)
  • Week 12: Chain complexes and homology, Projective resolutions. (notes)
  • Week 13: Tor and Ext (notes, PS7)
  • Week 14: Direct and Inverse limits. (notes)
Topics not covered: Tensor, Symmetric and Exterior algebras, Determinants. Length of Modules, Noetherian and Artinian modules. (These are covered in the MA877 Commutative Algebra.)