MAT802 - Module Theory (NISER, Odd Sem 2026-27)


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

Lectures: Tue 8:30, Wed 9:30, Thu 10:30, Fri 11:30 in M2 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:

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)