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Instructor: Chitrabhanu Chaudhuri (office M-227, email: chitrabhanu [at] niser [dot] ac [dot] in) |
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:
- Lang, S.; Algebra, Revised Third Edition, Springer, GTM 211.
- Weibel, Charles A.; An Introduction to Homological Algebra, Cambridge University Press, 1995.
- MIT Undergraduate Algebra II - Student Notes
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, problem set)
- Week 2: Module homomorphisms, Quotient modules, Isomorphism theorems, Direct sums, Generating sets. (notes, problem set)
- Week 3: Free modules, Presentations. (notes)
- Week 4: Modules over PID's. (notes, problem set)
- Week 5: Smith Normal Form and applications, Noetherian Rings, Hilbert basis theorem. (notes, problem set)
- Week 6: Tensor product of modules. (notes, problem set)
- 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, problem set)
- Week 11: Flat modules, Localisation. (notes, problem set)
- Week 12: Chain complexes and homology, Projective resolutions. (notes)
- Week 13: Tor and Ext (notes, problem set)
- Week 14: Direct and Inverse limits. (notes, problem set)
Topics not covered: Tensor, Symmetric and Exterior algebras, Determinants. Length of Modules, Noetherian and Artinian modules. (These are covered in the MA877 Commutative Algebra.)