MTWRF 11:30-12:25 JR (Journalism Building) 353
Instructor: Sasha Leibman
office:
MW (Math Tower) 406
e-mail:
leibman.1@osu.edu
phone: 614-620-7767
Textbook: D.S.Summit and R.M.Foote, Abstract Algebra, 3rd edition
Lecture notes: Modules
Handouts: The replacement theorem (implying that all bases in a vector space have the same cardinality)
Midterm – Solutions. Review problems
Homework:
| Homework 1 | – due by Wednesday, January 17. Solutions |
| Homework 2 | – due by Tuesday, January 23. Solutions |
| Homework 3 | – due by Tuesday, January 30. Solutions |
| Homework 4 | – due by Tuesday, February 6. Solutions |
| Homework 5 | – due by Tuesday, February 13. Solutions |
| Homework 6 | – due by Tuesday, February 20. Solutions |
| Homework 7 | – due by Tuesday, February 27. Solutions |
Calendar: [LN=Lecture Notes, TB=Text Book]
| January 8: |
Definition, examples, an properties of modules
[LN 1.1-3, TB 10.1]
Submodules [LN 1.4.1-3, TB 10.1] Intersection and sum of submodules [LN 1.4.3-5, TB 10.1] Submodules, generated by subsets [LN 1.5, TB 10.3] |
| January 9: |
Factorization of modules
[LN 1.6, TB 10.2]
Torsion elements of a module and the torsion submodule [LN 1.7, TB 10.1.8] Annihilators [LN 1.8, TB exercises 10.1.9-10] Homomorphisms of modules [LN 1.9.1-7, TB 10.2] |
| January 10: |
The kernel and cokernel of a homomorphism
[LN 1.9.8-11, TB 10.2]
Isomorphism theorems for modules [LN 1.10, TB 10.2] Finitely generated modules as factors of free modules [LN 1.11, TB 10.2] The module Hom(M,N) and the algebra End(M) [LN 1.12, TB 10.2] Schur's lemma [LN 1.13, TB exercises 10.3.9-11] |
| January 11: | Exercises from Sections 10.1, 10.2 – solutions to some exercises |
| January 12: |
Commutative diagrams and exact sequences of module homomorphisms
[LN 1.14.1-5, TB beginning of Section 10.5 ]
The short five lemma and the snake lemma [LN 1.14.6-7, TB 10.5, Proposition 10.24, and exercise 17.1.3] |
| January 16: |
The direct product/sum of two modules as universal objects
(LN 2.1-2, TB 10.3)
Categories and universal objects (LN 2.3, TB Appendix II) |
| January 17: |
The direct product and the direct sum of a family of modules
(LN 2.4, TB 10.3)
The internal direct sum of two submodules (LN 2.5.1-3, TB 10.3) |
| January 18: |
Direct summands and splitting short exact sequences
(LN 2.5.4-6, TB 10.5, pp.383-384)
The internal direct sum of a family of submodules (LN 2.6, TB 10.3) Free modules and bases (LN 3.1, 3.2, TB 10.3) |
| January 19: |
Exercises from Section 10.3
An example where R2≃R as R-modules (TB exercise 10.3.27) |
| January 22: |
Maximal linearly independent subsets and the rank of a module
(LN 3.3, TB 11.1)
ZN is not a free Z-module (TB exercise 10.3.24) |
| January 23: |
The dimension and subspaces of a vector space
(LN 3.4, TB 11.1)
Bilinear mappings of modules (LN 4.1) Tensor product of modules over a commutative ring (LN 4.2, TB 10.4) |
| January 24: | Properties of tensor product (LN 4.3.1-10, TB 10.4) |
| January 25: | Tensor product and torsion (LN 4.3.11-13, TB 10.4) |
| January 26: |
Extension of scalars
(LN 4.4, TB 10.4)
Tensor product of algebras (LN 4.5, TB 10.4) Exercises from Section 10.4 |
| January 29: |
Exercises from Section 10.4
A relation between Hom and tensor product (LN 4.3.14) |
| January 30: |
Tensor product of homomorphisms
(LN 4.6, TB 10.4)
Tensor product of several modules (LN 4.7, TB 10.4) Tensor algebra of a module (LN 4.8, TB 10.4) |
| January 31: |
The symmetric and the exterior algebras of a module
(LN 4.9, TB 10.4)
Symmetric and alternating tensors (LN 4.10, TB 10.4) |
| February 1: |
Differential forms.
Tensor products over non-commutative rings (LN 4.11, TB 10.4) |
| February 2: | Exercises from Section 11.5 |
| February 5: |
Functors. Exact functors
(LN 5.1, TB 10.5)
The functor ⊗K and flat modules (LN 5.2.1-3, TB 10.5) |
| February 6: | Flat modules (LN 5.2, TB 10.5) |
| February 7: |
The functor Hom(K,*) and projective modules
(LN 5.3, TB 10.5)
The functor Hom(*,K) and injective modules (LN 5.4, TB 10.5) |
| February 8: | Right exactness of the functor ⊗K (LN 5.4, TB 10.5) |
| February 9: | Exercises from Section 10.5 |
| February 12: | Dual modules (LN 5.5, TB 11.3) |
| February 13: |
Homomorphisms of free modules of finite rank and their matrices
(LN 6.1, TB 11.2)
Change of basis, transition matrix (LN 6.2, TB 11.2) |
| February 14: | The dual module of a free module of finite rank (LN 6.3, TB 11.3) |
| February 15: |
The rank of modules and of homomorphisms
(LN 6.4.1-7)
Exercises from Section 11.2 |
| February 16: | Exercises from Sections 11.2,3 |
| February 19: | Tensor products of free modules of finite rank (LN 6.5) |
| February 20: | Homomorphisms and bilinear forms as tensors (LN 6.6) |
| February 21: |
Tensor algebras of a free module of finite rank
(LN 6.7, TB 11.4)
The determinant of an endomorphism of a free module of finite rank (LN 6.8, TB 11.4) |
| February 22: |
The inverse endomomorphism of a free module of finite rank
(LN 6.8.5, TB 11.4)
Elementary matrix operations (LN 6.8.6-7, TB 11.4) |
| February 23: |
The cross product in R3
Submodules of free modules of finite rank over PIDs (LN 7.1, TB 12.1.1) |
| February 26: |
Submodules of free modules of finite rank over PIDs
(LN 7.1.1-2, TB 12.1)
Matrices of homomorphisms of free modules of finite rank over PIDs (LN 7.1.3-4, TB 12.1) |
| February 27: |
Reduction of matrices over EDs
(LN 7.1.5-7, TB Exercises 12.1.16-19)
The fundamental theorem of finitely generated modules over PIDs. Invariant factors and elementary divisors (LN 7.2.1-2, TB 12.1) |
| February 28: |
Uniqueness of invariant factors and elementary divisors
(LN 7.2.3-8, TB 12.1)
Computation of invariant factors via relations (LN 7.2.9, TB 12.1) |
| February 29: |
The rational normal form of a matrix over a field
(LN 7.3, TB 12.2)
The Smith normal form of the matrix xI-A. (LN 7.4, TB 12.2) |
| March 1: |
The minimal and the characteristic polynomials of a linear transformation
(LN 7.4, TB 12.2)
The Jordan normal form of a matrix (LN 7.5, TB 12.3) |
| March 4: | Exercises from Sections 12.1-3 |
| March 5: | Midterm |
Plans: