MTWRF 11:30-12:25 UH (University Hall) 074
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 (version of 3/5)
Additional texts: Zorn's lemma and dimension of vector spaces
Midterm on March 5. – Solutions
Homework:
| Homework 1 | – due by Wednesday, January 21. / Solutions. |
| Homework 2 | – due by Tuesday, January 27. / Solutions. |
| Homework 3 | – due by Tuesday, February 3. / Solutions. |
| Homework 4 | – due by Tuesday, February 10. / Solutions. |
| Homework 5 | – due by Tuesday, February 17. / Solutions. |
| Homework 6 | – due by Tuesday, February 24. / Solutions. |
| Homework 7 | – due by Tuesday, March 3. / Solutions. |
Calendar: [LN=Lecture Notes, TB=Text Book]
| January 12: | Definition, examples, and properties of modules [LN 1.1-3, TB 10.1] |
| January 13: |
Submodules
[LN 1.4.1-3, TB 10.1]
Intersections and sums of submodules [LN 1.4.3-5, TB 10.1] Submodules generated by subsets [LN 1.5, TB 10.3] Quotient modules [LN 1.6, TB 10.2] Torsion elements of a module and the torsion submodule [LN 1.7, TB 10.1.8] |
| January 14: |
Annihilators
[LN 1.8, TB exercises 10.1.9-10]
Homomorphisms of modules [LN 1.9.1-7, TB 10.2] The kernel and cokernel of a homomorphism [LN 1.9.8-11, TB 10.2] |
| January 15: |
Monomorphisms, epimorphisms, and isomorphisms of modules
[LN 1.9, TB 10.2]
Isomorphism theorems for modules [LN 1.10, TB 10.2] Finitely generated modules as factors of free modules of finite rank [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 16: | Exercises from Sections 10.1, 10.2 — solutions |
| January 20: |
Exercises from Section 10.2
Commutative diagrams and exact sequences of module homomorphisms [LN 1.14.1-5, TB beginning of Section 10.5 ] |
| January 21: |
The short five lemma and the snake lemma
[LN 1.14.6-7,
TB 10.5, Proposition 10.24, and exercise 17.1.3]
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 22: |
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 23: |
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] |
| January 26,27: |
The Chinese remainder theorem and p-primary components of modules
[LN 2.7, TB exercises in 10.3]
Free modules and bases [LN 3.1, 3.2, TB 10.3] The maximal free submodule and the rank of a module [LN 3.3, TB 10.3] Vector spaces and dimension [LN 3.4, TB 10.3] |
| January 28: |
The direct product of free modules need not be free
[TB exercise 10.3.24]
An example where R2≃R as R-modules [TB exercise 10.3.27] |
| January 29: |
Bilinear mappings of modules
[LN 4.1]
Tensor product of two modules over a commutative ring [LN 4.2, TB 10.4] Properties of tensor products [LN 4.3.1-5, TB 10.4] |
| January 30: | Properties and examples of tensor products [LN 4.3.6-12, TB 10.4] |
| February 2: |
The tensor product of a module over an ID and the field of fractions
[LN 4.3.13, TB 10.4]
The tensor product of two algebras [LN 4.5, TB 10.4] Extension of scalars [LN 4.4, TB 10.4] The isomorphism Hom(M1⊗M2,N)≅ Hom(M1,Hom(M2,N)) [LN 4.3.14, TB 10.4] |
| February 3: |
The tensor product of two homomorphisms
[LN 4.6, TB 10.4]
The tensor product of several modules [LN 4.7, TB 10.4] The tensor algebra of a module [LN 4.8, TB 11.5] |
| February 4: |
The symmetric and the exterior algebras of a module
[LN 4.9, TB 11.5]
Symmetric and alternating tensors [LN 4.10, TB 11.5] Tensor products over non-commutative rings [LN 4.11, TB 10.4] |
| February 5: |
Functors. Exact functors
[LN 5.1, TB 10.5]
Functor *⊗K and flat modules [LN 5.2.1-5, TB 10.5] |
| February 6: | Flat modules [LN 5.2, TB 10.5] |
| February 9: | Functor Hom(K,*) and projective modules [LN 5.3, TB 10.5] |
| February 10: |
Functor Hom(*,K) and injective modules
[LN 5.4, TB 10.5]
Right exactness of *⊗K [LN 5.4, TB 10.5] |
| February 11: | 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 16: |
The dual module of a free module of finite rank
[LN 6.3, TB 11.3]
The dimension of vector spaces and the rank of homomorphisms [LN 6.4.1] |
| February 17: |
The rank of modules over integral domains
[LN 6.4]
Tensor products of free modules of finite rank [LN 6.5] |
| February 18: | Homomorphisms as tensors [LN 6.6] |
| February 19: |
Bilinear forms
[LN 6.6]
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 20: | Properties of determinant [LN 6.8, TB 11.4] |
| February 23: | Exercises from Sections 11.2,3,4,5 |
| February 24: | Submodules of free modules of finite rank over PIDs [LN 7.1.1-2, TB 12.1] |
| February 25: |
Matrices of homomorphisms of free modules of finite rank over PIDs
[LN 7.1.3-7, TB 12.1]
The fundamental theorem of finitely generated modules over PIDs. Invariant factors and elementary divisors [LN 7.2.1-2, TB 12.1] |
| February 26: | Uniqueness of invariant factors and elementary divisors [LN 7.2.3-8, TB 12.1] |
| February 27: |
Computation of invariant factors via relations
[LN 7.2.9, TB 12.1]
The rational normal form of a matrix over a field [LN 7.3, TB 12.2] |
| March 2: |
"The elementary divisors 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] The minimal and the characteristic polynomials of a linear transformation [LN 7.4, TB 12.2] |
| March 3: | The Jordan normal form of a matrix [LN 7.5, TB 12.3] |
| March 4: | Exercises from Sections 12.1,2,3 |
| March 5: | Midterm |
Plans:
Midterm