| Tuesday |
Thursday |
Aug. 21
In Lecture:
GROUP THEORY
- Towers of subgroups, composition series
- Theorem of Schreier about refinements of towers
- Theorem of Jordan-Hölder
Book reference:
HOMEWORK 1
(due at the beginning of the lecture on 08/30)
| Aug. 23
In Lecture:
- PSL2(F) is simple if #F ≥ 4
Book reference:
|
Aug. 27
In Lecture:
- Solvable groups, derived series
- Nilpotent groups
- Central series
Book reference:
| Aug. 30
In Lecture:
- Recap central series, nilpotent groups
- Frattini's Argument
CATEGORY THEORY
- Categories
- Concrete categories
- Free objects
Book reference:
- D-F: 6.1 (nilpotent groups, Frattini's Argument)
- L: I.11
- D-F: Appendix II
HOMEWORK 2
(due at the beginning of the lecture on 09/11)
|
Sep. 04
In Lecture:
- Free objects
- Free groups
- Universal, couniversal objects
- Products, coproducts
Book reference:
| Sep. 06
In Lecture:
- Products and coproducts in the category R-Mod
- Products and coproducts in the category Grp
- Products (resp. coproducts) are couniversal (resp. universal)
- Functors
Book reference:
|
Sep. 11
In Lecture:
- Dual/opposite category, duality
- Natural transformations
- Equivalence of categories
- Skeleton of a category
MODULE THEORY
Book reference:
- L: I.11, I.12
- D-F: Appendix II
- D-F: 10.3 (module theory)
HOMEWORK 3
(due at the beginning of the lecture on 09/20)
| Sep. 13
In Lecture:
Please check on ICON to see whether it is your turn to work problems on the board.
|
Sep. 18
In Lecture:
- Free modules
- Products and Hom functors
- Bimodules and Hom functors
- Tensor products of modules
- The universal property of the tensor product
Book reference:
| Sep. 20
In Lecture:
- The universal property of the tensor product
- Tensor product functors
- Bimodules and tensor products
- The functors HomR(R,-) and R⊗R- are
naturally isomorphic
Book reference:
HOMEWORK 4
(due at the beginning of the lecture on 10/02)
|
Sep. 25
In Lecture:
- Coproducts and tensor products
- Tensor products of free modules
- Associativity of the tensor product
- Tensor product of algebras
- Exact sequences
Book reference:
| Sep. 27
In Lecture:
- Equivalent short exact sequences
- The Short Five Lemma
- Split exact sequences
- Exact functors
- HomR(M,-) and HomR(-,M) are left exact
Book reference:
|
Oct. 02
In Lecture:
- M⊗R- and -⊗RN are right exact
- Projective modules
- Injective modules
- Baer's criterion
Book reference:
| Oct. 04
In Lecture:
Please check on ICON to see whether it is your turn to work problems on the board.
TAKE-HOME MIDTERM EXAM (due at the beginning of the lecture on 10/18 - CANCELLED)
Please read the RULES before talking to anyone about this!!!
You should basically do this on your own or by talking to me!
THE MIDTERM EXAM HAS BEEN CANCELED ON FRIDAY, OCTOBER 12, 2018, at 1:40PM, DUE TO THE FACT THAT SOME OF YOU MISUNDERSTOOD THE RULES ON THE FIRST PAGE.
|
Oct. 09
In Lecture:
- Direct sums of projective modules are projective, direct products of injective modules
are injective. Direct sumands of projective modules are projective.
Direct sumands of injective modules are injective.
- Divisible modules
- Connection to injective modules
- Generators and cogenerators for R-Mod
- Q/Z is an injective cogenerator for Z-Mod
- R* = HomZ(R,Q/Z) is an injective
cogenerator for R-Mod
Book reference:
| Oct. 11
In Lecture:
- Every left R-module can be embedded into an injective left R-module
- Cofree modules
- Equivalent formulations for injective modules
- Flat modules
- Adjoints
Book reference:
HOMEWORK 5
(due at the beginning of the lecture on 10/23)
|
Oct. 16
In Lecture:
- Noetherian and Artinian rings and modules
- If R is left Noetherian (resp. left Artinian), then
every finitely generated R-module is Noetherian (resp. Artinian)
- More characterizations of Noetherian and Artinian rings and modules
- Composition series of modules
Book reference:
- L: Ch. X.1 & X.7
- R: Sect. 8.2
| Oct. 18
In Lecture:
- Schreier's refinement theorem
- Jordan-Hölder Theorem
- A module has a composition series iff it is Noetherian and Artinian
- The Jacobson radical
Book reference:
- L: Ch. X.1, X.7
- R: Sect. 8.2
|
Oct. 23
In Lecture:
- Nakayama's Lemma
- Nilpotent ideals
- If R is Artinian, the Jacobson radical is nilpotent
- Semisimple rings and modules
Book reference:
- L: Ch. X.1, X.7
- R: Sect. 8.2, 8.3
HOMEWORK 6
(due at the beginning of the lecture on 11/01)
| Oct. 25
In Lecture:
- A ring R is semisimple iff all left R-modules are semisimple
- A ring R is semisimple iff all left R-modules are projective
- If a ring R is semisimple then R is both left Artinian and left Noetherian
- A ring R is semisimple iff R is left Artinian and J(R)=0
Book reference:
|
Oct. 30
In Lecture:
- If R is left Artinian, then R is left Noetherian
- Semisimple group algebras (Maschke's Theorem)
- Semisimple matrix rings
- Idempotents: orthogonal, primitive, primitive central
- Decompositions of 1R
Book reference:
| Nov. 01
In Lecture:
- Connection between decompositions of 1R and
decompositions of R
as a left module over itself
and as a ring
- If RR has a composition series, then there is a primitive
central decomposition of 1R.
- Wedderburn's structure theorem for semisimple rings
Book reference:
|
Nov. 06
In Lecture:
- Wedderburn's structure theorem for semisimple rings
- Rings of fractions and localization
Book reference
- R: Sect. 8.3
- D-F: 18.2
- D-F: 15.4 (rings of fractions and localization)
| Nov. 08
In Lecture:
Please check on ICON to see whether it is your turn to work problems on the board.
|
Nov. 13
In Lecture:
- Rings of fractions and localization
- Extension and contraction of ideals
Book reference:
| Nov. 15
In Lecture:
- Extension and contraction of ideals
- Saturated ideals
- Prime ideals of localized rings
- Localization of modules
Book reference:
|
Nov. 20
THANKSGIVING BREAK
| Nov. 22
THANKSGIVING BREAK
|
Nov. 27
In Lecture:
| Nov. 29
In Lecture:
- Discussion of in-class test
- Localization of modules
- Localization defines an exact functor
Book reference:
|
Dec. 04
In Lecture:
- Local-globl principles
- Being zero, injectivity/surjectivity of homomorphisms, and flatness can be detected locally
- Being a free module, or a cyclic module, cannot be detected locally
Book reference:
| Dec. 06
In Lecture:
- Explicit example that being a free module, or a cyclic module, cannot be detected locally
- The p-adic integers - an amazing local ring
Book reference:
|