Spring 2014 Math 603 schedule

Monday Wednesday Friday
Jan. 13

No class
Jan. 15

In Lecture:
  • Outline of the second semester
  • Proof that if R is a UFD, so is R[x]; Gauss Lemma
  • Heights of prime ideals, dimension of a commutative ring

Associated Reading:
  • Lang, XII.6, II.5,
  • DF, p. 232, 8.2, 8.3
  • Hartshorne's "Algebraic Geometry," p. 5 - 7.

Video of class (downloadable)
Jan. 17

In Lecture:
  • No class
Jan. 20
No class, Martin Luther King holiday
Jan. 22

Classes at Penn cancelled due to weather.
Jan. 24 (9:00 a.m. in room 4C4 of DRL labs)

In Lecture:
  • Dimension theory, continued
  • Euclidean algorithm in a power series ring

Associated Reading:
  • Hartshorne's "Algebraic Geometry," p. 5 - 7.
  • Lang: 4.9

Video of class (downloadable)
Jan. 27

In Lecture:
  • Weierstrauss preparation theorem
  • Complete rings with respect to an ideal
  • Start of the proof that k[[x_1,...,x_n]] is a UFD.

Associated Reading:
  • Hartshorne's "Algebraic Geometry," p. 5 - 7.
  • D-F: 10.4 - 10.5
  • Lang: 4.9, 3.3 - 3.4, 16.1 - 16.2.

Video of class (downloadable)
Jan. 29

In Lecture:
  • Completion of the proof that k[[x_1,...,x_n]] is a UFD.
  • Bimodules
  • Tensor products of modules
  • Localization of modules

Associated Reading:
  • D-F: 16.2, 10.5
  • Lang: p. 88, 3.4, 20.4, 26.3
  • Hartshore's ``Algebraic Geometry," II.2, p. 143 - 145.

Video of class (downloadable)
Jan. 31 (9:00 a.m. in room 4C4 of DRL labs)

In Lecture:
  • Projective and injectives ( review) and flats
  • Review of homological algebra, Ext and Tor

Associated Reading:
  • D-F: 16.2, 10.5
  • Lang: p. 88, 3.4, 20.4, 26.3
  • Hartshore's ``Algebraic Geometry," II.2, p. 143 - 145.

Video of class (downloadable)
Feb. 3

In Lecture:
  • More on Tor
  • Criteria for flatness
  • The snake lemma

Associated Reading:
  • D-F: p. 788 - 795
  • Lang: Chapter 16, sectiions 16.1 - 16.3

Video of class (downloadable)
Feb. 5

In Lecture:
  • Completion of the proof of the characterization of flatness via tensuring with ideals
  • Grothendieck groups of modules,
  • Start of K-theory

Associated Reading:
  • D-F: 16.2, 10.5
  • Lang: p. 88, 3.4, 20.4, 26.3
  • Hartshore's ``Algebraic Geometry," II.2, p. 143 - 145.
  • Milnor, "Introduction to algebraic K-theory," chapter 1

Video of class (downloadable)
Feb. 7 (9:00 a.m. in room 4C4 of DRL labs)

In Lecture:
  • Local rings
  • Dedekind rings
  • Continuation of K-theory

Associated Reading:
  • D-F: 12.1, p. 795
  • Lang: 10.1, p. 171
  • Hartshore's ``Algebraic Geometry," II.2, p. 143 - 145.
  • Milnor, "Introduction to algebraic K-theory," chapter 1

Video of class (downloadable)
Feb. 10

In Lecture:
  • Dedekind rings, continued
  • Locally free modules

Associated Reading:
  • Hartshorne's "Algebraic Geometry," p. 5 - 7.
  • D-F: 10.4 - 10.5
  • Lang: 4.9, 3.3 - 3.4, 16.1 - 16.2.

Video of class (downloadable)
Feb. 12

In Lecture:
  • Completion of some proofs about projective modules over Dedekind rings
  • Noetherian modules and rings
  • Hilbert basis theorem

Associated Reading:
  • D-F: chapter 12.1,
  • Lang: chapter 10
  • Milnor, "Introduction to algebraic K-theory," chapter 1

Video of class (downloadable)
Feb. 14

No class
Feb. 17

No class
Feb. 19

No class
Feb. 21

No class
Feb. 24

In Lecture:
  • R[[x]] is Noetherian if R is commutative and Noetherian
  • Associated prime ideals
  • Primary decomposition

Associated Reading:
  • Lang: Sections 6.3 - 6.5

Video of class (downloadable)
Feb. 26

In Lecture:
  • Associated primes and primary decomposition, continued

Associated Reading:
  • Lang: Sections 6.3 - 6.5

Video of class (downloadable)
Feb. 28

No class
March 3

In Lecture:
  • Elliptic curves and the group law of Pic
  • Proof that primary decompositions exist

Associated Reading:
  • Lang: Sections 6.3 - 6.5

Video of class (downloadable)
March 5

In Lecture:
  • The machinery needed to show primes associated to a primary decomposition are uniquely determined
  • Associated primes of a module

Associated Reading:
  • TBA

Video of class (downloadable)
March 7

No class
March 17

In Lecture:
  • Radicals of annihilators, the support of modules, and the associated primes of modules

Associated Reading:
  • Lang: Sections 6.3 - 6.5

Video of class (downloadable)
March 19

In Lecture:
  • The fundamental theorem on finitely generated modules over a PID

Associated Reading:
  • D-F: 12.2
  • Lang: 3.7, 3.5

Video of class (downloadable)
March 21

No class
Mar. 24

In Lecture:
  • Rational and Jordan canonical forms

Associated Reading:
  • D-F: 12.1, 12.3
  • Lang: 5.1 - 5.3.

Video of class (downloadable)
Mar. 26

In Lecture:
  • More on canonical forms

Associated Reading:
  • D-F: 12.3, 11.5
  • Lang: 5.2. 26.7 - 26.8, 29.1

Video of class (downloadable)
March 28

No class
Mar. 31

In Lecture:
  • Symmetric and alternating tensors
  • Determinants

Associated Reading:
  • D-F: 12.3, 11.5
  • Lang: 5.2. 26.7 - 26.8, 29.1

Video of class (downloadable)
April 2

In Lecture:
  • Wedderburn's Theorem and group representation theory
  • Start of field theory
  • Straightedges versus rulers
  • Algebraic closures
  • Extending embeddings

Associated Reading:
  • D-F: 13.1, 13.2, 13.3
  • Lang: 5.1, 5.2

Video of class (downloadable)
April 4

No class
April 7

In Lecture:
  • Ruler and compass constructions, continued
  • Bezout's theorem
  • Splitting fields

Associated Reading:
  • D-F: 13.3, 13.4
  • Lang: 5.2
  • Hartshorne: p. 47

Video of class (downloadable)
April 9

In Lecture:
  • Regular n-gons and Fermat primes
  • Existence of splitting fields and algebraic closures
  • Normal, separable and Galois extensions

Associated Reading:
  • D-F: 13.5, 13.6, 14.1
  • Lang: 5.1, 5.2, 5.3, 5.4, 5.6, 6.1

Video of class (downloadable)
April 11

No class
April 14

In Lecture:
  • Statement of the main theorems of Galois theory
  • Examples

Associated Reading:
  • D-F: Chapter 14
  • Lang: Chapter 5

Video of class (downloadable)
April 16

In Lecture:
  • Proofs of some of the main Theorems of Galois theory
  • Determining Galois groups

Associated Reading:
  • D-F: Chapter 14
  • Lang: Chapter 5

Video of class (downloadable)
April 18

No class
April 21

In Lecture:
  • S_n acting on k(x_1,...,x_n) by permutations,
  • Discriminants

Associated Reading:
  • D-F: Chapter 14
  • Lang: Chapter 5

Video of class (downloadable)
April 23

In Lecture:
  • Determining Galois groups of polynomials
  • Example: Quartic extensions
  • Using reductions of polynomials

Associated Reading:
  • D-F: Chapter 14
  • Lang: Chapter 5

Video of class (downloadable)
April 25

No class
April 28

In Lecture:
  • Henselian rings, with applications to finding Galois groups
  • Kummer Theory, Artin Schreier Theory

Associated Reading:
  • D-F: Chapter 14
  • Lang: Chapter 5

Video of class (downloadable)
April 30

In Lecture:
  • Solvability by radicals

Associated Reading:
  • D-F: Chapter 14
  • Lang: Chapter 5

Video of class (downloadable)
May 2

No class

Last updated: 5/5/14
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