Commutative Algebra - Fall Semester 2020
Update: Due to the global state of emergency caused by COVID 19 , the IMM has suspended on-site lecturer visits until further notice and will perform the lessons of the Fall semester, 2020 through distance learning. This includes the Commutative Algebra course, which is taught through a combination of pre-recorded video lectures, Google Classroom assignments and interactive teacher-to-students Q&A/exercise sessions through the Zoom video conferencing tool. The Zoom interactive sessions are recorded and made available to the students, along with all other course materials.
Dates: 15/09/2020 - 31/01/2021
Faculty: Lothar Göttsche (Trieste, Italy), Volkmar Welker (Marburg, Germany), Sarfraz Ahmad (Lahore, Pakistan), Angela Tabiri (Tutor; Ghana)
Lothar Göttsche
Wolkmar Welker
Sarfraz Ahmad
Angela Tabiri
Syllabus:
Professor Göttsche
Groups: Definition of a group, Subgroups, Normal subgroups and quotient groups, Group homomorphisms, Group operations, The symmetric group, Operations on Subsets, The Sylow Theorems.
Rings: Definition of a ring, examples and first properties, Homomorphisms, Ideals and Quotient Rings, Prime Ideals and Maximal Ideals, Polynomial rings over a field, Euclidean rings and principal ideal domains, Irreducibility of polynomials.
Fields: Field extensions, degree theorem, Algebraic extensions and simple algebraic extensions, Algebraic closure, Splitting fields, Normal extensions, Separable extensions, Finite fields, Galois groups, The fundamental theorem of Galois theory, Quadratic, biquadratic and Cubic polynomials, Solvability by radicals.
Professors Welker and Ahmad
Monomial Ideals, Monomial Ordering, Division Algorithms in a Polynomial Ring in n Variables, Grobner Bases, Buchberger’s Algorithm, Macaulay Bases Theorem, Hilbert Basis Theorem, Edge Ideals, Simplicial complexes and Stanley Reisner Ring, Simplicial Homology, Hilbert Series of Monomial Algebra.
Course materials:
ICTP Postgraduate Diploma Programme - Algebra (video playlist, 2021) by Prof. Lothar Göttsche
N.B. Only the first 5 lectures are pertinent to this course.
ICTP Diploma Programme - Abstract Algebra (video playlist, 2014) by Prof. Lothar Göttsche
Course Schedule (Pakistan time):
Week 1 (05/10-09/10):
Watch recorded lectures 1+2
Read chapter 1 Sections 1,2,3 in the lecture notes
Exercise sheet 1 - due on 16/10, 23:59
Week 2 (12/10-16/10):
Watch recorded lectures 3+4
Read chapter 1 Section 4, and Section 5 until including Orbit stabilizer theorem
Exercise sheet 2 - due on 23/10, 23:59
Week 3 (19/10-23/10):
Watch recorded lectures 5+6 and Lecture 7 until 20:30
Read chapter 1 Section 5 starting with action by conjugation, Section 6, 7
Exercise sheet 3 - due on 30/10, 23:59
Week 4 (26/10-30/10):
Watch recorded lecture 7 starting 20:30, lecture 8, lecture 9 until 21:30
Read chapter 1 Section 8
Exercise sheet 4
Week 5 (02/11-06/11):
Watch recorded lecture 9 starting 21:30 lecture 10, lecture 11 until 20:00
Read chapter 2 Sections 1,2,3
Exercise sheet 5
Week 6 (09/11-13/11):
Watch recorded lecture 11 starting 20:00, Lecture 12, Lecture 13 until 18:00
Read chapter 2 Sections 4,5,6
Week 7 (16/11-20/11):
Break in lessons (preparation for the midterm exam)
Week 8 (23/11-27/11), beginning of part 2:
Live lecture on Tuesday from 13:00 to 15:00 and on Friday from 19:00 to 21:00
Problem set 1 - due on 30/11, 23:59
Week 9 (30/11-04/12):
Live lecture on Tuesday from 13:00 to 15:00 and on Friday from 19:00 to 21:00
Week 10 (07/12-11/12):
Live lecture on Tuesday from 13:00 to 15:00 and on Friday from 19:00 to 21:00
Week 11 (14/12-18/12):
Live lecture on Tuesday from 13:00 to 15:00 and on Friday from 19:00 to 21:00
Problem set 2 - due on 15/01, 23:59
Week 12 (21/12-25/12):
Live lecture on Tuesday from 13:00 to 15:00 and on Friday from 19:00 to 21:00
Week 13 (28/12-01/01):
Live lecture on Tuesday from 13:00 to 15:00
Complete class notes for part 2
MTH619 - Commutative Algebra
Course ContentsRings and Ring homomorphisms, Ideals and Quotients Rings, Zero Divisors, Nilpotent elements, Nilradical and Jacobson radical, Operations on Ideals, Modules and Module Homomorphisms, Direct sum and product of modules, Finitely generated modules, Exact Sequence, Tensor Product of Modules, Primary Decomposition, Noetherian Rings, Artinian Rings, Hilbert Functions, Dimension Theory of Noetherian Local Rings, Cohen-Macaulay Rings and Modules.
Monomial Ideals, Monomial Ordering, Division Algorithms in a Polynomial Ring in N-Variables, Grobner Bases, Buchberger’s Algorithm, Macualay Bases Theorem, Hilbert Basis Theorem, Edge Ideals, Cohen-Macaulay Graphs, Constructions of Cohen-Macaulay Graphs, Simplicial complexes and Stanley Reisner Ring, Simplicial Homology, Hilbert Series of Monomial Algebra.
Books
M.F Atiyah, I.G. Macdonald, Introduction to Commutative, Algebra-Wesley Publishing Company 1969
D. Eisenbud, “ Commutative Algebra with a view toward algebraic geometry” Springer 1994
H. Matsumura, Commutative Algebra, 2nd Edition, The Banjamin Publishing Company London, 1980
D.S Dummit and R.M. Foote, Abstract Algebra, John Wiley and Sons, Inc. 1999
R. Villareal “ Monomial Algebras”. Monographs and Textbooks in Pure and Applied Mathematics, Vol. 238, Marcel Dekker Inc. New York 2001
W. Bruns, J. Herzog” Cohen=Macaulay Rings”, Second Edition, Cambridge 1998