Credits: 5
Schedule: 16.04.2019 - 23.05.2019
Contact information for the course (applies in this implementation):
For any questions regarding the course please contact me at:
Mateusz Michalek
michalek@mis.mpg.de
mateusz.michalek@aalto.fi
Details on the course content (applies in this implementation):
The course will provide introduction to basic concepts of commutative algebra (the basic example is the ring of polynomials). We will start from basic objects like prime ideals and modules (generalizations of vector spaces). The course will be loosely based on a classical book "Introduction to commutative algebra" by Atiyah and Macdonald.
An interested student could take a look at a book under construction (especially first three chapters), as these provide a complementary perspective:
https://personal-homepages.mis.mpg.de/michalek/book.html
The student should be familiar with definition of a ring, vector space, field, ideal, group.
Elaboration of the evaluation criteria and methods, and acquainting students with the evaluation (applies in this implementation):
Each lecture will provide a variety of exercises on many levels of difficulty. Solutions to a number of exercises will be a sufficient condition to pass the course, but not a necessary one. (Other options would be discussions, indicating e.g. how the student tried to solve the exercise and why s/he failed.)
0 : Students not attending the course
1 : Students failing the course
2: Students who submitted at least one exercise every week
3: Students who submitted at least two exercises every week
4: Students who gathered at least 10 points (plus the requirement for 3)
5: Students who gathered at least 30 points (plus the requirement for 3)
Details on calculating the workload (applies in this implementation):
The time needed for the course will very much depend on student's background.
Details on the course materials (applies in this implementation):
Lecture notes are appearing in pdf in course materials.
Lecture 1. Ideals
-zero divisors, integral rings, characterisation of fields
-operations on ideals and their properties
-radicals
-types of ideals: maximal
-local rings
-Chinese reminder theorem (general version)
Lecture 2. Prime ideals
-Definition and characetrizaion
-Prime avoidance lemma
-Characterization of radical
-Spectrum of a ring as a topological space
-Definition and examples of primary ideals
Lecture 3. Graded rings and modules
-Definition of graded rings and homogeneous ideals
-Homogenization
-Projective spectrum
-Definition and examples of modules
-Submodules
-Graded modules
Lecture 4. Special Modules
-Free modules, bases and their properties
-McCoy Theorem
-Projective modules, finitely generated modules, finitely presented modules
-Nakayama's Lemma and Corollaries
-Projective modules over local rings are free (with a proof in a finitely generated case)
-Quillen-Suslin theorem (without proof)
Lecture 5. Tensor product
-Definition and universal properties
-Properties
-Flat modules
-Extension and restriction of scalars for modules
-Flatness
Lecture 6. A-algebras
-Definition and examples
-Finite type algebras, affine algebras
-Tensor product
-Direct and inverse limits
-Tensor product of limits and exactness
-p-adic numbers
Lecture 7. Fractions
-Localisation
-Properties of spectrum and quotient after localisation
-Modules of fractions
-Support of a module
-Local properties of modules
Lecture 8. Flatness
-Characterization of flatness
-Flat algebras
-Faithful flatness
Lecture 9. Noetherian and Artinian rings
-Length of a module
-Noetherian and Artinian modules
-Constructions and examples
-Akizuki Theorem
-Eakin-Nagata theorem and generalizations
Lecture 10. Primary Decomposition
-Associated primes
-Primary Decomposition Theorem
-Embedded and isolated components
-Krull's Intersection Theorem
Lecture 11. Integral dependence
Lecture 12. Nullstellensatz
Course Homepage (valid 01.08.2018-31.07.2020):
https://mycourses.aalto.fi/course/search.php?search=MS-E1998
Grading Scale (valid 01.08.2018-31.07.2020):
0-5
Additional information for the course (applies in this implementation):
Commutative algebra is the modern language used in algebraic geometry and number theory. I highly recommend this course to anyone who is thinking about possible future academic career.
Personally I can say that the course is based on a similar one I attended as a student myself. I loved that course and promised myself to run a similar one in future. It definitely affected my future career a lot - basically it was one of the main motivations for me to become an algebraic geometer. I hope it could be just as inspiring for you as it was for me!
I can promise that my office will be always open for students and anyone willing to learn more or coming with questions or solutions etc will be most welcome!
Details on the schedule (applies in this implementation): Preliminary schedule:
Lecture:
Ti 16.04.2019 klo 12:15 - 14:00, R001/Y229a
Ke 17.04.2019 klo 10:15 - 12:00, R001/M203
Ti 23.04.2019 klo 12:15 - 14:00, R001/Y229a
Ke 24.04.2019 klo 10:15 - 12:00, R001/M203
Ti 30.04.2019 klo 12:15 - 14:00, R001/Y229a
(1.5. on vappu)
Ti 07.05.2019 klo 12:15 - 14:00, R001/Y229a
Ke 08.05.2019 klo 10:15 - 12:00, R001/M203
Ti 14.05.2019 klo 12:15 - 14:00, R001/Y229a
Ke 15.05.2019 klo 10:15 - 12:00, R001/M203
Ti 21.05.2019 klo 12:15 - 14:00, R001/Y229a
Ke 22.05.2019 klo 10:15 - 12:00, R001/M203
exercise:
To 18.04.2019 klo 10:15 - 12:00, R001/M203
To 25.04.2019 klo 10:15 - 12:00, R001/M203
To 02.05.2019 klo 10:15 - 12:00, R001/M203
To 09.05.2019 klo 10:15 - 12:00, R001/M203
To 16.05.2019 klo 10:15 - 12:00, R001/M203
To 23.05.2019 klo 10:15 - 12:00, R001/M203
- Teacher: Hollanti Camilla
- Teacher: Orlich Milo