Fundamental concepts in commutative algebra,
- January 23. Some details
about the exam: it contains 5 problems for 25 points, and you should
choose 4. In addition there is a less typical
bonus problem for 10 points. In large, the theoretic part of the exam covers
the material of [AM] that was studied in class (more or less chapters 1—9)
and practical problems are related to this material. Due to your requests
I provide here a concrete exam’s
- November 28. I recommend to
look at the book on commutative algebra of Altman and Kleiman,
which is available on the web. It also includes nice sections on
- November 7. We got new rooms
for lectures. I updated the general info in accordance.
- October 31. Due to your
request, the deadline for submitting homework will be on Thursdays
midnight, starting with HW1 (which should now be submitted on November 2).
- October 22. The lecture on
October 24 will be given by Uri Brezner.
· Michael Temkin
· E-mail: temkinmath.huji.ac.il
· Office: Ros
78A (tel. 02-6584575)
· Office Hours: by appointment
· Lectures: Mondays 10-11:45 at Ros 70, Tuesdays 16-17:45 at Mathematics 110.
· Uri Brezner
· E-mail: uri.breznermail.huji.ac.il
· Office: Ros
26 (tel. 02-6584846)
· Office Hours: by appointment
· Recitations: Tuesdays 12-13:45 at Sprinzak 116.
· The commutative algebra part will be
due to "Introduction to Commutative Algebra" by Atiyah
· The final grade will be obtained as
follows: 50% of the exam grade plus 50% of the homework grade. The grade for
homework will be computed by averaging 10 best weekly homework grades.
The exam will be only
on the commutative algebra part of the course, and an exam syllabus will be
posted in the end of the course.
· The exercises are assigned on Tuesdays on this webpage, and should be
submitted to Uri's mailbox until the end of the next Thursday (any time before
24:00 is ok). Late (or non-submitted) homework will not be graded. Homework may
include a non-mandatory part marked by *, which is not for submission but can
be helpful for deeper understanding of the material. You are welcome to discuss
it with me or Uri.
More difficult problems are marked
- October 24. The first homework is here. Due date
is October 31. (Changed to November 2.)
- October 31. The second homework is here. Due date
is November 9.
- November 7. The third homework is here. Due date
is November 16.
- November 14. The fourth homework is here. Due date
is November 23.
- November 21. The fifth homework is here. Due date
is November 30.
- November 28. The sixth homework is here. Due date
is December 7.
- December 5. The seventh homework is here. Due date
is December 14.
- December 12. Eighth homework: solve problems 18,19,20 after
chapter 1 and problem 21 after chapter 3 in Atyah-Macdonald.
Due date is December 21.
- December 19. Ninth homework is here. Due date
is December 28.
- December 27. Tenth homework is here. Due date
is January 4.
- January 2. Eleventh homework is here. Due date
is January 11.
- January 9. Twelfth homework is here. Due date
is January 18.
- January 16. Thirteenth homework: solve problems 10, 11, 20,
21, 22, 35 after chapter 5. Due date is January 25.
- October 23. Some basic definitions from chapter 1, including
commutative rings with units, their homomorphism, R-algebras, polynomial
rings, maximal/prime/principal ideals, quotient rings, nilpotent elements
and zero divisors.
- October 24. End of chapter 1
and some exercises after it: extension and contraction of ideals with
respect to a homomorphism, Chinese remainder theorem, some properties of
- October 30. Beginning of
chapter 3: localizations of rings, its universal property, and behavior of
ideals under localization.
- October 31. Categories,
limits and colimits (including products and
coproducts, fibred products, equalizers and coequalizers),
complete and cocomplete categories. Criteria of
completeness and cocompleteness. Here
are my written notes about categories. (Two precautions: (1) I'll polish
it and maybe add some details, (2) The order of exposition is different. I
introduce representable functors first and
deduce the criteria from Yoneda lemma. We will
study (or at least formulate) this next week.)
- November 6. Adjoint functors, continuous
and cocontinuous functors.
- November 7. Started R-modules
(beginning of chapter 2). Definitions, submodules, quotients, limits and colimits, kernels and cokernels.
- November 13. Different versions
of Nakayama lemma, additive invariants and K_0 of subcategories of R-Mod,
exact sequences, the snake lemma.
- November 14. Bilinear maps,
tensor products, adjunction between tensor products and Hom's, exactness properties of tensor products and Hom's, further properties of tensor products.
- November 20. Extension of
scalars, localization of modules.
- November 21. Classification
of adjoint pairs of functors
between categories of modules, tensor products of algebras and basic
examples, including quotients, localizations, products of field
extensions. Definition of flat modules.
- November 27. Faithfully flat
modules and simple instances of faithfully flat descent.
- November 28. Affine algebraic
varieties over an algebraically closed field. Hilbert Nullstellensatz
and the correspondence between varieties and radical ideals.
- December 4. Morphisms between
affine varieties. Function (or coordinate) ring k[V] of an affine variety
V, and the equivalence of categories it induces: k-Aff^op
opposite functor Max(A).
- December 5. Examples,
products, fiber products, modification needed to construct varieties over
an arbitrary field.
- December 11. Affine space
over an arbitrary perfect field. Projective spaces P^n,
graded rings, and homogeneous ideals.
- December 12. Cones,
projective varieties, projective Nullstellensatz.
Started a new topic: general varieties and schemes. So far, gave a local
definition of regular functions on an affine variety V and proved the
theorem that it can be identified with an element of k[V] (i.e. agrees
with the old definition of polynomial functions).
- December 18. Preasheaves and sheaves. Arbitrary algebraic varieties
(V,O_V) over k, where V is a topological space
and O_V is the structure sheaf of regular functions from V to k. Morphisms
of varieties and the theorem that the category of affine varieties embeds
fully faithfully into the category of arbitrary varieties. Analogs for
affine schemes – regular functions and the structure sheaf.
- December 19. Noether and Artin modules and
rings. Modules of finite length, composition series and Jordan-Holder
- December 26. Noetherian rings
and Hilbert basis theorem.
- January 1. Primary ideals and
their properties in general rings and in Noetherian rings (chapter 4 and
the end chapter 7).
- January 2. Primary
decomposition and the two uniqueness theorems.
- January 8. Artin rings (chapter 8).
- January 9. Integral
dependence (started chapter 5): basic properties of integral and finite homomorphisms.
- January 15. Going down
theorem, integrally closed domains.
- January 16. Going up theorem,
Noether normalization theorem.
- January 22. Dimension theory
of finitely generated algebras over a field.
- January 23. Unique
factorization domains, valuation rings, Dedekind rings (chapter 9 without