Administrivia Prerequisites, text and syllabus HW sets HW solutions Outlines of lectures

Comuutative Algebra

 

Administrivia

The course meets from 2:30 to 3:20 MWF in OSC 201. Grades will be computed according to the formula 50 percent for homework, 20 percent for midterm, 30 percent for final. Homework will typically be set in each class meeting and will be due at the following class meeting, late homework will not be accepted. Exams will be in the open book take home format with a generous time allowance, dates to be negotiated. I have no regular office hours but am always happy to meet with students by appointment (send me email to make an appointment). My office is Wean Hall 7101 and my email address is jcumming@andrew.cmu.edu.

 

Prerequisites, text and syllabus

Prerequisites: a knowledge of the basic facts about rings, groups and fields. The Algebra I course is more than sufficient. I will aim to cover most of the material in the course text, Atiyah and Macdonald's ``Introduction to Commutative Algebra''. I will also aim to cover some of the arithmetic and geometry which motivates the subject. In particular we will see how highly abstract methods are useful in solving such concrete problems as counting the number of points in which two curves intersect or working out which integers are represented by a given quadratic form.

 

The following is a very tentative syllabus.

  1. Review of ring theory. Zorn's lemma
  2. Nilradical and Jacobson radical
  3. Modules
  4. Noetherian rings and modules, the Basissatz
  5. The Nullstellensatz. Affine varieties
  6. Integrality. Algebraic number fields
  7. Tensor products. Exact sequences
  8. Localisation. Bezout's theorem
  9. Dedekind rings. Prime factorisation of ideals
  10. Valuations. Completion. The p-adics
  11. Quadratic forms
  12. Elliptic curves

 

Homework sets and exams

  1. Chapter I Qs 2, 3, 4. Chapter III Q 6.
  2. I.5 II.7 VII.8
  3. I.10 I.14 VII.2
  4. The dreaded Midterm.

 

Homework solutions

  1. Homework 1.
  2. Homework 2.
  3. Homework 3.
  4. Homework 4.
  5. Homework 5.
  6. Homework 6.
  7. Homework 7.
  8. Homework 9.

 

Outlines of lectures

    1. Review of basic defns
    2. Ring = commutative ring with 1
    3. HMs and subrings preserve the 1
    4. Ideals, first isomorphism thm, ideals of R/I
    5. I prime iff R/I domain, maximal iff R/I field
    6. Zorn's lemma
    7. Every proper ideal contained in a maximal one
    8. Nilpotent elements form an ideal
    9. Nilradical is intersection of the prime ideals
    1. Multiplicatively closed sets
    2. A mult closed set not containing 0 is disjoint from a prime ideal
    3. Defn of the Jacobson radical
    4. Nilradical is a subset of the J radical
    5. For M maxl, a not in M iff some multiple of a is congruent to 1 mod M
    6. a is a nonunit iff a is in some max ideal
    7. a is in the J radical iff 1 + a b is a unit for all b
    8. Defn of R-module and some examples
    9. If f: R --> S is a ring HM, S has an R-module structure
    1. Submodules and quotient modules
    2. Generating sets for modules
    3. Free generating sets
    4. Free module on an arbitrary set X
    5. If X and Y have same cardinality implies free modules on X and Y are isomorphic
    6. If M gend by X, M is a quotient of free module on X
    7. Defn: Noetherian module, Noetherian ring
    8. The R-submodules of R are the ideals of R
    1. M is Noetherian iff all increasing chains of submodules stabilise iff every nonempty set of submodules has a maxl element
    2. For N a submodule of M, M Noetherian iff N and M/N both are
    3. R Noetherian implies R^n Noetherian all finite n
    4. R Noetherian implies all fg R-modules are Noetherian
    5. R Noetherian ring implies R[x] Noetherian ring (but NOT a Noetherian R-module)
    1. Defn of S being ring-finite and module-finite over subring R
    2. Ring and module finite are both transitive, module finite implies ring finite
    3. Quotient of a Noetherian ring is Noetherian
    4. R Noetherian implies R[x_1 .... x_n] Noetherian
    5. R Noetherian and S ring-finite over R implies S Noetherian
    6. Review of fields, structure of k(a)
    7. a alg/k implies k(a)=k[a] is module finite over k
    8. Defn of algebraic independence
    9. If k a field then k[x_1 .. x_n] is a Noetherian UFD with infinitely many irreducibles
    1. If a_1.....a_n are algebraically independent over k then the field k(a_1, ... a_n) is not ring finite over k
    2. V(X) for X a set of polys, I(Y) for Y a set of points
    3. V and I are order-reversing, X contained in IV(X), Y contained in VI(Y)
    4. Varieties are sets V(I). If V is a variety VI(V) = V
    5. I(V) equals its own radical
    6. The ideal of a point (a_1..a_n) is maxl ideal generated by the x_i - a_i
    7. Nullstellensatz: if k is algebraically closed all maxl ideals in ring of polys are as above
    8. Start of proof of Nullstellensatz: M maxl, a_i = x_i + M, ETS that each a_i is alg/M. If not arrange that a_1...a_m are alg indt and the rest are alg over k(a_1...a_m)
    9. End of proof of O-Satz: if A Noetherian, A subring of B, B subring of C, C is r-f over A and m-f over B then B is m-f over A. Apply to situation above with A = k, B = k(a_1...a_m) and C = k[x_1....x_m]/M = k[a_1....a_n]
  7. Assume k is algebraically closed until further notice.
    1. Any proper ideal in k[x_1.....x_n] gives a nonempty variety
    2. If I is an ideal, IV(I) = radical of I
    3. The maps V and I set up a 1-1 order-reversing correspondence between radical ideals and varieties
    4. Prime ideals are radical and correspond to irreducible varieties, that is thise that are not the union of two smaller subvarieties
    5. The prime spectrum and maximal spectrum of an arbitrary ring