# Commutative Algebra

Although the title of this course is commutative algebra, we are really going to do algebraic geometry. The plan of this yearlong course is to follow very closely the lecture notes of Ravi Vakil which you can find announced on this blog and actually posted on this page. Ravi also explains his philosophy on the blog and I am going to subscribe to this philosophy as much as possible. In particular, you will have to do a lot of reading and exercises yourself to keep up with this course.

Perhaps I will point out here myself that as this is a foundations course we will not actually get very far with the material. On the other hand, it will hopefully provide solid foundations which will allow you to speed up the learning process once you've gone through it.

It is strongly encouraged to go to the lectures, which are on Tuesday and Thursday 11:00-12:15 in Mathematics 507.

Problem sets will be announced in lecture on Tuesdays and on this web page. They are due in lecture on the next Tuesday. Please write out all arguments completely.

The TA for the course is Zachary Maddock. His email address is ---@math.columbia.edu and his office is Mathematics ---. He will have office hours on Monday probably between ---------.

Grades are computed by a weighted average between the scores on problem sets and final. The weights are 2/3 and 1/3 respectively.

The final will be a written exam.

Here are the weekly problem sets. Please hit the refresh button on your browser to make sure you have to latest list.

1. First set due Tuesday, September 14. Exercises 2.3.F, 2.3.K, 2.3.O, 2.3.T, 2.3.Y, 2.4.A, 2.4.C, 2.5.C, 2.5.E, 2.6.A, 2.6.B, 2.6.F from Ravi's notes.
2. Second set due Tuesday, September 21.
3. Third set due Tuesday, September 28. Exercises 3.4.O, 3.5.D, 3.5.E, 3.5.H, 3.6.B, 3.7.B from Ravi's notes. (General rule: while solving an exercise you may use results from the text preceding the exercise and also use results from exercises preceding the exercise.) Also: Please read and understand the section entitled "The spectrum of a ring" in the algebra.pdf, then pick 4 of the excercises in the section entitled "The spectrum of a ring" in the exercises.pdf and do them.
4. Fourth set due Tuesday, October 5. Describe A^1_Q (4.2.C), describe A^2_Q (4.2.E), maps and affine spaces over C (4.2.L), describe A^n_Z (4.2.M), ideal of coordinate axes in A^3 (4.7.B), and the following three:
• Let A = C[a, b, c, d]/(ac - bd, c^2 - ab^2, d^2 - a^3). Let X = Spec(A). Let 0 be the closed point of X corresponding to the maximal ideal (a, b, c, d). Let U = X - {0}. Compute O_X(U). Hint 1: Parametrize the points of X in a clever way, i.e, solve the equations. Hint 2: It is very unlikely you will do this exercise by just using algebra. It is hard even with the hint given above, but I hope you will really try to do it in all detail. If you finding yourself writing more than a couple of pages, then you probably have to step back and try to find a trick to shorten the solution.
• Let A be a ring and p a minimal prime ideal. Show that the maximal ideal of the local ring A_p consists entirely of nilpotent elements.
• Let A be a ring such that the set of minimal primes of A is not an open subset of Spec(A). Show that A has infinitely many prime ideals. Hint: Topology!
5. Fifth set due Tuesday, October 19.
• Show that the following three rules define topologies on Proj(S) and that they are in fact the same topology:
• Induced from Spec(S).
• Such that D_+(f), f homogeneous of postive degree, form a basis.
• Such that V_+(I) are closed.
• Let A be a ring and M an A-module. Let f_i be elements of A which generate the unit ideal in A. Assume each localization M_{f_i} is a finite A_{f_i}-module. Show that M is a finite A-module.
• From Ravi's notes do: 5.4.B, 6.1.E, 6.2.E.
6. Sixth set due Tuesday, October 26.
• Take a look at Lemma Tag 01ST in the stacks project. Prove it without looking at the proof given there. If your proof is better/shorter/more readable, TeX it up and email it to me.
• Do exercises 1.4, 11.1, 22.2, and 22.4 of the chapter on exercises in the stacks project. The tags are 02CJ, 02DT, 028Q, and 028R.
7. Seventh set due Tuesday, November 16. Ravi's notes 7.5.K, 8.2.D, 8.3.G, 8.3.H. (As usual, you can use any result from Ravi's notes preceding the actual exercuise you are doing.)
8. Eigth set due Tuesday, November 23. Do the following exercises:
1. Prove a Noetherian scheme is quasi-compact and quasi-separated.
2. If f : X -> Y is a finite type morphism and Y is Noetherian, prove that X is Noetherian and f is quasi-compact and quasi-separated.
3. Let k be a field and let B be a k-algebra of finite type. Assume the Hilbert Nullstellensatz in the following form: The residue field of any maximal ideal of B is a finite extension of k (we will prove this later in the course). Prove that the intersection of the maximal ideals of B is the nilradical of B.
4. Let k be a field and let B be a k-algebra of finite type. Prove that if B has finitely many prime ideals, then (a) all primes of B are maximal, and (b) B is finite over k. (Hint: Use previous exercise.)
5. Let A -> B be a ring map of finite type. Let p be a minimal prime of A such that there are finitely many primes of B lying over p. Show that there exists an f in A such that A_f -> B_f is a finite ring map. (Hints: Try to prove that each generator of B is integral over A_f for some f. Use that every element of pA_p is nilpotent, hence every element of pB_p is nilpotent too, and use the previous exercise.)
The 5th exercise says that the morphism of schemes Spec(B) -> Spec(A) is finite over an open neighbourhood of the point p. This kind of statement is much harder to prove if p is not a minimal prime (and it isn't quite true -- the correct statement is a version of Zariski's Main Theorem).