Math 746, Fall 2013, Professor Kustin

Announcements

Course Announcement and References

• Course Announcement: Commutative Algebra Course, Fall 2013 or .ps format. The course announcement also serves as the syllabus.
  
  
• Some of my favorite references are:
  • Bruns and Herzog, "Cohen-Macaulay rings".
  • Eisenbud, "Commutative Algebra with a view toward Algebraic Geometry".
  
  
• My favorite software is Macaulay2. Here is a small Macaulay2 session where I calculated the primary decomposition of an ideal. I used a session like this as I prepared the lecture on primary decomposition. All of the commands I used are explained on-line. I find the Macaulay2 Table of Contents to be very helpful.
  
  
• Many people post their pre-prints on the arXiv.

Homework

• 1. The problems on page 6 in Matsumura all look reasonable.
  
  
• 2. There is a meta-theorem that every ideal maximal with respect to any reasonable property is automatically a prime ideal. You can find and prove results of this type. I have put many of these on the Quals over the years. Maybe you came upon them as you were studying for your Qual. (I will tell you my reference for these problems at a later date.)
  
  
  • a. Let A be a ring and M be an A-module. Let I be an ideal in A that is maximal among all annihilators of non-zero elements of M. Prove that I is prime ideal. This result is particularly interesting.
    
    
  • b. Let I be an ideal in the ring A. Suppose that I is not finitely and is maximal among all ideals that are not finitely generated. Prove that I is prime ideal.
    
    
  • c. In a ring A, let I be an ideal which is maximal among non-principal ideals. Prove that I is prime ideal.
    
    
  • d. In a ring A, let I be an ideal which is maximal among ideals that are not countably generated. Prove that I is prime ideal.
    
    
• 3. Matsumura gives an example on the bottom of page 5 of a domain from Number Theory which is not a UFD. Of course, this ring is historically important. (It helped to inspire Dedekind to create the notion of ideals (that is "idealized elements") and to prove that in a Dedekind domain there is a Unique Factorization of ideals into prime ideals.) Nonetheless, I am partial to the following example from Geometry. Give a complete proof that the ring k[x,y,z,w]/(xy-zw) is not a UFD. (In this problem k is a field.)
  
  
• 4. My comments in problem 3 cause me to ask: "What is the (unique) factorization of the ideal (6) into prime ideals in the Dedekind Domain Z[√-5]?"
  
  
• 5. It turns out that every ideal in a Dedekind domain A is a projective module. (Surely, we will prove that a finitely generated module is projective if and only if it is locally free. Every ideal in a Dedekind ideal is locally principal; hence locally free.) Take your favorite Dedekind domain A (say Z[√-5]) and exhibit an ideal which is not a free A-module; but which is direct summand of a free A-module. Give details.
  
  
• 6. Let M be an A-module of finite length. Prove that all composition series for M have the same length. (This is usually called the Jordan-Hölder Theorem.)
  
  
• 7. Fill in the details concerning the filtration of Remark 2 on page 2 of Avramov-Miller paper. This is an important and interesting paper. I think that my question is equivalent to recording an explicit composition series for k[x_1,...,x_c]/(x_1^q,...,x_c^q).
  
  
• 8. Consider the Segre embedding of the direct product of projective r space and projective s space into projective (r+1)(s+1)-1 space. What is the ideal of polynomials which vanishes on the image of this embedding? Can you prove that you have found the entire ideal? (I will lecture about this embedding on September 3.)
  
  
• 9. At the bottom of page 14, Matsumura writes: (A subring of a Noetherian or Artinian ring does not necessarily have the same property: why not?) Please answer this question. It is very easy.

Possible topics for the Optional Supplemental Student Seminar

• 1. In class I will prove that every finitely generated projective module over a local ring is free. The result remains true if the hypothesis "finitely generated" is removed. In class, I will not prove the result for modules which are not finitely generated; but this proof sits on pages 10 and 11 of Matsumura if anyone wants to present it in seminar.
  
  
• 2. In a similar manner, there are relatively elementary proofs of the Quillen-Suslin theorem that every finitely generated projective module over a polynomial ring is a free module. The most elementary proofs are due to Vaserstein.
  
  
• 3. I think that "Grothendieck's Generic Freeness Lemma" is fairly accessible; see for Example, Eisenbud's book section 14. This is an important and very useful result.
  
  
• 4. I think that Buchsbaum-Eisenbud criterion for the exactness of a complex is fairly accessible. One can find it in Eisenbud's book or Bruns and Herzog, or the original "What makes a complex exact?". These ideas are a starting point for the study of Free Resolutions, which is an area of particular interest to me.
  
  
• 5. "Eventually" we will study "Duality, Canonical modules, and Gorenstein rings"; but if anyone is impatient, it might be possible to make an interesting lecture out of Eisenbud's presentation of this topic (Chapter 21) or Bruns and Herzog's presentation (Section 18).
  
  
• 6. The arXiv is a source of brand new ideas in Commutative Algebra (and many other fields).
  
  
• 7. Boij-Söderberg theory is only a few years old and is very popular; parts of it are very accessible. See, for example, the first paper on the subject "Graded Betti numbers of Cohen-Macaulay modules and the multiplicity" by Boij and Söderberg in J. Lond. Math. Soc. (2) 78 (2008), no. 1, 85--106. (The easiest way to download a paper is to find it on MathSciNet and then click on "Article". You probably have to be on campus to do this, or have VPN on your laptop.)
  
  
• 8. Combinatorial Commutative algebra provides another source of topics that do not appear to require a lot of technical background. The first theorem on this topic is Macaulay's theorem about lex-segment ideals. The modern continuation of Macaulay's theorem is the Bigatti-Hulett Theorem.
  
  
• 9. Here is an example of Combinatorial Commutative algebra. A couple of years ago, an undergraduate student asked me if she should go to graduate school in Mathematics. I suggested that we read: Distributive Lattices, Bipartite Graphs and Alexander Duality, by Herzog And Hibi or arXiv version. It was fun. She went to graduate school.
  
  
• 10. I think that it is possible to create a seminar out of the concept "The branches of a curve singularity". One can think of this topic either as a completetion question (that is factoring formal power series) or as a blowing-up question (that is basically normalization.) The shortest version is a 4 page paper by Dan Katz. There also are some handwritten notes of a lecture I gave on this topic. I became interested because of a paper I wrote with Cox, Polini, and Ulrich, see Proposition 1.12. (This paper has appeared as a Memoir of the AMS. The arXiv is the same and has open access.) . An earlier version of the [CKPU] paper has a more complete argument of the correspondence between the minimal prime ideals of the completion and the maximal ideals of the normalization.
  
  
• 11. We talked a little bit about Bezout's Theorem. One could make a seminar out this topic. This is a very important topic: it is the basis for addition on elliptic curves; I expect you will use it regularly in the Algebraic Geometry course.
  
  
• 12. I will say a little bit about the arithmetic rank of an ideal, set-theoretic complete intersections, local cohomology, and the Cech complex. One could make a seminar out of fleshing these things out. If Gennady Lyubeznik (or Margherita Barile) has written a survey article on these topics that would be a great place to start.
  
  
• 13. Describe all ideals in k[x,y] which are generated by homogeneous polynomials. This is a special case of the Hilbert-Burch Theorem, which is a Theorem I use all of the time.
  
  
• 14. I don't think that I will prove the Hilbert basis theorem in class. Would anyone like to prove it in seminar? The Hilbert basis theorem states that if A is a Noetherian ring, then A[x] is also a Noetherian ring.
  
  
• 15. There are many "characterizations of noetherian rings" in section 3 that I will skip. I like all of them; any given one would make a fine seminar; Thm. 3.4 answers HW 2b (I think). I skip 3.2, 3.4, 3.5, 3.6, 3.7.
  
  
• 16. In class I will define tensor product and state some of its properties; but I won't actually prove the properties. Someone could prove these properties in seminar. I am thinking of these properties:
  • a. The tensor product of two A-modules, where A is a commutative ring, exists and is an A-module.
  • b. The tensor product A\otimes_A N is canonically isomorphic to N.
  • c. Tensor prodcut commutes with direct sum.
  • d. Tensor product is right exact.
  
  
  17. Cameron and Tyler both expressed interest in "Noether Normalization". (Cameron used this Theorem in his seminar talk and Tyler is cooking up a seminar talk which uses this Theorem.) So naturally, they asked, when will we do it? In some sense we covered the same ground when we did section 5. We proved that if A is a domain which is finitely generated as a ring over the field k, then A is an algebraic extension of a polynomial ring P. We used this to prove that the Krull dimension of A is equal to the transcendence degree of A over k. Noether normalization is similar but fussier. It picks the polynomial ring P in such a way that A is not only algebraic over P, but is even integeral. (Each element of A satisfies a monic polynomial with coefficients in the carefully chosen polynomial ring P.) This theorem is particularly lovely because there is a very tight relation between the prime ideals of P and the prime ideals of an integral extension of P. It does not look like Matsumara's book does Noether Normalization at all. Maybe somebody could prove it in seminar. It looks like a very nice treatment is given in 13.2 and 13.3 of Eisenbud. It also looks like a very nice, self-contained version is given on Noether-Normalization-from-Wikipedia.pdf or Wikipedia.
  
  18. These exercises lead one through the functor "Tor". They serve as a wonderful introduction to homological algebra. They refer to a course I taught many years ago; but I think they still work.