Dear ○○,

I have made my decision. Please come and join us. We shall study mathematics together. Please do your best at the final exam of this month. Say, if necessary, that Goto is willing to accept you. But, I must say also that I am now old and it is getting more difficult and difficult to give (and find) good problems for students. So I would like to let you know about this fact also in advance, before you will make your own decision.

After coming to Japan, you shall enter, apart from your background, a new world of commutative algebra. Your background will never be lost and will never become useless, but at first, you cannot count its help too much.

Let me now explain a little bit about how and what I regard modern commutative algebra as.

Commutative algebra has many sources. For instance, algebraic geometry, number theory, singularity theory, invariant theory, and analysis, too. D. Buchsbaum (he is one of my teachers) told me like this. “We are in debt very much to differential equations also.” I am sorry I do not understand very well what he wanted to say. Probably he wanted to mention that modern mathematics has a common root originated in the 19-th century.

According to my opinion also, commutative algebra was started by D. Hilbert at the beginning of the 20-th century from his famous works on invariants. This is the reason, why we meet his names so many times in commutative algebra. Hilbert basis theorem (Every finitely generated algebra over Noetherian rings are Noetherian.), Hilbert’s zero points theorem in algebraic geometry, Hilbert characteristic functions and series, the Hilbert-Burch type theorem of finite free resolutions, and so on.

After Hilbert and E. Noether, people were deeply interested in the structure of rings (commutative and even non-commutative) from the view-point of (so-called) “classical” ideal theory. Even nowadays, it is a very important view-point to analyze the structure of rings to see how many different kinds of ideals are contained in the given rings. W. Krull was one of the famous mathematicians of that time (for us), who built up the frame work of ideal-theoretic commutative algebra, which we frequently use even now; Krull’s altitude theorem, Krull’s intersection theorem, and Krull domains. You must remember also F. S. Macaulay. He discovered, around 1915, many important theorems on the ideals (homogeneous or even monomial ideals) in the polynomial rings k[X_1, X_2, … , X_n] over a field k. Since his languages are quite different from ours (e.g., modular systems = ideals), for us it is very difficult to follow his arguments, but his researches were recovered around 1980’s and recognized to be very deep and important. However the real revolution was made by J.-P. Serre in 1955. He introduced, at the famous conference of Tokyo Nikko, homological algebra into commutative algebra and proved that the localization A_P of a regular local ring A (i.e., a Noetherian local ring A whose maximal ideal generated by d elements, where d = dim A, the Krull dimension of A, so that the local rings of Dedekind domains are regular local rings of dimension at most 1) is again regular for every prime ideal P in A. This result settled the long-standing difficult open problem; even now we have no proof of his result without using homological method.

Homological algebra is essentially linear algebra over rings. Serre’s method provides, in some sense, a kind of representation theory of rings. For example, ideal theory intends to clarify the structure of rings from inside, but Serre showed that there is some possible method to analyze rings and algebras (by their behaviors among or against the others) from outside of them!

After Serre, people rushed to learn and wanted to use homological method. The people working on classical ideal theory were left alone behind them, so that the mathematical generations made a drastic change.

There also many excellent mathematicians appeared together with Serre; Nagata, Northcot, Samuel, Rees, etc. You must remember Zariski, who was the most important algebraic geometer of the 20-th century. The book “Zariski-Samuel” is still an excellent text book for us (I like the book.). It was written a little after Serre’s work, so that only in the appendix the book has cited to Serre’s work on homological methods and the rings, so-called Cohen-Macaulay rings.

On the other hand, 1960’s were years of A. Grothendieck, who gave a drastic revolution in algebraic geometry. Grothendieck began to re-write classical algebraic geometry into abstract algebraic geometry, introducing the concept of scheme, based on homological algebra. While his great work was developed, it was completely recognized that commutative algebra is exactly the theory of affine schemes (local theory of schemes), and then commutative algebra fell down into the position of menservants of algebraic geometry.

Even during these unhappy days of commutative algebra, the giants, say Bass, Auslander, and Buchsbaum, etc. prepared for the next days/chances for commutative algebra and finally, for the last 20 years of the 20-th century, the people like Hochster, Huneke, Bruns, and Herzog, and so on, succeeded in rebuilding commutative algebra, mainly constructing the deep theory of Cohen-Macaulay rings (appeared in invariant theory, singularity theory, and conbinatorics, etc.).

This brief history might explain the reason why I strongly recommend you to study the other kind of commutative algebra apart from your own (and certainly favorite) background. At first you might feel not happy, but soon you will find big freedom.

I am very afraid if my explanation is not good enough. I wish if you could speak and read Japanese!

But anyway after coming to Japan, you will automatically enter this world. Even in your country I think you can do so. But, I feel (I am not sure, but) that probably you are isolated in your home city; so at this moment it might be better to come to Japan.

Please let me recommend you as follows. You already finish the book of Atiyah-Macdonald. That is wonderful. This book is the ordinary textbook in my lab. You already know about Hilbert series of modules and the basic theory of dimension of rings and modules, systems of parameters of finitely generated modules over Noetherian local rings, etc.

Some of my students finish the book of Atiyah-Macdonald while they are in undergraduate courses, but the others finish it in the middle of the first year of graduate courses (Master courses, I mean).

Usually in the first semester of the first year of master courses, I give a lecture of introductory homological algebra, which consists of the following contents, where everything is discussed over commutative rings.

(1) Projective modules and injective modules (See Maclane.)
(2) Projective resolutions injective resolutions of modules (See Maclane.)
(3) Definition of Ext and Tor (See Maclane.)
(4) Ext and Tor under localization and flat base changes (See Northcot’s book: Introduction of homological algebra, or think by yourself.)
(5) Associated prime ideals of modules (See Bourbaki: Commutative Algebra, Ch.4)
(6) Depth of finitely generated modules over Noetherian local rings (See Bruns-Herzog or Eisenbud.)
(7) Cohen-Macaulay rings and modules (See Bruns-Herzog.)
(8) Regular local rings and Serre’s theorem (See Bruns-Herzog or Eisenbud.)
(9) Explicit resolutions of ideals and Koszul complexes (See Bruns-Herzog or Eisenbud.)
(10) Structure theorem of injective modules and injective resolutions over Noetherian rings (See Bruns-Herzog or Sharp’s papers.)
(11) Gorenstein rings (See Bruns-Herzog or a series of Sharp’s papers.)
(12) Local cohomology and local duality theorem (See Bruns-Herzog or a series of Sharp’s papers, or Herzog-Kunz’s book of Lecture Notes in Mathematics, 238, Springer Verlag.)
(13) Graded rings and modules (See Bruns-Herzog or my paper jointly wit K.-i. Watanabe, On graded rings, I.)
(14) Theory of multiplicities (See Bourbaki: Commutative Algebra, Ch. 8, 9.)
(15) Dualizing complexes (See Sharp’s papers)

After studying these, I usually recommend my students to read several papers or Serre’s book “Local Algebra”, to attack concrete objects like numerical semi-group rings in order to find or solve concrete problems, or just to deepen their knowledge of commutative algebra. Recent days, I give problems on Rees algebras, associated graded rings, and Hilbert coefficients of ideals something like that.

Even if you do (or, can) not join us, I would like to recommend you to go ahead in this way and begin to study these theme. I agree that it is a long way. But you are very young. If you do not lose your destination, you can certainly reach there within 1 or 2 years.

Also please let me know if you cannot find papers or books cited above. I shall arrange what I can do for you, so that you will access them.

Finally, let me give you some information about the recent topics on commutative algebra, which you might not understand at this moment, but I am sure it is worthy for you to keep in your mind towards your further study.

(1) Rees algebras of ideals and modules and related topics on graded rings associated to ideals
(2) Integral closures of ideals, modules, and rings
(3) Hilbert functions and polynomials
(4) Castelnuovo-Munford regularity
(5) Homological conjectures
(6) Singularity theory
(7) Tight closures
(8) Representation theory of local rings
(9) Combinatorial commutative algebra (Stanley-Reisner rings and ideals, monomial ideals)
(10) Computational commutative algebra
(11) Interfaces of all above

With best wishes for your fruitful stay in my lab,

Sincerely yours,

Shiro Goto