The Grimoire Project

The Grimoire is a textbook on commutative algebra for advanced master students and graduate students that I am currently writing. Some of its last chapters could be interesting also to more advanced researchers. It is written in French. Here you can download the current version :

  • Grimoire d'Algèbre Commutative

  • Your comments are welcome and gratefully acknowledged!

    Status Update :

  • November 8, 2014 : work on the Grimoire project has resumed. Lectures Bélier, Taureau et Gémeaux are complete; currently working on Cancer.

  • December 7, 2014 : Lectures Bélier, Taureau, Gémeaux et Cancer are complete; currently working on Lion.

  • March 6, 2015 : Lectures Bélier, Taureau, Gémeaux, Cancer and Lion are complete; currently working on Vierge.

  • May 23, 2015 : The Grimoire Project enters Summer recess : until the end of September I will be working on a different project, so there will be not much action for the Grimoire : I might make the occasional correction, and maybe toy with the index once and then, but other than that don't expect much until October. The Vierge lecture is not quite finished yet, but it is well advanced.

  • October 14, 2015 : work on the Grimoire project has resumed. Currently working on Vierge.

  • October 23, 2015 : Lectures Bélier, Taureau, Gémeaux, Cancer, Lion et Vierge are complete; currently working on Balance.

  • November 1, 2015 : I've corrected the proof of Zariski's Main Theorem, that was confusing and slightly broken in various places. Please download the revised version, and keep on sending me your remarks. I've also added a result on UFDs with some applications, at the end of section 11.4. Now, back to Balance.

  • December 15, 2015 : Lectures Bélier, Taureau, Gémeaux, Cancer, Lion, Vierge and Balance are complete; currently working on Scorpion. I've also made some small corrections to the proof of Swan's theorem in Lecture 5 (Lion), and I've added some exercices at the end of section 6.5.

  • February 1, 2016 : I've taken the important decision to add a section (a revised section 5.5 that I'm writing now) on the language of sheaves and a few basic notions of scheme theory. This has entailed some reorganization in lectures Lion, Vierge and Balance. The reason to include this material is that eventually I'm planning to give a presentation of a valuation-theoretic proof due (essentially) to Fujiwara of Gruson-Raynaud's important theorem on flattening of morphisms by admissible blow-ups. This proof is more elementary than Gruson and Raynaud's original argument, but of course to properly state the theorem and to give a clean presentation of its proof it is convenient to be able to manipulate some non-affine schemes (such as blow-ups).

  • April 7, 2016 : I've finally completed the digression on sheaves and schemes announced in February; it is longer than originally planned and is now split over two sections : the new section 3.4 on sheaves and the new section 5.5 on schemes. This entailed some rearranging of material around the text. Now I'm ready to return to Scorpion.

  • April 29, 2016 : The Grimoire Project enters Summer recess : until late September I will be working on a different project, so not much will happen on this page. Scorpion is not quite finished, but is well advanced. See you in a few months!

  • November 21, 2016 : The Grimoire Project is active again! I was very busy until now, but at last I am free. I've just started a new section in Scorpion, dedicated to the real spectrum of a ring : it doesn't contain much at present, but watch it grow!

  • January 16, 2017 : Lectures Bélier, Taureau, Gémeaux, Cancer, Lion, Vierge, Balance and Scorpion are complete; next up is Sagittaire.

  • April 13, 2017 : Lectures Bélier, Taureau, Gémeaux, Cancer, Lion, Vierge, Balance, Scorpion and Sagittaire are complete. I've also made several minor fixes in earlier parts of the text, and added several new entries to the index. There will be a few further updates next week, then the Grimoire Project will enter Summer recess.

  • December 12, 2017 : work on the Grimoire Project has finally resumed! Currently working on Capricorne.

  • March 4, 2018 : today the Grimoire Project has reached an important milestone : the proofs of both the Gruson-Raynaud theorem and of elimination of quantifiers for fields with non trivial valuations are complete : see theorems 10.57 and 10.62 (I give a geometric form of elimination of quantifiers : see the remarks before paragraph 10.4.3). Still working on Capricorne : section 4 is almost done, and it remains (only) to write section 5.

  • Mai 5, 2018 : Lectures Bélier, Taureau, Gémeaux, Cancer, Lion, Vierge, Balance, Scorpion, Sagittaire and Capricorne are complete. Today the Grimoire also enters officially into summer recess. I might do some little cleaning up, and also add a few entries to the index, but that will be it until October at the earliest, I expect.

  • June 27, 2018 : I took advantage of some free time to make a thorough reorganization of the last 4 chapters of the Grimoire (and I've also added an exercise in what is now section 9.2). The upshot is that, for the time being, the Grimoire is complete only up to Sagittaire, so in a sense this is a step backward, but it's only in order to prepare a leap forward that I'll take after the summer : then hopefully I'll be able to complete both Capricorne and Verseau in a relatively short time. Also, you can start seeing now the general shape that Poissons will have in the end : more material is planned, but the basic scaffolding is in place.

  • September 23, 2018 : work on the Grimoire Project has resumed! Currently working on the revised version of Capricorne.

  • November 2, 2018 : I'm finally done with the revision of Capricorne! It took slightly longer and slightly more effort than I had anticipated, because while working on it, I realized that the previous draft was rather (and embarassingly) rough : now it is much better. Hence : lectures Bélier, Taureau, Gémeaux, Cancer, Lion, Vierge, Balance, Scorpion, Sagittaire and Capricorne are again complete, and I'll start working on Verseau in a matter of days, after I rest a little bit with a short break.

  • December 30, 2018 : During the holidays I'm not working on Verseau, but I'm not quite idle either : I just finished fixing a very old and embarrassing mistake : the homological and cohomological notations for complexes were all mixed up, throughout the Grimoire. I have been aware of the problem for years, but I had never found the time and the nerve to correct this : now it's done. I'll resume working on Verseau around the second week of January : stay tuned!

  • January 26, 2019 : Lectures Bélier, Taureau, Gémeaux, Cancer, Lion, Vierge, Balance, Scorpion, Sagittaire, Capricorne and Verseau are complete. I'm taking a well deserved break from the Grimoire, to take care of a bunch of long-delayed other smaller projects.

  • April 22, 2019 : I've just completed a major revision of Bélier : two of the previous sections (1.3 and 1.4) have fused into a single longer one, and a new section 1.4 has been added, containing a self-contained treatment of the elementary theory of fields; moreover, some material on unique factorisation domains has been moved from section 5.2 to section 1.1.

  • May 8, 2019 : today the Grimoire project enters officially into summer recess : I might add some entries to the Index, but do not expect any substantial work until October at the earliest. The last addition to the text so far is problem 12.27, detailing Nagata's example of an infinite dimensional noetherian domain. See you in a few months!

    September 29, 2019 : work on the Grimoire Project has resumed! Currently working on Poissons. The first new result is a proof of the classical Molien formula for the Hilbert-Poincaré series of the ring of polynomial invariants for the action of a finite group on a finite-dimensional vector space : see Problem 12.55.