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Algebra Tips for reading Grothendieck's EGA/SGA/FGA trilogy

  1. Feb 8, 2017 #1
    Dear Physics Forum advisers,

    I am very interested in studying the art of algebraic geometry, motivated by its applications in the machine learning and data analytics. I recently came across Grothendieck's EGA/SGA/FGA saga, and I am really interested in reading it as I like how it presents the AG in most general setting, starting with orders sets (I prefer learning in most abstract setting and trying myself to deduce it into concrete examples); I also heard it has a lot of useful theorems and ideas (few of them were actually used in the cryptography). Where should I begin to read it? I am not sure how EGA, SGA, and FGA are correlated with one another; it seems that they are voluminous. What books could I use as good supplements for Grothendieck?

    My background in algebra: I finished reading Isaacs and Aluffi in abstract algebra. I have been reading Eisenbud's book in the commutative algebra. Although my background is not most complete, I prefer to learn and investigate topics as I read books, rather than waiting to finish all necessary prerequisites (my interest dies if I do later).
     
  2. jcsd
  3. Feb 9, 2017 #2
    Anyone?
     
  4. Feb 10, 2017 #3

    Demystifier

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    You seem very ambitious. :wideeyed:
    Do you know how many books (and how many pages) the complete saga has? And do you really think that it is useful for applications in the machine learning and data analytics?
     
  5. Feb 10, 2017 #4
    I know that EGA has at least five volumes, SGA has 7 volumes, and FGA has 4 volumes, with different sub volumes for each. I know it is useful since many data points do not adhere to specific types of fields, and EGA deals with AG from most general perspective.

    I could not find hardcopies for EGA and first volumes for SGA (I aways prefer hardcopy over PDF)....I am currently searching for foreign websites for purchasing EGA.
     
  6. Feb 10, 2017 #5

    Demystifier

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    Are you searching for copies in english language? (If you want pdf/djvu in Franch, I can help.)
     
  7. Feb 10, 2017 #6
    The language choice does not matter, but French would be better since I believe even the most sophisticated translation has some loss of original flavors. However, I would like to acquire hardcopies, the original publications. I know there are PDF/djvu in both languages, but I do have some vision problems that discourage me from reading electronic files. I have been searching for foreign websites that sell those books, but so far I have no luck.
     
  8. Feb 10, 2017 #7

    Demystifier

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    Speaking of Grothendieck, is there something like Grothendieck For Dummies? I mean, something like a book with the most important ideas of Grothendieck explained at a semi-popular non-technical level.
     
  9. Feb 10, 2017 #8
    I do not think so. I just want to find hardcopies of EGA and SGA. I really ant to read them, but avoid vision problem from electronic reading too.
     
  10. Feb 10, 2017 #9

    Demystifier

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    Well, if you have pdf/djvu, then a good printer should solve your problem. :smile:
     
  11. Feb 10, 2017 #10
    That is true. I am going to try it. By the way, it seems that different copies exist for EGA in online, with different page numbers even for original volumes. I am confused.
     
  12. Feb 13, 2017 #11

    martinbn

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    You would need to be very strong at commutative algebra and comfortable at reading very general and abstract mathematics to start with the EGA. If it doesn't work at first, try some other textbook in algebraic geometry. A classic one is Hartshorn's book, but there are others including many lecture notes.
     
  13. Feb 13, 2017 #12

    MathematicalPhysicist

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    I didn't find a hardcopy of these texts, try a printer.

    What a coincidence I had thought a few days ago of reading these texts by Grothendieck, but I cannot find the time for it;

    First you need to read Macdonald's and Atiyah's book on commutative algebra, or also Jacobson's volumes; there's also Eisenbud.

    If French isn't your mother tongue (I have two mother tongues, but one isn't really good for technical subjects such as maths and physics), you should use a dictionary, one of my goals is to translate this to English in latex (it would take sort of 4-5 years just spending on this translation and understanding the material.)
     
  14. Feb 13, 2017 #13

    mathwonk

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    I would just go to a library and sit down and try to read a few pages. I myself would not recommend those sources as useful for most people, although they were valuable sources for some of my most brilliant friends. I myself am a (retired) professional algebraic geometer, who became initiated into the field only after abandoning the attempt to read those ponderous tomes, and just begin to work on concrete problems. "Grothendieck for dummies" does exist in a sense, namely Hartshorne's book, although the cutoff for those dummies is pretty high. Another source is some lecture notes on my shelf from Univ of Maryland, lecture notes #1 1962, "available from Harvard math dept" (probably not any longer), where Dieudonne' gave a short survey of the content of EGA. David Mumford's "red book" is one of the very best introductions to "schemes" which is Grothendieck's basic concept underlying modern algebraic geometry. His later version of this book, edited by Oda, is available online. If you want to learn schemes, I would suggest beginning with the original "red book" by Mumford.

    My own short essay is not to be compared with these previously recommended authoritative works, but is aimed at a naive young student, so may interest a beginner. It is only a few minutes' read, compared to these other deeply informative sources which require long study:
    http://alpha.math.uga.edu/~roy/introAG.pdf
     
  15. Feb 13, 2017 #14
    Thanks for your valuable advice! I actually studied the basic grammars and vocabulary of French (SparkCharts and books). I believe it is enough to read Grothendieck as I believe EGA is written in simple French/ I am reading Eisenbud, which is easy and detailed enough for me to do all imagination (I do not know if Jacobson has a book in the commutative algebra; for the basics of algebra, I read Isaacs and Aluffi, which I believe is better than Lang).
     
  16. Feb 13, 2017 #15
    Dear Professor mathwonk, thank you for your valuable advice! I am going to read your essay about the AG.
    I thought Hartshorne is actually lite version of EGA, where many details and proofs from EGA are skipped, resulting in the elevated difficulty of reading of Hartshorne. Anyway, I really like Eisenbud's book in the commutative algebra, so I bought his second volume (Geometry of Scheme). I hope it is as good introduction to schemes as Red Book; I can buy that book too of course (got the research funding!).

    One of the motivation for my study is that since the data space has no clear structures (at least in pseudocode approach), so I thought it would be better to start approaching it with AG from most general perspective, assuming only the basic structures of sets. I believe Grothendieck generalized AG to the level of partially ordered sets, rather than standard introduction to algebraic fields.

    Could you send me Dieudonne's lecture notes you mentioned?
     
  17. Feb 13, 2017 #16
    Also, what are key differences between The Red Book (2nd Edition) and Algebraic Geometry I-II by Mumford? My impression is that AG I-II is a follow-up to Red Book since former treats the AG in complex manifolds, but I am not sure.
     
  18. Feb 14, 2017 #17

    mathwonk

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    the original red book, issued in a second edition, were notes for a course by Mumford introducing the ideas of schemes, beginning with a brief sketch first of more classical "varieties". Then he discusses basic concepts of scheme theory, with motivation for taking a categorical perspective for instance in defining products. One usually thinks of a product of X and Y as a set of ordered pairs of elements (x,y), but this fails for schemes because the "points" are more sophisticated, and can have various dimensions, i.e. not all points in a scheme are zero dimensional. So the right approach is to realize that a product of X and Y is another object Z equipped with "projection" maps Z-->X and Z-->Y such that every pair of maps A-->X and A-->Y is induced from a unique map A-->Z via the projections. Then he discusses the consequences and complications arising from changing the ring of scalars, not just doing geometry over R or over C, and how that introduces Galois theoretic matters and leads to a concept of a variety as a "functor". Then he discusses the local theory and how to algebraicize matters via consideration of coherent and quasi coherent modules, somewhat analogous to vector bundles in differential geometry. Finally he discusses some classical results of Zariski, such as the "main thorem" describing the geometry of "normal" varieties.

    The later version (AG II?), edited by Oda, omits the discussion of classical varieties and begins as I recall with affine schemes. (Affine schemes are to schemes as coordinate neighborhoods are to manifolds.) He also includes some more sophisticated versions of his discussion of complex geometry as introduced in his Algebraic varieties I. Finally he discusses in some detail the main topic omitted from the red book, namely sheaf cohomology, especially in the case of the Cech construction, including a discussion of spectral sequences, for which he tries to "debunk" their reputation for being so difficult.

    The little volume Alg Var I, is a very terse and jam packed volume on complex algebraic geometry in projective space with many useful results proved carefully but succinctly that are often omitted in other books. He distills there a proof of desingularization of curves from the monumental argument by Hironaka for the general n dimensional variety, and he gives a complete proof that every non singular cubic surface has exactly 27 lines on it. This is a concise and deep treatment of many classical topics, including a very clear and useful account of the classical Riemann Roch theorem for curves.

    So the books are logically ordered as : 1) AGI, 2) redbook, 3) AGII (Oda), but I would not completely postpone reading the later volumes if they are your interest, since you could conceivably spend years just mastering the first one. I believe you can reasonably begin on either of the first two, but it seems more challenging to begin with the third.

    By the way, many people feel the best beginning book on algebraic geometry is the book of Shafarevich, but it is not primarily aimed at Grothendieck's theory.

    (sorry, I have no easy way to send copies of the maryland notes, but they should exist in libraries.)

    there is another source for an intro to grothendieck style AG, a translation of three volumes from the Japanese, namely AG 1, 2 and 3, by Kenji Ueno. Japanese works tend to be rather complete, e.g. there are solutions to exercises given. This work covers transition from varieties to schemes, schemes, and cohomology.
    https://www.amazon.com/s/ref=nb_sb_noss?url=search-alias=stripbooks&field-keywords=kenji+ueno,+algebraic+geometry


    If you want my advice, for an introduction to Grothendieck's ideas, begin with Mumford's red book. Then progress to Hartshorne. But if you are attracted to the task of reading Grothemdieck-Dieudonne' (it really is a joint work, probably largely written by Dieudonne'), keep a copy of EGA or SGA handy and try it from time to time.

    The paradox is that a more detailed work is not necessarily easier to read, because it is just too long. as they say, you don't see the forest for the trees.

    in the special but important case of curves, there is a fantastic book by George Kempf, explaining and using Grothendieck's ideas to treat Jacobians of curves, called Abelian Integrals, that used to be available from the University of Mexico Autonoma. Some libraries have this as well, but it is not as well known as it deserves to be due probably to being published only by the Univ of Mexico.

    https://www.amazon.com/Abelian-integrals-George-Kempf/dp/B007FD5XKW/ref=sr_1_1?s=books&ie=UTF8&qid=1487120828&sr=1-1&keywords=george+kempf,+abelian+integrals


    By the way, George Kempf himself is one of my brilliant friends who read EGA.
     
    Last edited: Feb 15, 2017
  19. Feb 14, 2017 #18

    MathematicalPhysicist

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    I tried reading Kenji Uneo's series; but it's hard to understand, he skips on some definitions in the book or there's no algorithmic explanation of how to calculate for example the local intersection multiplicity;
    you can read one of my questions from the first volume over there:
    http://math.stackexchange.com/questions/1070309/local-intersection-multiplicity

    It's good that there are solutions to the problems even if they are too short to really understand why they are correct.
     
  20. Feb 15, 2017 #19

    mathwonk

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    thanks for this testimony. notice i did not really recommend Ueno's books, but since the request was for sources that try to introduce Grothendieck ideas, I mentioned it. algebraic geometry is such a huge subject it is hard to know what to recommend to a beginner. the super general gigantic tomes like EGA are essentially the last things I would recommend, since one has virtually no chance of grasping their importance without knowledge of more classical and specific topics. Still, one cannot understand something without opening it so one might as well go for it, if that is the attraction. just don't expect to either understand what you read there or to understand the larger subject. so again, i really suggest Mumford's red book for an intro to schemes, and Serre's FAC as an intro to cohomology in algebraic geometry. For cohomology in complex manifold theory, I like Hirzebruch's Topological methods in algebraic geometry.

    I admit that even as a retired professional algebraic geometer, when I moved and had to pare down my library, I parted with my copies of EGA, as books I had never benefited from. At the moment, I do have some regrets, but that was my choice at the time. I did not part with any of Mumford's works, nor Shafarevich, nor Hartshorne. I also kept my copies of George Kempf's works, who himself learned from EGA. I also recommend anything by Arnaud Beauville. I kept also my copies of Euclid, Archimedes, Gauss, Riemann, Galileo, Euler, Hilbert, Goursat, Serre, E. Artin, Siegel, Weyl, M. Artin, Bott - Tu, Segre, Arbarello, Cornalba, Griffiths, Harris, Gunning, Walker, Fulton, Dold, Brieskorn, Manin, Lang, Zariski-Samuel, Courant, Cartan, Jacobson, A.A. Albert, Van der Waerden, Atiyah, Hirzebruch, Spivak, Mackey, Steenrod, Dieudonne', Berberian, ....
     
    Last edited: Feb 15, 2017
  21. Feb 15, 2017 #20
    Thanks! I decided go with Mumford's Red Book along with Shafarevich, and try EGA later. I am surprised to see that Shafarevich has virtually no prerequisites, and I like how he slowly builds the intuition for varieties. I might skip his second volume as I believe Mumford covers it.
     
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