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

[bacte2013]

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).

 

[bacte2013]

Anyone?

 

[Demystifier]

You seem very ambitious.
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?

 

[bacte2013]

You seem very ambitious.
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?

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.

 

[Demystifier]

Are you searching for copies in english language? (If you want pdf/djvu in Franch, I can help.)

 

[bacte2013]

Are you searching for copies in english language? (If you want pdf/djvu in Franch, I can help.)

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.

 

[Demystifier]

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.

 

[bacte2013]

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.

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.

 

[Demystifier]

I just want to find hardcopies of EGA and SGA. I really ant to read them, but avoid vision problem from electronic reading too.

Well, if you have pdf/djvu, then a good printer should solve your problem.

 

[bacte2013]

Well, if you have pdf/djvu, then a good printer should solve your problem.

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.

 

[martinbn]

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.

 

[MathematicalPhysicist]

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.)

 

[mathwonk]

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/%7Eroy/introAG.pdf

 

[bacte2013]

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.)

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).

 

[bacte2013]

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/%7Eroy/introAG.pdf

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?

 

[bacte2013]

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/%7Eroy/introAG.pdf

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.

 

[mathwonk]

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

edit: I tried reading a few pages in vol. 1 of this series just now and was not favorably impressed. The style seems to emphasize complicated and unilluminating formulas, as opposed to clear insights, and the formulas I just checked (p.31) even seem to be somewhat in error, (notice on line -14, p.31, that n+1 quantities have been substituted into a polynomial f of only n variables - the term x^(k),j+1/x^(k),j+1 should have been omitted), as is apparently also the summary of their meaning at the bottom of page; i.e. I believe one does not replace xi/xj by xi/xk, but rather by (xi/xk)/(xj/xk), as is visible in the line -14 just referred to. Otherwise the substitution would not make sense, as the corresponding functions would not be equal. The substitution is supposed to make different representations of the same function correspond to each other. Still I am sure there are many things to be learned from this source, e.g. in the form of explicit details. E.g. the discussion of existence of fibre products on pages 120-127 of Ueno's book seems more detailed than the sketch on pages 84-85 in Mumford's red book.

further edit: I just tried to read even the introductory parts of that treatment of fibre products, the discussion of representable functors, and found it also very confusing, due again to an apparent lack of facility in the use of the English language by the translator. Again the formulas are mostly correct, with a few typos, but the discussion surrounding their meaning is quite confusing, i.e. these formulas do not seem to be showing what he says they are. Or rather it takes some familiarity with what is going on to understand just what the brief choice of words is meant to convey. So unless one already knows what is going on, it is hard to find out from the text. This is too bad, as Ueno is a world renowned expert.

So I cannot recommend the English translation of Ueno's book to a beginner, although very likely Japanese readers with access to the Japanese original will no doubt have a completely different experience.

Actually, having struggled to figure out what was meant by the explanations in Ueno, and having corrected the typos, I appreciate the beauty of this section on representable functors more. The mathematics is well chosen, and details are provided for some helpful parts, and the occasional remark as to the importance of a topic is helpful. I have actually learned something. But it is probably crucial that I already knew what was going on. And I am clearly being unfair and ungrateful to the translator, since with effort I can learn from his translation, whereas the original would be impenetrable to me.

Unfortunately such minor but troublesome errors are often introduced when a work is translated from another language. Of course it is also true that many typographical errors were introduced into Mumford's red book as well, when it was re-typeset by Springer, so the original notes from Harvard's math department are preferable, but now quite rare. (I bought mine there over 50 years ago, but also have the Springer version which I am actually reading now. A bonus of the Springer version, at least the second expanded edition, is that it includes the wonderful 1974 lectures on Curves and their Jacobians, delivered by Mumford at Univ. of Michigan.) Another advantage of the original redbook is the more generous spacing of the type on the page, which clues you in as to when a new idea is being begun. This is often obscured in the more crowded Springer formatting.

So my advice, for an introduction to Grothendieck's ideas, is to begin with Mumford's red book, consulting other books for details. Then progress to Hartshorne. But if you are attracted to the task of reading Grothendieck-Dieudonne' (it really is a joint work, probably largely written by Dieudonne'), keep a copy of EGA or SGA handy and try reading it from time to time. Or read Hartshorne and use EGA to fill details. I myself am still plowing through Mumford's redbook, after more than 2 years, off and on, and it is to me at least, not at all easy sailing. Still it is wonderfully eye-opening, but every line requires thought, and sometimes outside study. This why I was led to try Ueno for more detail on fiber products, but may now turn instead to Hartshorne, or maybe back to Ueno, since I do roughly know what is the point of it. So I need about as long to figure out what Ueno's book is trying to say as I do to understand what Mumford is not saying. Perhaps oddly, I still prefer a cryptic remark by an expert, to a detailed explanation from a novice, (not thinking of Ueno here, but others, more or less like myself, who have written treatments), which latter may have actually missed the entire point.

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/dp/B007FD5XKW/?tag=pfamazon01-20

This is not necessarily easy reading, but provides important material in a form not available elsewhere.
By the way, George Kempf himself is one of my brilliant friends who read EGA. George is also author of an amazingly terse but informative book Algebraic varieties, which treats sheaf cohomology in the context of varieties as opposed to schemes, but which is also subject to numerous typographical errors.
https://www.amazon.com/dp/0521426138/?tag=pfamazon01-20

 

[MathematicalPhysicist]

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 schems, schemes, and cohomology.

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.

 

[mathwonk]

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, ...

 

[bacte2013]

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, ...

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.

 

[bacte2013]

By the way, it seems that IHES sells the complete volume of EGA at their website. The price is around 300 dollars, including the shipping cost from France.

 

[MathematicalPhysicist]

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, ...

What do you think of Hodge's classical methods of algebraic geometry? I don't think schemes is dealt over there, but perhaps it's better to first try to go through classical algebraic geometry (I read through Harris and Griffiths which covers some classical theorems such as Cayley's 9 points theorem (not the exact wording of the theorem), I should read through it once more in the future, after I'll finish reading the algebraic topology books (I took a course in algebraic topology, but I don't believe I have a firm and rigorous grasp of the subject; I aced the course, but it doesn't mean nowadays that you understood the subject :-) ).

I also finished reading Ulrich Krengel's book called Ergodic Theorems, but I feel I should have finished reading Bogachev's 2 volumes before trying to attack ergodic theory. (I know the basics of measure theory, but I haven't yet got to the Radon Nykodim theorem, which seems important also in ergodic theory.)

 

[bacte2013]

What do you think of Hodge's classical methods of algebraic geometry? I don't think schemes is dealt over there, but perhaps it's better to first try to go through classical algebraic geometry (I read through Harris and Griffiths which covers some classical theorems such as Cayley's 9 points theorem (not the exact wording of the theorem), I should read through it once more in the future, after I'll finish reading the algebraic topology books (I took a course in algebraic topology, but I don't believe I have a firm and rigorous grasp of the subject; I aced the course, but it doesn't mean nowadays that you understood the subject :-) ).

I also finished reading Ulrich Krengel's book called Ergodic Theorems, but I feel I should have finished reading Bogachev's 2 volumes before trying to attack ergodic theory. (I know the basics of measure theory, but I haven't yet got to the Radon Nykodim theorem, which seems important also in ergodic theory.)

Could you answer my questions to you? I have not heard back from you about your unsupportive remarks.

 

[MathematicalPhysicist]

Could you answer my questions to you? I have not heard back from you about your unsupportive remarks.

What questions have you asked me? I don't see any questions in this thread addressed to me.

BTW, thanks for telling us that IHES are selling copies of EGA.

 

[MathematicalPhysicist]

What questions have you asked me? I don't see any questions in this thread addressed to me.

By the way, it seems that IHES sells the complete volume of EGA at their website. The price is around 300 dollars, including the shipping cost from France.

Can you provide a link?

I tried searching through google and I didn't find a suitable webpage from which I can purchase a copy of it.

 

[Demystifier]

"Grothendieck for dummies" does exist in a sense, namely Hartshorne's book, although the cutoff for those dummies is pretty high.

Hartshorne certainly does not fit my definition of "Grothendieck for dummies", but I found something that does:
http://www.dam.brown.edu/people/mumford/blog/2014/Grothendieck.html

 

[bacte2013]

Hartshorne certainly does not fit my definition of "Grothendieck for dummies", but I found something that does:
http://www.dam.brown.edu/people/mumford/blog/2014/Grothendieck.html

I bought the hardcopy of EGA from IHES in France. They sell the whole volume for 270 USD, including shipping fees. I am so grateful that they decided to reopen the publishing.

Speaking of Mumford, I am currently deciding whether I should read his The Red Book or Qing Liu's Algebraic Geometry and Arithmetic Curves. I believe they are good preparation for EGA, but I am not sure which one I should go for. Also, someone from MathOverFlow said that Heartshorne is really a lite version of Grothendieck, which makes Hartshorne more difficult to read then EGA.

 

[bacte2013]

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_...ji+ueno,+algebraic+geometry&tag=pfamazon01-20If 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/dp/B007FD5XKW/?tag=pfamazon01-20By the way, George Kempf himself is one of my brilliant friends who read EGA.

Dear Professor mathwork,

Have you read a book called "Algebraic Geometry and Arithmetic Curves" by Qing Liu? I heard it resembles EGA a lot, so I borrowed it and read side to side with Mumford's The Red Book. Which one is better for building knowledge for EGA?

 

[mathwonk]

no i have not read qing liu but it sounds appealing. the virtue even of shafarevich's second volume is the easy reading level, the skill for choosing things that are accessible. e.g. the subject of holomorphic versus algebraic varieties is very nice there. so even though his second volume is not comprehensive on anything, that is actually a plus. i.e. shafarevich chooses every topic according to how easy it is to present and understand.

 

[bacte2013]

no i have not read qing liu but it sounds appealing. the virtue even of shafarevich's second volume is the easy reading level, the skill for choosing things that are accessible. e.g. the subject of holomorphic versus algebraic varieties is very nice there. so even though his second volume is not comprehensive on anything, that is actually a plus. i.e. shafarevich chooses every topic according to how easy it is to present and understand.

Thanks for your advice! I am just going to read both Liu and Shafarevich. How much do I have to know about varieties before moving on to sheaves and schemes? It seems that Shafarevich's first volume has an extensive treatment of the varieties on the projective space, but I am not sure if all of the are relevant for AG from schemes. I read through the preface of second volume, but it did not specify topics from the 1st volume I need to learn well.

 

[martinbn]

I've looked at Liu's book some time ago, and then I found it very nice.

 

[mathwonk]

Pretty much all of Shafarevich vol. 1 is basic. I.e., dimension, non singularity, normality, divisors and differentials, and intersection numbers.

There is one serious error in chapter I.6.3, i.e. the corollary to theorem 7 is false unless the map is assumed "proper". This forces one to modify the proof later on the argument that the set of singular points of a variety is closed in chapter 2.1.4. Either replace the affine space there by projective space, or use the correct argument in the original (one volume) edition of Shafarevich's book, before it was subdivided into two volumes.

I also believe the argument for the existence of the normalization of a projective curve to have a gap, i.e. Th. 6, chapter 2.5.3.

There are other small errors, but overall Shafarevich is really excellent and readable, and the other flaws do not detract much.

 

[mathwonk]

If this still has interested readers, let me give an example I have just tested out myself. I am now reviewing the proof that projective space satisfies the algebraic version of compactness, i.e. that it is complete or "universally closed". In particular, every regular map from projective space to affine space is constant, and every regular map from a projective variety to any variety has closed image.

I want to refer you to 4 sources where the proofs of this fact are all essentially the same, but I assure you they look very different. Please try reading these to see which one are more readable by you. I personally like the first two, i.e. less abstract, ones, and it took me a significant amount of thought to see they are all essentially the same.

notice also that the determinant of a linear map is a significant tool in the proof of both the properties of surjective linear maps and of the nakayama lemma.

1) Shafarevich, Basic algebraic geometry I, chapter I.5.2.
2) Mumford, Algebraic geometry I, Complex projective varieties, p.34.
3) Mumford, redbook of algebraic geometry, chapter I.9.
4) Grothendieck, EGA II.5.5.3.

There is a small variation in these proofs at the end, wherein the first two use openness of the condition that a linear map is surjective, while the last two use Nakayama's lemma, but the key first part of all the proofs is the same. I admit I also like the last two proofs now that I understand them, but my point is they could not possibly be understood (by me) until after understanding the first two.

why are there lineouts through my statements?

 

[FourEyedRaven]

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.

The biographical articles on the AMS

https://www.ams.org/notices/200409/fea-grothendieck-part1.pdfhttps://www.ams.org/notices/200410/fea-grothendieck-part2.pdf
or the following mathematical biography

www.amazon.com/Alexandre-Grothendieck-Mathematical-Portrait-English/dp/1571462821/

 

[mathwonk]

In old fashioned algebraic geometry a point of an algebraic variety is an n tuple of "numbers" at which all of a given collection of equations vanish. In modern scheme theory, a "point" of an affine scheme is a prime ideal in the ring of regular functions on that scheme. I have tried below to explain how the old concept evolved into the new one. Notice this whole discussion discusses only the concept of a "point" in scheme theory. In general the discussion in EGA is quite unmotivated, and I believe cannot be appreciated without extensive knowledge of classical geometry. Shafarevich is excellent in regard to motivation, as well as being mostly self contained, even more so in its earlier editions.

Algebraic geometry arises from the problem of solving systems of polynomial equations. In modern times, this study has become the theory of “schemes”. You may know the definition of an “affine scheme” as the set of prime ideals of a commutative ring with identity. To see what these have to do with one another, we want to retrace, in the classical case of polynomials over a field, how prime ideals arise in looking for common solutions of systems of equations.

Let k be any field, and {fi} in k[T1,...,Tn] = k[T] a collection of polynomials in n variables over k. A “solution” or “common zero” of the system {fi=0} will be a vector, or point, p = (p1,...,pn) in En, where E is a field extension of k, such that for all i, fi(p) = 0, in E. The key to connecting a solution vector p with a prime ideal of k[T] is to look at the associated evaluation map π:k[T]—>E, taking f(T) to f(p), and especially at its kernel ker(π) = Ip. Then p is a common zero of the polynomials {fi} if and only if {fi} ⊂ Ip. Moreover since E is a field, the ideal Ip is prime in k[T]. Thus a “solution” of the system {fi} in k[T], which might be thought of as a “point” of the algebraic variety defined by this system, determines a prime ideal of k[T] containing {fi}, or equivalently, a prime ideal of the quotient ring k[T]/({fi}), where ({fi}) is the ideal generated by the set {fi}.

Conversely, if {fi} ⊂ I ⊂ k[T] where I is a prime ideal, then there is an associated k algebra map π:k[T]—>k[T]/I ⊂ E where E is the field of fractions of the domain k[T]/I. If we define the point p in En by setting pi = π(Ti) in E, then this map π is evaluation at p, in the sense above.

Thus prime ideals of the quotient ring k[T]/({fi}) correspond to common zeroes of the system {fi}, with components in extensions of k. This correspondence is not however one-one, since e.g. both i and -i in C define the same prime ideal in R[T], since R[T]/(T2+1) has two isomorphisms with C, taking T to either i or -i. Thus considering solutions in a field extension of k involves some Galois theory over k. Indeed if E is an algebraic closure of k, then maximal prime ideals m ⊂ k[T] correspond to orbits of Gal(E/k) in En, parametrizing different embeddings of k[T]/m in E. (See Mumford, II.4., pp. 96 - 99.)

Natural questions:

1) Which prime ideals correspond to solutions in kn, i.e. when do the solutions have coordinates in the original field k itself?

2) Which prime ideals correspond to solution vectors p = (p1,...,pn) with coordinates pi lying in some algebraic extension E of k?

3) What is the geometric meaning of solution vectors whose coordinates lie in transcendental extensions of k?

Question 1). Let p in kn be a solution of the system {fi}. Then the evaluation map at p, taking k[T1,...,Tn]—>k, is surjective since it equals the identity on k, hence its kernel is a maximal ideal m ⊃ {fi}. Thus p in kn corresponds to a maximal ideal m ⊃ {fi} such that the composition k—>k[T]/m—>k is an isomorphism, and since k[T]/m—>k is injective, equivalently such that k—>k[T]/m is an isomorphism.

Conversely let {fi} ⊂ m ⊂ k[T] and assume the map k—>k[T]/m is an isomorphism. Thus for each i, there is a unique pi in k that maps to Ti, mod m. Thus the composition with the inverse isomorphism k[T]—>k[T]/m—>k takes Ti to pi, and this is a k algebra map, and by definition equals evaluation at p = (p1,..., pn). Moreover it takes every element of m to 0. Thus p is a common zero of the ideal m, and hence of the system {fi} having coefficients in k.

Thus common solutions with coefficients in k, of the system {fi}⊂ k[T], correspond one - one with maximal ideals m ⊂ k[T], with {fi} contained in m, and such that k—>k[T]/m is an isomorphism. Equivalently they correspond to maximal ideals m ⊂ k[T]/({fi}) such that k—>k[T]/m is an isomorphism. Of course there might not be any such ideals, as in the case of {T2+1} in R[T], since ({T2+1}) is already maximal and has quotient larger than R. This is because as we know, many systems of equations have no solutions in the coefficient field.

Note that a maximal ideal of k[T] with p = (p1,...,pn) in kn as common zero, contains the functions Ti-pi. But the ideal (T1-p1,...,Tn-pn) is already maximal since the composition k—>k[T]/(T-p) takes pi to Ti, thus surjects, so is isomorphic.

So arguments like those above give one - one correspondences between the following sets:

{common solutions (a1,...,an) in kn of {fi} ⊂ k[X1,...,Xn]}

≈ {k algebra maps π:k[X1,...,Xn]—>k with π(fi) = 0, for all i}

≈ {k-algebra maps π:k[X1,...,Xn]/({fi})—>k}

≈ {maximal ideals M: {fi} ⊂ M⊂k[X1,...,Xn] and such that k—>k[X1,...,Xn]/M is an isomorphism}

≈ {ideals of form {fi} ⊂ (X1-a1,...,Xn-an) ⊂ k[X1,...,Xn]}

These correspondences are as follows: a point a in kn at which all fi = 0 yields the k algebra map π = evaluation at a, which takes all fi to 0; a k algebra map π:k[X1,...,Xn]—>k is always surjective since it already is on k, so π has a maximal ideal kernel M such that the composition

k—>k[X1,...,Xn]—>k[X1,...,Xn]/M —> k is an isomorphism, and since π(fi) = 0 for all i, thus {fi} ⊂ M; a maximal ideal M ⊃ {fi} such that the composition

k—>k[X1,...,Xn]/M is an isomorphism is always of form (X1-a1,...,Xn-an) where ai = the unique element of k such that Xi = ai mod M; finally a maximal ideal of form M = (X1-a1,...,Xn-an) ⊃ {fi} determines the point a by setting aj equal again to the unique element of k congruent mod M to Xj, and since the point a belongs to M, all fi vanish at a.

Question 2). The question is more complicated but the answer is simpler: solutions p of the system {fi}, with coordinates pi in some algebraic extension of k, correspond to all maximal ideals of k[T]/({fi}), i.e. to those maximal ideals of k[T] containing the system {fi}. This time however the correspondence is not one to one, since several solutions can correspond to the same maximal ideal. To prove it will take a little work, but one direction is elementary.

Let p be a common zero of all fi, where each pj is in E, and E is algebraic over k. Consider the evaluation map π:k[T]—>E taking f(T) to f(p). The image is a domain k[p1,...,pn], since it is contained in the field E, and its fraction field

k(p1,...,pn) is a finitely generated algebraic extension field of k, hence a finite dimensional k vector subspace of E. Multiplication by any non zero element u in the domain k[p1,...,pn] is an injective k linear map of the finite dimensional k vector space k[p1,...,pn] to itself, hence also surjective. Thus there is some v in k[p1,...,pn] such that uv = 1. Thus the image k[p1,...,pn] of the evaluation map is actually a field, i.e. k[p1,...,pn] = k(p1,...,pn). Hence the kernel, ker(π) is a maximal ideal of k[T] containing {fi}.

The converse is less elementary, but we claim it is true: i.e. if {fi} ⊂ m⊂ k[T] is any maximal ideal, then the quotient k[T]/m is always a finite dimensional algebraic extension field of k, hence embeds in E = algebraic closure of k, via some embedding π:k[T]/m—>E. If pi = π(Ti), then the map π with kernel m, is evaluation at p in En, which is a common solution of the system {fi} with coordinates in E. Here we see that different embeddings of k[T]/m into E give rise to different solutions p corresponding to the same ideal m.

It seems we get correspondences analogous to those above as follows, where E is an algebraic extension of k:

{common solutions (a1,...,an) in En of {fi} ⊂ k[X1,...,Xn]}

≈ {k algebra maps π:k[X1,...,Xn]—>E with π(fi) = 0, for all i}

≈ {k-algebra maps π:k[X1,...,Xn]/({fi})—>E}

≈ {maximal ideals M: {fi} ⊂ M⊂k[X1,...,Xn] and such that k[X1,...,Xn]/M is k-isomorphic to a subfield of E}.

If E is an algebraic closure of k, this last set is the set of all maximal ideals of k[X1,...,Xn].

The simplest statement arises if k = E is already algebraically closed. Then we can combine the statements in 1) and 2) and see that there is a one one correspondence between solutions of the system {fi} with coordinates in k, and the set of all maximal ideals of k[T] containing {fi}. In particular, if the set {fi} does not generate the unit ideal, then there must be some maximal ideals containing the system, hence there are some common solutions. This is a several variables analog of the fundamental theorem of algebra. In its simplest form it says if k is algebraically closed, there is a one one correspondence between points of kn and maximal ideals in k[T1,...,Tn], where p = (p1,...,pn) in kn corresponds to (T1-p1,...,Tn-pn) in k[T]. This is Hilbert’s famous (“weak”) nullstellensatz, the foundation result of the whole subject of algebraic geometry, the precise dictionary between geometry and polynomial algebra.

If {fi} ⊂ k[T] where k is any field and E its algebraic closure, maximal ideals of k[T]/({fi}) correspond to (finite) Gal(E/k) orbits of the common solution set of {fi} in En. Solutions of the system {fi} lying in kn are the fixed points of the action. [see Mumford’s picture in the redbook chapter II.4, of the scheme associated to the circle X^2+Y^2 = 1, essentially consisting of the closed disc bounded by the usual circle of real points, with the interior of the disc corresponding to algebraic but non real points.]

Question 3), the interpretation of points on the algebraic variety V({fi}) with coordinates in non algebraic (“transcendental”) extensions of k. For simplicity we assume k is algebraically closed, as we will henceforth always assume in this course. We know from the discussion above, such points p correspond to prime ideals Ip in k[T]/({fi}), hence the maximal ideals containing this prime ideal constitute a certain subcollection of the k valued points of the variety V({fi}) ⊂ kn. I.e. a point p with values in a transcendental extension E of k corresponds to a subcollection V(Ip) of k valued points.

Thus if {fi}⊂ Ip ⊂ k[T], then the prime ideal Ip corresponds to a subvariety V(Ip) ⊂ V({fi}) ⊂ kn, and this subvariety V(Ip) is our geometric interpretation, in kn, of the “point” p with values in the transcendental extension E of k. Moreover, if we think of a k - valued point, i.e. a maximal ideal, as having dimension zero, then a “point” p with coordinates pi in E ⊃ k which generate a transcendence degree r extension k(p1,...,pn) of k in E, corresponds to a prime ideal Ip ⊂ k[T]/({fi}) of “coheight r”, so the variety V(Ip) has dimension r.

I.e. the prime ideal Ip can be joined to a maximal ideal by a chain of prime ideals of length r, but not longer, and since k(p1,...,pn) = fraction field of k[p1,...,pn] ≈ k[T]/Ip, thus as commutative algebra students may know, the transcendence degree of k(p1,...,pn) equals the Krull dimension of k[p1,...,pn], equals the coheight of Ip ⊂ k[T]. (We will prove this later.) The subvariety V(I) ⊂ V({fi}) thus has dimension r. Note that “dimension” is a relative term, and here it is taken relative to the base field k. Since not all fields are algebraic over k, not all “points” are zero dimensional over k.

To sum up, if k is algebraically closed, and {fi} ⊂ I ⊂ k[T], where I is a prime ideal of coheight r, then I corresponds equivalently to an (“irreducible”) r dimensional subvariety of the variety V({fi}), and to a point with coordinates in the extension field E = fraction field(k[T]/I) of k of transcendence degree r. (Since a prime ideal is not the proper intersection of two other ideals, here “irreducible” means the variety it defines is not the proper union of two other varieties.)

Grothendieck’s theory of schemes makes this correspondence more intrinsic. I.e. if I is a prime ideal of k[T], then I represents a point of the scheme spec(k[T]) (= set of prime ideals of k[T]), and the fraction field E of k[T]/I is just the residue field of the local ring at the “point” I. The fact that a function in k[T] vanishes at this point iff it lies in I, iff it vanishes at every point of V(I) with coordinates in k, is why this point is called a generic point for V(I). This is reflected in the topology for spec(k[T]) where the generic point of V(I) is dense in V(I). Thus we see more clearly that in scheme theory, even though the residue field at a closed point is the algebraically closed base field k, the residue field at a non closed point is transcendental over k.

So it seems that in a scheme, an abstract “point”, represented by a prime ideal of a ring in the case of an affine scheme, corresponds to an embedded point with coefficients in the residue field of the scheme theoretic point. I.e. if the affine ring is k[X], then the point represented by the prime ideal I, defines the point ([X1],...,[Xn]) in E^n, where E is the residue field of the point I in spec(k[X]).