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As for learners in algebraic geometry in 21st century, is there a textbook, lecture note or anything like that to introduce algebraic geometry utilizing the language of derived categories and stacks?

My primary concern is that since these languages are more or less standard in many (if not all) aspects of algebraic geometry, why not introduce them as early as possible? Someone might argue these are not motivated very well at early stages of learning. But considering the amount of commutative algebra and classical homological algebra being used by (or at least developed within) a rigorous abstract algebraic geometry textbook (e.g. Hartshorne's, Qing Liu's, etc.), it is no fault to introduce the modern common language in the first place once and for all and leave to the beginners for years' digestion.

Ricardo Andrade
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W.Z.
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    I assume you know of http://stacks.math.columbia.edu/? – jmc Dec 15 '14 at 18:44
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    See the related question : http://mathoverflow.net/questions/12765/algebraic-stacks-from-scratch – prefaisceau Dec 15 '14 at 20:55
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    You should talk to some professional algebraic geometers, as (IMHO) your perception of how one should go about learning the subject is misguided. The idea that derived categories and stacks should be included as part of an introduction to algebraic geometry is badly mistaken. It is like advocating that introductory physics should include General Relativity and quantum mechanics, since those are more or less standard in many aspects of physics at the professional level. The challenges of education are serious. – user74230 Dec 15 '14 at 22:11
  • @user74230 I just want everything to be conceptually concise at as early stage as possible. To get a real feel of how machinery works, it consumes time, so choosing the more advanced concepts to mess with might be saving time since the earlier students get exposed to them the better as soon as a proper introduction is provided. It's just my own thought. – W.Z. Dec 15 '14 at 22:48
  • @jmc Stacks Project is not a textbook. – W.Z. Dec 15 '14 at 22:52
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    Just my opinion (and I am not an expert in these modern homological ideas) but I think that there's a real risk in learning these ideas without first going through the grind of learning classical algebraic geometry and Hartshorne-type material (I guess today Hartshorne is classical algebraic geometry too!). Of course if you have already mastered Hartshorne, this comment does not apply. – Ashwath Rabindranath Dec 15 '14 at 23:29
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    Ultimately, it really really really depends on what you want to do. You may be the sort of person who can just start memorizing a lot of terminology and abstract nonsense, but if you have no intuition for what a scheme is, you're unlikely to be able to prove very much about schemes. On the other hand, I came from topology, and it was very useful for me to be able to just thing about a stack from the category theory perspective. So it might be worth it to provide some context in your question. – Jonathan Beardsley Dec 16 '14 at 01:09
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    I might add that without a good understanding of homological algebra, really important concepts like sheaf cohomology and derived functors aren't going to make much sense. For that, I strongly recommend Charles Weibel's book. It's encyclopedic and well written. The only drawback is that some of his terminology is non-standard. – Jonathan Beardsley Dec 16 '14 at 01:12
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    @W.Z.: I don't wish to get into an extended discussion about this; please just follow my advice to talk in person with a professional algebraic geometer. If the goal is to actually understand things in a serious way and to become a creative user of these ideas then what you have in mind is a very very bad idea. I have nothing more to say. – user74230 Dec 16 '14 at 04:06
  • @Jon I couldn't agree more with you! (I think this is reflected in the last paragraph of my answer below.) –  Dec 16 '14 at 22:43
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    I disagree with your comment about higher sheaves not being motivated well. As soon as you start talking about functor of points and moduli problems, stacks / groupoid valued functors pop up pretty fast – Daniel Barter Dec 19 '14 at 01:05
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    I like the general spirit of this question even if it doesn't have a good answer. There are many applications of algebraic geometry to pure algebra and representation theory that require certain machinery not well elucidated in books like Hartshorne's or Liu's (derived categories are an example). For people more interested in these applications instead of pure algebraic geometry, the "classic texts" are very unmotivated. – Jason Polak Dec 19 '14 at 16:39

2 Answers2

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If you have already learned classical algebraic geometry and are planning to study how it's been studied through stacks, one of the best places to learn from is the Stacks Project. While I understand that it is not a textbook, it is a collaborative mega-project that uses stacks to study algebraic geometry. You may also like to read Toen's course notes, Demazure's book, and Anton's notes.

Just one note - you may not be very well motivated to study derived categories and stacks without first learning classical algebraic geometry, eg., Hartshorne. While they are used in research in algebraic geometry, they might seem like very complex things that cannot be used correctly. (Of course, one can go into a whole discussion about why this is usually discouraged, but this is not the scope of the question.)

  • I think for what the OP is asking, Toën's notes are unbeatable (at least for a first introduction to the language). After that you might as well pick up Lurie's thesis. http://dspace.mit.edu/handle/1721.1/30144 – bananastack Dec 16 '14 at 00:21
  • @user125763 That's true. Lurie's thesis is on derived algebraic geometry, though, and I don't really think that the OP should read that unless he/she has a firm grip of algebraic geometry and knows (some) algebraic topology, like homotopy theory. –  Dec 16 '14 at 00:27
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    I don't think anyone should start learning algebraic geometry from either source. Certainly, after spending a fair amount of time over Eisenbud-Harris, Hartshorne and friends, I profited immensely from Toën's notes when trying to understand the basics of stacks (I just think that the functorial point of view makes things more transparent). After that, Lurie's thesis (or at least the small part I actually read) was just a very pleasant read. We already had a famous MO user who thought she should do away with classical AG, I don't think we need to rekindle the flame wars. – bananastack Dec 16 '14 at 00:32
  • @Ring Spectra Thank you for your generous answer with links. Could you also please comment a little bit more on your last parenthesized statement? I am just curious. – W.Z. Dec 16 '14 at 00:34
  • @RingSpectra: actually, I'd like to ask you a question, since you seem to know a fair bit of derived technology. Do you know of any book on homological algebra which uses the derived point of view? For example discussing cup products via Ext and maybe results like in this blog post? https://amathew.wordpress.com/2012/05/15/re-learning-homological-algebra/ – bananastack Dec 16 '14 at 00:34
  • @user125763 Of course, I agree with you on starting to learn AG from either source, but I don't think we want flame wars either (I don't know who the user is, though)! :-) RE your next comment - I'm not an expert, simply a beginner, but I don't recall any book that studies homological algebra from the derived context. I'm sure there's something out there, though! –  Dec 16 '14 at 00:35
  • @W.Z. Imagine trying to read Lurie's Higher Algebra without first understanding what $\infty$-categories actually are and what they could be used for. It's not a good idea to rush through the basics in order to reach some great goal, like AG from stacks, but that final goal should/could be something to keep in mind when reading, so you know what you'll be up against once you reach that point. –  Dec 16 '14 at 00:46
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    @bananastack If I understand what you are looking for, Methods of Homological Algebra by Gelfand and Manin fits the bill. Beware the many minor typos. – David E Speyer Dec 19 '14 at 02:11
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    @DavidSpeyer: thanks. That's actually the book where I first learned homological algebra and loved it. Although there are a lot of topics there in the algebro-geometric side, I was hoping to expand my knowledge in the algebro-topological. – bananastack Dec 23 '14 at 23:18
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You might want to try Sheaves On Manifolds by Kashiwara and Schapira especially chapter 2.

loift
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    Can you explain a little why this book (and this chapter) is good and meets the original poster's (OP's) requirements? This can be a good answer, but it would certainly benefit from more details. – Joonas Ilmavirta May 14 '16 at 18:26