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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.

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    $\begingroup$ I assume you know of stacks.math.columbia.edu? $\endgroup$ – jmc Dec 15 '14 at 18:44
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    $\begingroup$ 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. $\endgroup$ – user74230 Dec 15 '14 at 22:11
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    $\begingroup$ 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. $\endgroup$ – Ashwath Rabindranath Dec 15 '14 at 23:29
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    $\begingroup$ 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. $\endgroup$ – Jonathan Beardsley Dec 16 '14 at 1:09
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    $\begingroup$ @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. $\endgroup$ – user74230 Dec 16 '14 at 4:06
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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.)

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  • $\begingroup$ 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. dspace.mit.edu/handle/1721.1/30144 $\endgroup$ – bananastack Dec 16 '14 at 0:21
  • $\begingroup$ @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. $\endgroup$ – user62675 Dec 16 '14 at 0:27
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    $\begingroup$ 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. $\endgroup$ – bananastack Dec 16 '14 at 0:32
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    $\begingroup$ @bananastack If I understand what you are looking for, <i>Methods of Homological Algebra</i> by Gelfand and Manin fits the bill. Beware the many minor typos. $\endgroup$ – DES-SupportsMonicaAndTransfolk Dec 19 '14 at 2:11
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    $\begingroup$ @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. $\endgroup$ – 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.

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    $\begingroup$ 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. $\endgroup$ – Joonas Ilmavirta May 14 '16 at 18:26

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