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I know little about algebraic geometry, however while studying noncommutative geometry some results showed that a category I understand well (holomorphic vector bundles over noncommutative tori) was equivalent to the derived category of coherent sheaves over an elliptic curve. So, what is the category of coherent sheaves on an elliptic curve, what is its derived category, and why are these important for algebraic geometry? Edit: Is the category of coherent sheaves on a higher dimensional abelian variety much more complicated than the category of coherent sheaves on an elliptic curve?

I don't necessarily need the most technical account; I'm really just looking to get a sense of why these things are important and what information they encode.

Edit: Since the original post seemed to imply that I did no prior research, here is what I know. The category of coherent sheaves is an expansion of the category of holomorphic vector bundles on an elliptic curve so that the category becomes abelian. This is what allows you to take the derived category. There is also an intrinsic characterization of coherent sheaves, which is like the characterization of a holomorphic vector bundle as a locally free sheaf of $O_X$ modules but loosens the locally free condition. I wasn't looking for textbook definitions, I was looking for intuition that would help me to understand why we care about this, including applications in classical complex geometry.

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    $\begingroup$ So why should we give you all the definitions (elliptic curves, coherent sheaves, derived categories) which you can look up in hundreds of books, articles, blog posts etc.? Please do some research, and then you can ask specific questions here. Otherwise we can only answer which you can read also elsewhere. $\endgroup$ – Martin Brandenburg Mar 14 '13 at 0:34
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    $\begingroup$ There is definitely an analog for "higher dimensional abelian varieties." In fact, since coherent sheaves form an abelian category in great generality (work with an arbitrary scheme or even weaker) and the derived category is a general construction you can do to abelian categories you can take the derived category of coherent sheaves on a scheme. Most algebraic geometers stick to the case of the derived category of coherent sheaves on smooth varieties over algebraically closed fields of characteristic $0$ though. $\endgroup$ – Matt Mar 14 '13 at 0:52
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    $\begingroup$ As stated, this is not a question answerable in this forum, really. Book have been written to explain this! $\endgroup$ – Mariano Suárez-Álvarez Mar 14 '13 at 0:53
  • $\begingroup$ I have edited the question, and hopefully it is more answerable now. I was never asking anyone to regurgitate textbook definitions. Often I find algebraic geometry texts incredibly abstract and they do not get to applications until a lot of machinery has been developed. Since I likely will not need that machinery, I was hoping people here could help me cut to the chase, so to speak. $\endgroup$ – mck Mar 14 '13 at 1:23
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    $\begingroup$ I should add that if you care about the elliptic curve case, then the question "what type of information do they encode" is easy. If E and F are two elliptic curves with equivalent derived categories $D(E)\simeq D(F)$ then you can conclude that they are isomorphic as elliptic curves. This is special due to the low dimension. In general, this is definitely not the case. $\endgroup$ – Matt Mar 14 '13 at 1:42
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The following survey article by Orlov is perhaps the best introduction to this subject.

D. O. Orlov, Derived categories of coherent sheaves and equivalences between them, Uspekhi Mat. Nauk, 2003, Vol. 58, issue 3(351), pp. 89–172, English translation (PDF)

There are also many other accounts, for example the book by Huybrechts.

D. Huybrechts, Fourier-Mukai Transforms in Algebraic Geometry, Oxford University Press, USA, 2006.

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  • $\begingroup$ Thanks for the reference, I'll check it out! $\endgroup$ – mck Mar 14 '13 at 1:29
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    $\begingroup$ The Huybrechts book is absolutely fantastic. It really builds every single concept, definition, and tool used in practice up from nothing. It is extremely thorough with all the details there too. I look in this book for things every other week it seems. $\endgroup$ – Matt Mar 14 '13 at 1:40
  • $\begingroup$ Well, for details on triangulated and derived categories you'd better look somewhere else (like Verdier's thesis or Kashiwara-Schapira), but otherwise most of the details are there. $\endgroup$ – user314 Mar 14 '13 at 2:42
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This is more a comment than an answer but I dont have enough points to add a comment. You asked:

"I'm really just looking to get a sense of why these things are important and what information they encode."

Coherent sheaves over scheme are locally just (finitely generated) modules over a ring call it R. People are typically interested in understanding rings, or on the geometric level, schemes, and one can study them directly, but for some reason it turns out to be very helpful to study them indirectly by studying \emph{modules } over $R$. This I think is Morita theory or even just the whole philosophy behind representation theory. If you are going to study modules, then you often can understand them by taking free resolutions, etc, and do homological algebra, and then from homological algebra you are led fairly directly to the derived category of modules (or their global analogue, (quasi)coherent sheaves). And so we can recover a lot of information about a scheme by studying it noncommutative shadow, the derived category of coherent sheaves on it. And for example, there are various famous results like if two smooth schemes $X, Y$ have isomorphic derived categories of coherent sheaves, then are $X$ and $Y$ isomorphic? Well, no in general, but in various cases i.e. under some hypotheses on dimension or canonical bundle, the answer is yes.

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    $\begingroup$ I think you mean ampleness or anti-ampleness of the canonical divisor and not Calabi-Yau for the last sentence. Higher than dimension 1 you will essentially always have non-isomorphic, derived equivalent varieties when the canonical is trivial. $\endgroup$ – Matt Jun 18 '13 at 23:23

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