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.
Best Answer
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.