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My question asks for a concrete (and hopefully easy) example, why one wants to derive things in algebraic geometry. I heard, that a resolution of an object by free ones behaves much better than the original object but until now these are only meaningless phrases to me. Here is the setting, I would like to understand the term 'derived' in:

Let $X$ be a scheme and $X_Z$ (or $X_{ét}$, if you like) the small Zariski (or the small étale) Grothendieck site on $X$. Consider the abelian category $Shv^{Ab}(X_Z)$ of sheaves of abelian groups on $X_Z$. What is a concrete example why one likes to build complexes $Ch(Shv^{Ab}(X_Z))$ on that category and then the derived category $D(X_Z)$ in the sense of homological algebra (i.e. inverting the quasi-isomorphisms)?

If this setting is not appropriate enough, one can consider the ring object $\mathcal{O}_X$ in $Shv^{Ab}(X_Z)$ and modules over it. This category is again abelian, one forms chain complexes on it and the derived category $D(X_Z,\mathcal{O}_X)$. Why? What is an concrete example for the necessity of this complicated process?

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    $\begingroup$ One takes resolutions all the time to compute cohomology, yet cohomology is not a "complete invariant": the quasi-isomorphism type of the resolution carries more information. $\endgroup$ – Zhen Lin Dec 1 '13 at 7:40
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    $\begingroup$ Yes, that's a point. Thank you for pointing that out! Anyway, I would appreciate a concrete example like something with 'intersection theory' behaving well in the derived setting but nor in the classical one. $\endgroup$ – user8463524 Dec 1 '13 at 10:16
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    $\begingroup$ +1 since I also would like to see a concrete example. I think that the formalism of derived categories simplify a lot. For example, the complicated Grothendieck spectral sequence is essentially the trivial equation $R(F \circ G) = R(F) \circ R(G)$ in the derived category, following from the universal property of $R(F)$. I think Zhen has explained this somewhere else nicely. $\endgroup$ – Martin Brandenburg Dec 1 '13 at 10:32
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One motivation is deformation theory, see MO:111084. Basically, the cotangent complex turns out to be the left derived functor of the Kähler differentials functor $\Omega^1$.

In intersection theory, Serre's intersection formula is another example, see MO:12236. The Tor's in this formula are simply the homotopy groups of the derived tensor product, taken in the derived category of simplicial commutative algebras. Hence the intersection multiplicity is nothing but the length of this derived tensor product, viewed as a simplicial module.

Both of these examples also lead quite naturally into the world of derived algebraic geometry in the sense of Toën–Vezzosi or Lurie.

However, I should mention that more traditionally (i.e. in non-derived geometry), the derived category of (quasi-)coherent sheaves is not usually used for computing resolutions or such, but rather as a kind of package of classically interesting geometric data about your variety. In fact, in some cases this triangulated category contains sufficient information to reconstruct your original variety up to isomorphism. For this perspective, see papers of Bondal and Orlov, like arXiv:math/0206295.

In fact in noncommutative algebraic geometry or derived noncommutative algebraic geometry, varieties are completely replaced by their derived categories of (quasi-)coherent sheaves or stable (∞,1)-categorical enhancements thereof. Thus a noncommutative space in this sense is a kind of stable $k$-linear (∞,1)-category (often a DG category). A good introduction to this subject is arXiv:1108.3787.

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    $\begingroup$ I have never seen an answer with such a great quotient # nlab-links / # lines. $\endgroup$ – Martin Brandenburg Dec 1 '13 at 12:18

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