# What is a concrete example of why one wants to have a *derived category* in algebraic geometry?

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?

• 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. – Zhen Lin Dec 1 '13 at 7:40
• 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. – user8463524 Dec 1 '13 at 10:16
• +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. – Martin Brandenburg Dec 1 '13 at 10:32

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 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.