It seems to me that a lot of people do deformation theory (of schemes, sheaves, maps etc) in derived category (of an appropriate abelian category). For example, the cotangent complex of a morphism $f:X\rightarrow Y$ of schemes is defined as an object in the derived category of coherent sheaves $X$. I would like to understand why derived category is an appropriate category to do deformation theory. I would appreciate it if someone could give me a good example or motivation.
[Math] Why are derived categories natural places to do deformation theory
ag.algebraic-geometrydeformation-theoryderived-categories
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One crude answer is that passing to derived functors fixes one obstruction to being an equivalence. Any equivalence of abelian categories certainly is exact (i.e. it preserves short exact sequences), though lots of exact functors are not equivalences (for example, think about representations of a group and forgetting the G-action).
What derived functor does is fix this problem in a canonical way; you have to replace short exact sequences with exact triangles, but you get a functor which is your original "up to zeroth order," exact, and uniquely distinguished by these properties.
So, what BMR do is take a functor which is not even exact (and thus obviously not an equivalence), and show that the lack of exactness is "the only problem" for this being an equivalence.
EDIT: Let me just add, from a more philosophical perspective, that derived equivalences are just a lot more common. There are just more of them out in the world. Given an algebra A, Morita equivalences to A are classified essentially by projective generating A-modules, whereas derived Morita equivalences of dg-algebras are in bijection with all objects in the derived category of $A-mod$ which generate (in the sense that nothing has trivial Ext with them): you look at the dg-Ext algebra of the object with itself. If you have an interesting algebra (say, a finite dimensional one of wild representation type), there are a lot more of the latter than the former in a very precise sense. Of course, the vast majority of these are completely uncomputable an tell you nothing, but there are enough of them in the mix to make things interesting.
This will be a short overview on techniques I am familiar with. For simplicity, I will talk about bounded t-structures, which are determined by their heart $\mathcal{A} = D^{\le 0} \cap D^{\ge 0}$; and on the bounded derived category of coherent sheaves $D^b(X)$ on a variety/stack.
Tilting is in principle extremely powerful: $\mathcal{A}_1$ is obtained by tilting from $\mathcal{A}_2$ whenever objects of $\mathcal{A}_1$ have only two cohomologies with respect to $\mathcal{A}_2$, e.g. $\mathcal{A}_1 \subset \langle \mathcal{A}_2, \mathcal{A}_2[1]\rangle$. (See e.g. Lemma 1.1.2 in Link)
Tilting can be iterated. As an example, using 1. it is a not too difficult exercise to see that Bezrukavnikov's t-structures of perverse coherent sheaves can be constructed by iterated tilting. When $D^b(X)$ is equivalent to a derived category of quiver representations, iterated tilting can often be descibed by iterated quiver mutations.
It is not very difficult to construct torsion pairs. For example, when $\mathcal{A}$ is Noetherian, then any subcategory $\mathcal{T} \subset \mathcal{A}$ that is closed under extensions and quotients is the torsion part of a torsion pair $(\mathcal{T}, \mathcal{F} = \mathcal{T}^{\perp})$. Alternatively, any notion of stability condition on a heart $\mathcal{A}$, say, induced via a slope function: $\mathcal{T}_{> \mu_0}$ is the extension-closed subcategory generated by stable objects $E$ with $\mu(E) > \mu_0$. (See Link, section 6.)
Given a semi-orthogonal decomposition of a triangulated category, one can construct a t-structure on the full category by gluing t-structures on the components - this is all in the original BBD.
It is much easier to construct (unbounded) t-structures in the unbounded derived category $D_{qc}(X)$ of quasi-coherent sheaves. (Any subcategory closed under [1], extensions and small coproducts is the $D^{\le 0}$-part of a t-structure.) Sometimes one can use this to construct bounded t-structures on $D^b(X)$ by showing that they restrict - see e.g. section 2 of Link. But in general t-structures on $D_{qc}(X)$ do not descend, and even if they do, it might be hard to prove.
As an example of the latter techniques, when $G$ acts freely on $X$, then $G$-invariant t-structures on $X$ are in 1:1-correspondence with t-structures on the quotient $X/G$ satisfying an additional assumption: tensoring with $f_* \mathcal{O}_X$ is right-exact.
Any derived equivalence $D^b(X) \cong D^b(\mathcal{A})$ induces a t-structure on $D^b(X)$ by pull-back - I guess you already knew that!
I realize that I talked mostly about tilting - I do think it's a very powerful method.
Best Answer
I'll post an answer to what I think is a reasonable question, hoping that someone more expert will improve on this answer. My apologies if this answer is too chatty.
First, a very simple reason you might hope that there is something like a cotangent complex and that it should be an object in the derived category of quasi-coherent sheaves. Given a morphism $f: X \rightarrow Y$ of schemes (over some base, which I leave implicit), you get a right exact sequence of quasi-coherent sheaves on $X$:
$$f^{*}\Omega^{1}_{Y} \rightarrow \Omega^{1}_{X} \rightarrow \Omega^{1}_{X/Y} \rightarrow 0.$$
Experience has taught us that when we have a functorial half-exact sequence, it can often be completed functorially to a long exact sequence involving 'derived functors', and in abelian contexts such a long exact sequence is usually associated to a short exact sequence (or exact triangle) of 'total derived functor' complexes, the complexes being well-defined up to quasi-isomorphism and hence objects in a derived category. This is the semi-modern point of view on derived functors, which one can learn for instance from Gelfand-Manin's Methods of Homological Algebra. Experience has also shown that not only is the cohomology of the total derived functor complex important in computations, but that the complex itself, up to quasi-isomorphism, contains strictly more information and is often easier to work with, until the very last moment when you want to compute some cohomology.
Once you've seen this work a number of times (say for global sections, for $Hom$, and for $\otimes$), one might ask if there is a total derived functor of $\Omega^{1}$, call it $\mathbb{L}$, which among other things produces an exact triangle
$$f^{*}\mathbb{L}_{Y} \rightarrow \mathbb{L}_{X} \rightarrow \mathbb{L}_{X/Y}$$
such that the long exact sequence of cohomology sheaves begins with the original right exact sequence.
If you believe that such a thing should be useful, then you might go about trying to construct it. One way to do this involves interpreting $\Omega^{1}$ as representing derivations which in turn correspond to square-zero extensions, and this is where deformation theory comes in. So one might begin to ask what 'derived square-zero extensions' should be, and you might guess that you should try to extend not just by modules but by bounded above complexes of modules. When you do this, such a square-zero extension becomes not just a commutative algebra but some kind of derived version thereof, such as a simplicial commutative algebra. In these terms, the cotangent complex $\mathbb{L}$ of a commutative algebra turns out to be nothing but K\"ahler differentials of an appropriate resolution of our commutative algebra in the world of simplicial commutative algebras. Following this idea through and figuring out how descent should work leads, after a long song and dance, to the desired theory of the cotangent complex.
Once you set this all up, it becomes clear that there was no reason to restrict oneself to classical commutative algebras in setting up algebraic geometry, but that one could have worked with simplicial commutative algebras to begin with, and this leads to `derived algebraic geometry'. In some sense, this is the natural place in which to understand the cotangent complex, and here the higher cohomology of the cotangent complex has a natural geometric interpretation.
One should also point out that Quillen's point of view on the cotangent complex was as a homology theory for commutative algebras (search for Andre-Quillen homology), which in a precise sense is an analogue of the usual homology of a topological space. This is described in the last chapter of Quillen's Homotopical Algebra and is also discussed in Goerss-Schemmerhorn's Model Categories and Simplicial Methods.
To summarise. Deformation theory is about square-zero extensions (as well as about other more general infinitesimal extensions). K\"ahler differentials corepresent derivations which in turn correspond to square-zero extensions. For each morphism of schemes $X \rightarrow Y$, there is a natural right exact sequence involving K\"ahler differentials, which it would be useful to complete to a long exact sequence. Even better, we'd like this long exact sequence to come from an exact triangle of objects in the derived category of quasi-coherent sheaves. Realising this goal naturally leads to derived or homotopical algebraic geometry.