Here is an example mentioned in passing by user ali's answer, but I think it is cute (and powerful) enough to be worth fleshing out the details.

**Lifting from characteristic $p$ to characteristic zero**

In short, studying a geometric object (say, a scheme) $X$ in characteristic $p$ often involves lifting it to characteristic zero. For example, if $X$ is a smooth projective variety over $\mathbf{F}_p$, we may try to find a (flat) lift $\mathcal{X}$ over the $p$-adic numbers $\mathbf{Z}_p$. Now, $\mathbf{Z}_p$ embeds into $\mathbf{C}$ (in some completely noncanonical way), and we can apply powerful methods such as Hodge theory to the complex manifold underlying $\mathcal{X}_\mathbf{C}$.

Now, recall that
$$ \mathbf{Z}_p = \varprojlim_n \mathbf{Z}/p^{n+1}. $$
Thus lifting $X_0=X$ over $\mathbf{Z}_p$ involves finding compatible liftings $X_n$ over $\mathbf{Z}/p^{n+1}$ for all $n$. The system $\mathfrak{X} = \{X_n\}$ (or its inductive limit in locally ringed spaces) is a "$p$-adic formal scheme," and the next step involves checking that it is *algebraizable*, i.e. that it comes from an actual scheme $\mathcal{X}/\mathbf{Z}_p$ by the obvious "formal completion" functor.

Now the first step, finding the successive liftings $\{X_n\}$, is completely controlled by deformation theory. In our situation, it says the following:

If $X_0$ is a scheme over $\mathbf{F}_p$, and $X_n$ is a flat lifting of $X_0$ over $\mathbf{Z}/p^{n+1}$, there there exists an obstruction class
$$ {\rm obs}(X_n, \mathbf{Z}/p^{n+2}) \in {\rm Ext}^2(\mathbf{L}_{X_0/\mathbf{F}_p}, \mathcal{O}_{X_0}) = {\rm Hom}_{D(X_0)}(\mathbf{L}_{X_0/\mathbf{F}_p}, \mathcal{O}_{X_0}[2]), $$
which vanishes if and only if there exists a flat lifting $X_{n+1}$ of $X_n$ over $\mathbf{Z}/p^{n+2}$. It is functorial in the sense that for $f_n\colon X_n\to Y_n$ lifting $f_0\colon X_0\to Y_0$ we have a commutative square
$$\require{AMScd} \begin{CD}
\mathbf{L}_{Y_0/\mathbf{F}_p} @>>> \mathcal{O}_{Y_0}[2]\\ @VVV @VVV\\
Rf_{0, *}\mathbf{L}_{X_0/\mathbf{F}_p} @>>> Rf_{0, *}\mathcal{O}_{X_0}[2]
\end{CD}$$

In case the obstruction class vanishes, the set of isomorphism classes of such liftings $X_{n+1}$ is in a natural way a torsor under
$$ {\rm Ext}^1(\mathbf{L}_{X_0/\mathbf{F}_p}, \mathcal{O}_{X_0}) = {\rm Hom}_{D(X_0)}(\mathbf{L}_{X_0/\mathbf{F}_p}, \mathcal{O}_{X_0}[1]). $$

The group of automorphisms of any lifting $X_{n+1}$ restricting to the identity on $X_n$ is naturally isomorphic to
$$ {\rm Hom}(\mathbf{L}_{X_0/\mathbf{F}_p}, \mathcal{O}_{X_0}). $$

There is a similar story for lifting morphisms $f_0\colon X_0\to Y_0$.

So if you can show that ${\rm Ext}^2(\mathbf{L}_{X_0/\mathbf{F}_p}, \mathcal{O}_{X_0})$ vanishes, then you know that $X_0$ admits a formal $p$-adic lifting $\mathfrak{X}$. For example, if $X_0$ is a K3 surface, then this group can be identified with the space of global vector fields on $X_0$, and its vanishing is a difficult theorem due to Rudakov and Shafarevich. (And the fact that there is an *algebraizable* formal lifting, i.e. that an ample line bundle can be lifted to all $X_n$'s for a good choice of $\mathfrak{X}$, was shown later by Deligne.)

**Perfect schemes and Witt vectors**

Recall that for every perfect field $k$ of characteristic $p>0$ there exists a unique complete discrete valuation ring $W(k)$ (its ring of *Witt vectors*) with residue field $k$ whose maximal ideal is generated by $p$. It is a functor of $k$, and we have $W(k) \simeq k^{\mathbf{N}}$ as functors into sets. The addition and multiplication laws on $k^{\mathbf{N}}$ obtained this way are given by complicated universal formulas, e.g.
$$ (x_0, x_1, \ldots) + (y_0, y_1, \ldots) = (x_0 + y_0, x_1 + y_1 - \sum_{0<i<p} \frac 1 p \binom p i x_0^i y_0^{p-i}, \ldots). $$
We define $W_n(k) = W(k)/p^n$ and call these *Witt vectors of length $n$*.

For example, $W(\mathbf{F}_p) = \mathbf{Z}_p$, $W_n(\mathbf{F}_p) = \mathbf{Z}/p^n$.

In fact, the above can be defined for any ring $R$. If $R$ is a *perfect* $\mathbf{F}_p$-algebra, meaning that its Frobenius
$$ F_R \colon R\to R, \quad F_R(x) = x^p $$
is an isomorphism, then $W(R)$ is a flat lifting of $R$ over $W(\mathbf{F}_p) = \mathbf{Z}_p$.

Here is a beautiful argument (I think due to Bhargav Bhatt) employing the cotangent complex to show the existence of Witt vectors for perfect rings (or schemes) without using any strange-looking universal formulas for addition and multiplication.

**Theorem.** Let $X$ be a perfect $\mathbf{F}_p$-scheme. There exists a unique up to unique isomorphism formal $p$-adic lifting $\mathfrak{X} = \{X_n\}$ of $X_0=X$. Moreover, every morphism $f\colon X\to Y$ admits a unique lifting $\mathfrak{X}\to \mathfrak{Y}$.

The above implies that $\mathfrak{X}$ is a functor of $X$, denoted $W(X)$. It is not difficult to prove that it indeed coincides with the Witt vectors.

*Proof.* Consider the cotangent complex $\mathbf{L}_{X_0/\mathbf{F}_p}$ and the map
$$ F_X^* \colon \mathbf{L}_{X_0/\mathbf{F}_p}\to F_{X, *} \mathbf{L}_{X_0/\mathbf{F}_p} $$
induced by the absolute Frobenius $F_X\colon X\to X$. Since $F_X$ is an isomorphism, the map $F_X^*$ is an isomorphism too. The complex $\mathbf{L}_{X_0/\mathbf{F}_p}$ is defined by locally resolving $\mathcal{O}_X$ by free $\mathbf{F}_p$-algebras and considering their Kaehler differentials. And $F_A$ acts as zero on $\Omega^1_{A/\mathbf{F}_p}$ for every $\mathbf{F}_p$-algebra $A$:
$$ F_A^*(dx) = dF_A(x) = dx^p = px^{p-1} dx = 0. $$
Therefore the map $F_X^*$ above is the zero map. Since it is also an isomorphism, we conclude that $\mathbf{L}_{X_0/\mathbf{F}_p} = 0$!

Now by deformation theory, the obstructions to lifting lie in the zero group (and hence the successive liftings exist), the isomorphism classes of different successive liftings are permuted by the zero group (and hence the liftings are unique), and their automorphism groups are trivial (so the liftings are unique up to a unique isomorphism). Similarly, one handles the lifting of morphisms. $\square$

sheaf-- differential forms -- whose SpecSym is the tangent bundle. The cotangent complex degenerates to that in the smooth case, but is still very nicely behaved in case of mild singularities (e.g. local complete intersections). It was originally designed by Grothendieck et al in the guise of the "virtual tangent bundle" (its induced K-theory class) to formulate GRR formulas in the singular case. $\endgroup$ – crystalline Sep 20 '20 at 5:485more comments