The Stacks project

Example 97.22.4 (Key example). Let $S = \mathop{\mathrm{Spec}}(\Lambda )$ for some Noetherian ring $\Lambda $. Say $\mathcal{X} = (\mathit{Sch}/X)_{fppf}$ with $X = \mathop{\mathrm{Spec}}(R)$ and $R = \Lambda [x_1, \ldots , x_ n]/J$. The naive cotangent complex $\mathop{N\! L}\nolimits _{R/\Lambda }$ is (canonically) homotopy equivalent to

\[ J/J^2 \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} R\text{d}x_ i, \]

see Algebra, Lemma 10.134.2. Consider a deformation situation $(x, A' \to A)$. Denote $I$ the kernel of $A' \to A$. The object $x$ corresponds to $(a_1, \ldots , a_ n)$ with $a_ i \in A$ such that $f(a_1, \ldots , a_ n) = 0$ in $A$ for all $f \in J$. Set

\begin{align*} \mathcal{O}_ x(A') & = \mathop{\mathrm{Hom}}\nolimits _ R(J/J^2, I)/\mathop{\mathrm{Hom}}\nolimits _ R(R^{\oplus n}, I) \\ & = \mathop{\mathrm{Ext}}\nolimits ^1_ R(\mathop{N\! L}\nolimits _{R/\Lambda }, I) \\ & = \mathop{\mathrm{Ext}}\nolimits ^1_ A(\mathop{N\! L}\nolimits _{R/\Lambda } \otimes _ R A, I). \end{align*}

Choose lifts $a_ i' \in A'$ of $a_ i$ in $A$. Then $o_ x(A')$ is the class of the map $J/J^2 \to I$ defined by sending $f \in J$ to $f(a_1', \ldots , a'_ n) \in I$. We omit the verification that $o_ x(A')$ is independent of choices. It is clear that if $o_ x(A') = 0$ then the map lifts. Finally, functoriality is straightforward. Thus we obtain an obstruction theory. We observe that $o_ x(A')$ can be described a bit more canonically as the composition

\[ \mathop{N\! L}\nolimits _{R/\Lambda } \to \mathop{N\! L}\nolimits _{A/\Lambda } \to \mathop{N\! L}\nolimits _{A/A'} = I[1] \]

in $D(A)$, see Algebra, Lemma 10.134.6 for the last identification.

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