Definition 92.22.1. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. The cotangent complex $L_ f$ of $f$ is $L_ f = L_{\mathcal{O}_\mathcal {C}/f^{-1}\mathcal{O}_\mathcal {D}}$. We sometimes write $L_ f = L_{\mathcal{O}_\mathcal {C}/\mathcal{O}_\mathcal {D}}$.
92.22 The cotangent complex of a morphism of ringed topoi
The cotangent complex of a morphism of ringed topoi is defined in terms of the cotangent complex we defined above.
This definition applies to many situations, but it doesn't always produce the thing one expects. For example, if $f : X \to Y$ is a morphism of schemes, then $f$ induces a morphism of big étale sites $f_{big} : (\mathit{Sch}/X)_{\acute{e}tale}\to (\mathit{Sch}/Y)_{\acute{e}tale}$ which is a morphism of ringed topoi (Descent, Remark 35.8.4). However, $L_{f_{big}} = 0$ since $(f_{big})^\sharp $ is an isomorphism. On the other hand, if we take $L_ f$ where we think of $f$ as a morphism between the underlying Zariski ringed topoi, then $L_ f$ does agree with the cotangent complex $L_{X/Y}$ (as defined below) whose zeroth cohomology sheaf is $\Omega _{X/Y}$.
Lemma 92.22.2. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{B}), \mathcal{O}_\mathcal {B})$ be a morphism of ringed topoi. Then $H^0(L_ f) = \Omega _ f$.
Proof. Special case of Lemma 92.18.6. $\square$
Lemma 92.22.3. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_1), \mathcal{O}_1) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_2), \mathcal{O}_2)$ and $g : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_2), \mathcal{O}_2) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_3), \mathcal{O}_3)$ be morphisms of ringed topoi. Then there is a canonical distinguished triangle in $D(\mathcal{O}_1)$.
\[ Lf^* L_ g \to L_{g \circ f} \to L_ f \to Lf^*L_ g[1] \]
Proof. Set $h = g \circ f$ so that $h^{-1}\mathcal{O}_3 = f^{-1}g^{-1}\mathcal{O}_3$. By Lemma 92.18.3 we have $f^{-1}L_ g = L_{f^{-1}\mathcal{O}_2/h^{-1}\mathcal{O}_3}$ and this is a complex of flat $f^{-1}\mathcal{O}_2$-modules. Hence the distinguished triangle above is an example of the distinguished triangle of Lemma 92.18.8 with $\mathcal{A} = h^{-1}\mathcal{O}_3$, $\mathcal{B} = f^{-1}\mathcal{O}_2$, and $\mathcal{C} = \mathcal{O}_1$. $\square$
Lemma 92.22.4. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{B}), \mathcal{O}_\mathcal {B})$ be a morphism of ringed topoi. There is a canonical map $L_ f \to \mathop{N\! L}\nolimits _ f$ which identifies the naive cotangent complex with the truncation $\tau _{\geq -1}L_ f$.
Proof. Special case of Lemma 92.18.10. $\square$
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