The Stacks project

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.

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 $L_ f = L_{\mathcal{O}_\mathcal {C}/\mathcal{O}_\mathcal {D}}$.

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 ├ętale 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

\[ Lf^* L_ g \to L_{g \circ f} \to L_ f \to Lf^*L_ g[1] \]

in $D(\mathcal{O}_1)$.

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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