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. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 08SQ. Beware of the difference between the letter 'O' and the digit '0'.