The Stacks project

92.20 The cotangent complex of a morphism of ringed spaces

The cotangent complex of a morphism of ringed spaces is defined in terms of the cotangent complex we defined above.

Definition 92.20.1. Let $f : (X, \mathcal{O}_ X) \to (S, \mathcal{O}_ S)$ be a morphism of ringed spaces. The cotangent complex $L_ f$ of $f$ is $L_ f = L_{\mathcal{O}_ X/f^{-1}\mathcal{O}_ S}$. We will also use the notation $L_ f = L_{X/S} = L_{\mathcal{O}_ X/\mathcal{O}_ S}$.

More precisely, this means that we consider the cotangent complex (Definition 92.18.2) of the homomorphism $f^\sharp : f^{-1}\mathcal{O}_ S \to \mathcal{O}_ X$ of sheaves of rings on the site associated to the topological space $X$ (Sites, Example 7.6.4).

Lemma 92.20.2. Let $f : (X, \mathcal{O}_ X) \to (S, \mathcal{O}_ S)$ be a morphism of ringed spaces. Then $H^0(L_{X/S}) = \Omega _{X/S}$.

Proof. Special case of Lemma 92.18.6. $\square$

Lemma 92.20.3. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of ringed spaces. Then there is a canonical distinguished triangle

\[ Lf^* L_{Y/Z} \to L_{X/Z} \to L_{X/Y} \to Lf^*L_{Y/Z}[1] \]

in $D(\mathcal{O}_ X)$.

Proof. Set $h = g \circ f$ so that $h^{-1}\mathcal{O}_ Z = f^{-1}g^{-1}\mathcal{O}_ Z$. By Lemma 92.18.3 we have $f^{-1}L_{Y/Z} = L_{f^{-1}\mathcal{O}_ Y/h^{-1}\mathcal{O}_ Z}$ and this is a complex of flat $f^{-1}\mathcal{O}_ Y$-modules. Hence the distinguished triangle above is an example of the distinguished triangle of Lemma 92.18.8 with $\mathcal{A} = h^{-1}\mathcal{O}_ Z$, $\mathcal{B} = f^{-1}\mathcal{O}_ Y$, and $\mathcal{C} = \mathcal{O}_ X$. $\square$

Lemma 92.20.4. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. There is a canonical map $L_{X/Y} \to \mathop{N\! L}\nolimits _{X/Y}$ which identifies the naive cotangent complex with the truncation $\tau _{\geq -1}L_{X/Y}$.

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Your email address will not be published.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 08UT. Beware of the difference between the letter 'O' and the digit '0'.