The Stacks project

90.20 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 90.20.1. 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 90.18.2) 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 90.20.2. 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 90.18.6. $\square$

Lemma 90.20.3. 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 90.18.3 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 90.18.8 with $\mathcal{A} = h^{-1}\mathcal{O}_ Z$, $\mathcal{B} = f^{-1}\mathcal{O}_ Y$, and $\mathcal{C} = \mathcal{O}_ X$. $\square$

Lemma 90.20.4. 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 90.18.10. $\square$


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