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}.
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.
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
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