The Stacks project

76.50 Exact sequences of differentials and conormal sheaves

In this section we collect some results on exact sequences of conormal sheaves and sheaves of differentials. In some sense these are all realizations of the triangle of cotangent complexes associated to composable morphisms of algebraic spaces.

In the sequences below each of the maps are as constructed in either Lemma 76.7.6 or Lemma 76.15.8. Let $S$ be a scheme. Let $g : Z \to Y$ and $f : Y \to X$ be morphisms of algebraic spaces over $S$.

1. There is a canonical exact sequence

   \[ g^*\Omega _{Y/X} \to \Omega _{Z/X} \to \Omega _{Z/Y} \to 0, \]

   see Lemma 76.7.8. If $g : Z \to Y$ is formally smooth, then this sequence is a short exact sequence, see Lemma 76.19.12.

2. If $g$ is formally unramified, then there is a canonical exact sequence

   \[ \mathcal{C}_{Z/Y} \to g^*\Omega _{Y/X} \to \Omega _{Z/X} \to 0, \]

   see Lemma 76.15.13. If $f \circ g : Z \to X$ is formally smooth, then this sequence is a short exact sequence, see Lemma 76.19.13.

3. if $g$ and $f \circ g$ are formally unramified, then there is a canonical exact sequence

   \[ \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to g^*\Omega _{Y/X} \to 0, \]

   see Lemma 76.15.14. If $f : Y \to X$ is formally smooth, then this sequence is a short exact sequence, see Lemma 76.19.14.

4. if $g$ and $f$ are formally unramified, then there is a canonical exact sequence

   \[ g^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0. \]

   see Lemma 76.15.15. If $g : Z \to Y$ is a local complete intersection morphism, then this sequence is a short exact sequence, see Lemma 76.48.13.