The Stacks project

37.57 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 a pair of composable morphisms of schemes.

Let $g : Z \to Y$ and $f : Y \to X$ be morphisms of schemes.

  1. There is a canonical exact sequence

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

    see Morphisms, Lemma 29.32.9. If $g : Z \to Y$ is smooth or more generally formally smooth, then this sequence is a short exact sequence, see Morphisms, Lemma 29.34.16 or see Lemma 37.11.11.

  2. If $g$ is an immersion or more generally 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 Morphisms, Lemma 29.32.15 or see Lemma 37.7.10. If $f \circ g : Z \to X$ is smooth or more generally formally smooth, then this sequence is a short exact sequence, see Morphisms, Lemma 29.34.17 or see Lemma 37.11.12.

  3. If $g$ and $f \circ g$ are immersions or more generally 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 Morphisms, Lemma 29.32.18 or see Lemma 37.7.11. If $f : Y \to X$ is smooth or more generally formally smooth, then this sequence is a short exact sequence, see Morphisms, Lemma 29.34.18 or see Lemma 37.11.13.

  4. If $g$ and $f$ are immersions or more generally 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 Morphisms, Lemma 29.31.5 or see Lemma 37.7.12. If $g : Z \to Y$ is a regular immersion1 or more generally a local complete intersection morphism, then this sequence is a short exact sequence, see Divisors, Lemma 31.21.6 or see Lemma 37.56.23.

[1] It suffices for $g$ to be a $H_1$-regular immersion. Observe that an immersion which is a local complete intersection morphism is Koszul regular.

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