# Differentials

The sheaf of differentials Ω_{X/S} of one scheme X over another scheme S is the target of the universal O_S-derivation d_{X/S} : O_X —> Ω_{X/S}. I remember being surprised to learn that people habitually define this sheaf using the conormal sheaf C_{X/Xx_SX} of the diagonal morphism of X over S[1].

Why is it not the “right thing” to do? The reason is that both the conormal sheaf and the sheaf of differentials have a natural functoriality, and that the identification of C_{X/Xx_SX} with Ω_{X/S} is not compatible with this! Namely, consider the morphism that flips the factors on Xx_SX. This should clearly act by -1 on the conormal sheaf C_{X/Xx_SX} and by +1 on Ω_{X/S}. So there you go!

When X —> S is a morphism of algebraic spaces, then the diagonal morphism isn’t an immersion in general so the conormal sheaf is harder to define. In this case defining Ω_{X/S} as the target of the universal O_S-derivation d_{X/S} : O_X —> Ω_{X/S} on the small etale site of X works fine, see Tag 04CR.

Finally, suppose that X —> S is a morphism of algebraic stacks. We have yet to choose (in the stacks project) which site to use to define quasi-coherent sheaves on X. But in order to study differentials the only reasonable choice seems to be the lisse-etale site X_{lisse, etale}. Again there is a universal O_{S_{lisse, etale}}-derivation d : O_{X_{lisse, etale}} —> Ω. Now, (I think) Ω is not a quasi-coherent O_{X_{lisse, etale}}-module, and it is not what authors on algebraic stacks define as Ω_{X/S}, but for some purposes it might be the right thing to look at (e.g., deformation theory?).

Footnote 1: Yes, currently the stacks project also introduces sheaves of differentials for morphisms of schemes using this method. The first result is then that d_{X/S} is a universal derivation, see Lemma Tag 01UR. Having proven this, maps involving Ω_{X/S} are defined using the universal property.