Tangent sheaf - Knowlpedia

Definition

For a morphism of schemes $X\to S$ , the relative tangent sheaf is

\mathcal T_{X/S} =\mathcal Hom_{\mathcal O_X}(\Omega^1_{X/S},\mathcal O_X),

the $\mathcal O_X$ -dual of the sheaf of relative Kähler differentials. Equivalently, $\mathcal T_{X/S}$ is the sheaf $\operatorname{Der}_S(\mathcal O_X,\mathcal O_X)$ of $S$ -linear derivations.

When $S=\operatorname{Spec}k$ , it is written $\mathcal T_X$ . If $X$ is smooth over $S$ of relative dimension $n$ , then $\Omega^1_{X/S}$ and $\mathcal T_{X/S}$ are locally free of rank $n$ .

Lie bracket

The commutator of derivations gives $\mathcal T_{X/S}$ a Lie bracket. Along with its action on $\mathcal O_X$ , this makes it the Lie algebroid whose enveloping algebra is the sheaf $\mathcal D_X$ of differential operators in characteristic $0$ .

References

Robin Hartshorne, Algebraic Geometry, Springer, 1977, Chapter II, §8. DOI.