Tangent sheaf of a scheme (${\mathcal{T}}_{X/S}=\mathcal{H}om({\Omega }_{X/S}^1,{\mathcal{O}}_X)$)

The relative tangent sheaf is the dual of a sheaf of Kähler differentials. Its sections are derivations of the structure sheaf that kill base scalars. For a smooth morphism it is locally free and describes first-order deformations of sections into the scheme.