The tangent sheaf of an algebraic variety 
over a field 
is the sheaf
of 
-linear derivations
of its structure sheaf. Equivalently, it is 
, where 
is the sheaf of Kähler
differentials. On an affine variety 
, it corresponds to the module 
. Its sections
obey the Leibniz rule 
. For a smooth
variety of dimension 
it is a locally free sheaf
of rank 
and gives the algebraic tangent bundle.
On a singular variety this sheaf is still defined even though it need not be a locally free sheaf. The Lipman-Zariski conjecture asks whether being a locally free sheaf excludes singularities in characteristic zero.
