Tangent sheaf

in algebraic geometry

The sheaf $ \theta _ {X} $ on an algebraic variety or scheme $ X $ over a field $ k $, whose sections over an open affine subspace $ U = \mathop{\rm Spec} ( A) $ are the $ A $- modules of $ k $- derivations $ \mathop{\rm Der} _ {k} ( A, A) $ of the ring $ A $. An equivalent definition is that $ \theta _ {X} $ be the sheaf of homomorphisms $ \mathop{\rm Hom} ( \Omega _ {X/k} ^ {1} , {\mathcal O} _ {X} ) $ of the sheaf of differentials $ \Omega _ {X/k} ^ {1} $ into the structure sheaf $ {\mathcal O} _ {X} $( see Derivations, module of).

For any rational $ k $- point $ x \in X $, the stalk $ \theta _ {X} ( x) $ of the sheaf $ \theta _ {X} $ is identical to the Zariski tangent space $ T _ {K,x} $ to $ X $ at $ x $, that is, to the vector $ k $- space $ \mathop{\rm Hom} _ {k} ( \mathfrak M _ {x} / \mathfrak M _ {x} ^ {2} , k) $, where $ \mathfrak M _ {x} $ is the maximal ideal of the local ring $ {\mathcal O} _ {K,x} $. Instead of the tangent sheaf $ \theta _ {X} $ one can use the sheaf of germs of sections of the vector bundle $ V ( \Omega _ {X/k} ^ {1} ) $ dual to $ \Omega _ {X} ^ {1} $( or the tangent bundle to $ X $). In the case when $ X $ is a smooth connected $ k $- scheme, $ \theta _ {X} $ is a locally free sheaf on $ X $ of rank equal to the dimension of $ X $.

References

[1]	I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
[a1]	R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 MR0463157 Zbl 0367.14001