Tangent sheaf
in algebraic geometry
The sheaf 
on an algebraic variety or scheme 
over a field 
, whose sections over an open affine subspace 
are the 
-modules of 
-derivations 
of the ring 
. An equivalent definition is that 
be the sheaf of homomorphisms 
of the sheaf of differentials 
into the structure sheaf 
(see Derivations, module of).
For any rational 
-point 
, the stalk 
of the sheaf 
is identical to the Zariski tangent space 
to 
at 
, that is, to the vector 
-space 
, where 
is the maximal ideal of the local ring 
. Instead of the tangent sheaf 
one can use the sheaf of germs of sections of the vector bundle 
dual to 
(or the tangent bundle to 
). In the case when 
is a smooth connected 
-scheme, 
is a locally free sheaf on 
of rank equal to the dimension of 
.
References
| [1] | I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) |
Comments
References
| [a1] | R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 |
Tangent sheaf. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tangent_sheaf&oldid=23990