Difference between revisions of "Tangent sheaf"
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− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR></table> |
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− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table> |
Revision as of 21:57, 30 March 2012
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) MR0447223 Zbl 0362.14001 |
Comments
References
[a1] | R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 MR0463157 Zbl 0367.14001 |
Tangent sheaf. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tangent_sheaf&oldid=13919