View source for Tangent sheaf

''in algebraic geometry''

The sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t0921901.png" /> on an [[Algebraic variety|algebraic variety]] or [[Scheme|scheme]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t0921902.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t0921903.png" />, whose sections over an open affine subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t0921904.png" /> are the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t0921905.png" />-modules of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t0921906.png" />-derivations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t0921907.png" /> of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t0921908.png" />. An equivalent definition is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t0921909.png" /> be the sheaf of homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219010.png" /> of the sheaf of differentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219011.png" /> into the structure sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219012.png" /> (see [[Derivations, module of|Derivations, module of]]).

For any rational <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219013.png" />-point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219014.png" />, the stalk <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219015.png" /> of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219016.png" /> is identical to the [[Zariski tangent space|Zariski tangent space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219017.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219018.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219019.png" />, that is, to the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219020.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219022.png" /> is the maximal ideal of the local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219023.png" />. Instead of the tangent sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219024.png" /> one can use the sheaf of germs of sections of the vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219025.png" /> dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219026.png" /> (or the tangent bundle to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219027.png" />). In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219028.png" /> is a smooth connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219029.png" />-scheme, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219030.png" /> is a locally free sheaf on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219031.png" /> of rank equal to the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092190/t09219032.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.R. Shafarevich,   "Basic algebraic geometry" , Springer  (1977)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Hartshorne,   "Algebraic geometry" , Springer  (1977)  pp. Sect. IV.2</TD></TR></table>