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If were given a variety $X$ over an algebraically closed field $k$ together with some sheaf $O$.

What's then the definition of it's space of tangent global sections, say $T(X,O)$?

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The unhelpful answer would be: "The global sections of the tangent sheaf." If you read Hartshorne Section II.8, there are some definitions and exact sequences there. –  Thom Tyrrell Oct 8 '12 at 17:16
1) Do not use $\mathcal O$ (and even less O) for "some" sheaf: $\mathcal O$ is the structural sheaf. 2) Sheaves do not have "tangent global sections": they have global sections and these are part of the datum of a sheaf. 3) There is indeed a tangent sheaf to a variety: it is called $T$ (for example) and its global sections are denoted by $T(X)$ or $\Gamma(X,T)$. Your seem to be mixing-up several concepts and notations. –  Georges Elencwajg Oct 8 '12 at 17:44
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