First a word of warning: I am not a trained algebraic geometer, so it is possible (likely) that these questions are inappropriate for MO, if so: my apologies.


Said this: As far as I understand the tangent bundle $TX$ of a scheme $X$ is defined as the spectrum of the symmetric algebra of its sheaf of differentials, and so (I guess) pointwise nothing else than the associated scheme to the Zariski tangent space.
The "projection map" $\pi: TX \rightarrow X$, which is a morphism of schemes, gives a local description of the tangent bundle, by $TU \cong \pi^{-1}(U)$, where $U$ is an affine open set in $X$.


My (first) question is the following: (1) is there a "global" description of the vector space structure (which I suppose exists?!) on the tangent bundle of a scheme. I mean addition should somehow be defined by a scheme morphism $TX \times\_X TX \rightarrow TX$, where $TX \times\_X TX$ is the scheme fibre product, and scalar multiplication by a scheme morphism $\mathbb{C} \times TX \rightarrow TX$. Does there exist such a description? Or what other basic properties does the tangent bundle of scheme have? (A reference would also be appreciated)


As a second question: (2) Is the total space associated to the tangent sheaf (defined as the $\mathcal{O}\_X$ dual of the cotangent sheaf, so the total space associated to a quasicoherent sheaf) the same as the tangent bundle? I guess there could be some issues with "taking the dual of sheaves"?!



Vector space structure on the tangent bundle of a scheme and relation to the tangent sheaf
The tangent bundle of a scheme is not in general a vector bundle. You need your sheaf of differentials to be locally free. This is automatic if the scheme is smooth over a field.


However, the additive group structure arises canonically from the symmetric algebra construction of the tangent scheme with no smoothness hypotheses, because symmetric algebras are Hopf algebras, and the addition operation comes from applying the relative Spec functor to the comultiplication map. Generalities on relative Spec and symmetric algebras are in EGA 2 and 1, respectively (maybe there's a more accessible reference?).


The process of taking duals is not involutive in general. For example, if your scheme is a pro-finite dimensional affine space (i.e., the spectrum of a polynomial ring in infinitely many generators), you can get an extra big total space. The constructions are identical if your scheme is smooth (and I imagine you can weaken the smoothness condition a bit, but I don't know how much).


