My question is vague and general, and, if you want, naive.

$\bullet$ Given $X/S$ a scheme. We denote by $X_{red}$ the reduced scheme associated to $X$.

Grothendieck defines in EGA 4 (16.5.12.1) the tangent bundle $T_{X/S}$ of $X$ relatively to $S$. Is there any relation between $T_{X_{red}/S}$ and $T_{X/S}$ ? More precisely, we know that there is a morphism $$ f : T_{X_{red}/S} \to T_{X/S}\times_{X} X_{red} ; $$ what can be said of $f$ ? I am interested, in particular, in the case when $S=Spec \ k$, with $k$ a field.

$\bullet$ A related question (but in which I am less interested) is the comparison of the Zariski tangent space at one point (as defined in Hartshorne p. 80) of $X$ and $X_{red}$. In general, if $\pi : X_{red} \to X$ denotes the canonical morphism, and if $x\in X_{red}$, the linear map $$ T_x \ \pi : T_x \ X_{red} \to T_{\pi(x)} \ X $$ is injective, since $\pi$ is a closed immersion. Is it an isomorphism ?

$\bullet$ Finally, a related question is the comparison of $\Omega^1_{X/S}$ and $\Omega^1_{X_{red}/S}$.

$\bullet$ Ideally, I would like to transport sections of $T_{X_{red}/S}$ to sections of $T_{X/S}$.