The following question is based on some remarks in section I.2 of Deligne's book *Equations Différentielles à Points Singuliers Réguliers*.


Let $X$ be a smooth complex variety and $X\_1$ the first infinitesimal neighborhood of the diagonal in $X \times X$, so there is a natural morphism $X \to X\_1$. If we write $p\_1,p\_2 : X\_1 \to X$ for the two projections, then the first-order jet bundle of a vector bundle $V$ on $X$ is defined to be $J^1(V) = p\_{1\*} p\_2^\*V$. Here "upper star" is used in the sense of $\mathcal{O}$-modules. This allows for a convenient way of expressing the notion of a connection: this is just an isomorphism $p\_1^\*V \to p\_2^\*V$ which restricts to the identity over $X$, which is the same as an $\mathcal{O}$-linear map $V \to J^1(V)$ such that the composition $V \to J^1(V) \to V$ is the identity.


Here Deligne says something I don't understand: he refers to a first-order differential operator $j^1 : V \to J^1(V)$ "which associates with any section a first-order jet" (I'm translating from the French). What is he talking about, and what makes the map he's talking about a differential operator? I don't have much intuition for jets, although I know they have something to do with taking Taylor expansions, so any general explanation of that would be greatly appreciated as well.



Grothendieck connections and jets
According to EGA IV, section 16.7 we may denote 1-jets by $\mathcal{P}^1\_X(V)$ (In fact you may even take for $V$ a coherent sheaf). There is a canonical morphism $V \to\mathcal{P}^1\_X(V)$. Notice that it is not $\mathcal{O}$-linear but just $\mathbb{C}$-linear. Now Proposition 16.8.4 tells us
$$
\mathcal{H}om\_X(\mathcal{P}^1\_X(V), W) = \mathcal{D}iff\_X(V,W)
$$
where $\mathcal{D}iff\_X(V,W)$ denotes the module of (first order) differential operators from $V$ to $W$. And the map is induced by precomposing with the canonical map $V \to\mathcal{P}^1\_X(V)$ which plays the role of "universal first order differential operator" taking values in $V$. In EGA everything is slightly more general but you may stick to order 1 and all the ideas are already there.


As for the meaning of jets, I suggest to look at the mentioned chapter in EGA, together with the corresponding part in chap. 0 of EGA IV. If differentials are first order approximation, then first order differential operators are approximations up to first order, i.e. preserving the value of the function, not just the slope of the tangent.


