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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? 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.

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    This question is a rather old. But maybe this is useful. The map is given on affine schemes by $j^{1} (s) = 1 \otimes f$ in $J^1(V) \cong \mathscr{O}_{X_1} \otimes_\mathscr{O}_{X_0} V$. The point is that expanding $f$ in Taylor series and passing to the left component of the tensor product kills the higher order terms.2015-08-19

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