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Apologies in advance if this question is too basic. Let $X$ be a scheme. One defines the $n$-th jet space of $X$ to be the scheme representing the functor $$ A\mapsto X(A[t]/t^{n+1}) . $$ It is well-known that this functor is represented by a scheme $J^n X$. There are a number of short notes proving existence of $J^n X$, basic properties, ....

My understanding is that if $\mathscr{E}$ is a locally free $\mathscr{O}$-module on $X$, then for each $n$ there is a $\mathscr{O}$-module $J^n\mathscr{E}$ of "$n$-jets in $\mathscr{E}$." This is mentioned in passing in many places, but I have yet to run across a good functorial definition, let alone proofs of basic properties.

Is there a good reference which carefully defines $J^n\mathscr{E}$ and proves its basic properties?

Ideally, I'm interested in references that treat existence in maximal generality (e.g. arbitrary coherent $\mathscr{O}$-module on any scheme $X$).

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Two good references are [EGA IV4, 16.7-8] and [SGA 31, VIIA.1]. Let $f:X\to S$ be a morphism of schemes. Let $\mathscr I_{X/S}\subset \mathscr O_{X\times_S X}$ be the ideal defining the diagonal, and let $X^{(n)}\subset X\times_S X$ be the closed subscheme cut out by $\mathscr I_{X/S}^{n+1}$. There are morphisms $p_i^{(n)}:X^{(n)}\to X$ coming from the two projections $p_1,p_2:X\times_S X\to X$. For an $\mathscr O_X$-module $\mathscr F$, put $$ \mathrm{jet}^n\mathscr F = p_{1\ast}^{(n)} p_2^{(n)\ast} \mathscr F . $$ Then $\mathrm{jet}^n\mathscr F$ is quasi-coherent (resp. coherent) if $\mathscr F$ is. Moreover, the natural map $\mathrm d^n:\mathscr F\to \mathrm{jet}^n\mathscr F$ induces an isomorphism $$ \mathrm{Diff}_{X/S}^n(\mathscr F,\mathscr G)=\hom_{\mathscr O_X}(\mathrm{jet}^n\mathscr F,\mathscr G) . $$ Here, $\mathrm{Diff}^n_{X/S}(\mathscr F,\mathscr G)$ is defined inductively by $\mathrm{Diff}_{X/S}^0(\mathscr F,\mathscr G)=\hom_{\mathscr O_X}(\mathscr F,\mathscr G)$, and by $\mathrm{DIff}_{X/S}^{n+1}(\mathscr F,\mathscr G)$ being the sub-$\mathscr O_X$-module of $\hom_{\mathscr O_S}(\mathscr F,\mathscr G)$ consisting of those $D$ such that for all $a\in \Gamma(\mathscr O_X)$, the map $x\mapsto D(a x)-aD(x)$ is an element of $\mathrm{Diff}_{X/S}^n(\mathscr F,\mathscr G)$.

To sum it up:

If you define "differential operator of order $\leqslant n$ in the obvious way, then $\mathrm{Diff}_{X/S}^{\leqslant n}(\mathscr F,\mathscr G)=\hom_{\mathscr O_X}(\mathrm{jet}^n\mathscr F,\mathscr G)$.

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