Let $B\to X$ be a surjective submersion over the smooth integral scheme $X$ over $\mathbb{C}$. Associated to this we have in the $C^\infty$ world the notion of the $k$ jet-bundles $J\_k(B)$, which are affine bundles over $X$. I wonder what the best way to define this notion in the setting of algebraic geometry is. If $B=TX$ there is an algebraic description on Wikipedia which makes sense, but I cannot figure out how to give a satisfactory definition in general.


I am fine with restricting to the case where $B$ is the spectrum of a sheaf of $\mathcal{O}\_X$ algebras.


Edit: I am talking here about jet bundles in the sense of e.g. [this wikipedia article](https://en.wikipedia.org/wiki/Jet_bundle).



How to define jet bundles algebraically?
Let $S$ be a scheme, e.g., $\text{Spec}\ \mathbb{C}$. Let $f:X\to S$ be a morphism that is separated and smooth. Denote the associated relative diagonal morphism by $$\Delta\_{X/S}:X \to X\times\_S X.$$ This is a closed immersion whose ideal sheaf $\mathcal{I}$ is everywhere locally generated by a regular sequence. For every integer $e\geq 0$, the ideal sheaf $\mathcal{I}^{e+1}$ is the defining ideal sheaf of a closed subscheme of $X\times\_S X$. Denote this closed subscheme by, $$\Delta\_{e,X/S}:X\_e \to X\times\_S X.$$ Because $f$ is smooth, each associated projection morphism to $X$ is a finite and flat morphism (in particular, it is proper), $$p\_i: X\_e \to X, \ \ i=1,2.$$ By representability of the functor from Part IV.4.c, p. 267 (p.20 of the NUMDAM edition) in *[Fondements de la Géometrie Algébrique](https://en.wikipedia.org/wiki/Fondements_de_la_G%C3%A9ometrie_Alg%C3%A9brique)*, for every scheme $X$, for every flat and projective morphism, $$p:Y\to X,$$ for every finitely presented, quasi-projective morphism, $$q:Z\to Y,$$ there exists a universal pair, $$(r:\Pi\_{Z/Y/X} \to X, \ s:\Pi\_{Z/Y/X}\times\_X Y \to Z),$$ of a morphism from a scheme $T$ to $X$ and a $Y$-morphism from $T\times\_S Y$ to $Z$. In particular, for the flat and finite morphism $p\_2$ from $X\_e$ to $X$, for every smooth, quasi-projective morphism $q$ from a scheme $Z$ to $X\_e$, there is such a pair, $$(r:\Pi\_{Z/X\_e/X} \to X, \ s:\Pi\_{Z/X\_e/X}\times\_X X\_e \to Z).$$ Finally, for every finitely presented, quasi-projective morphism, $$\pi:B\to X,$$ the base change morphism is also finitely presented and quasi-projective, $$B\times\_{X,\text{pr}\_1} (X\times\_S X) \to X\times\_S X.$$ Thus, the pullback of this morphism over the closed subscheme $X\_e$ is also a finitely presented and quasi-projective morphism. Denote this pullback by $$\pi\_e:B\_e\to X\_e.$$ The "relative truncated sections" parameter space is the universal pair, $$(r:\Pi\_{B\_e/X\_e/X} \to X,\ s:\Pi\_{B\_e/X\_e/X}\times\_X X\_e \to B\_e).$$ If the morphism $q$ is smooth, then every "truncated section" parameterized by $\Pi\_{B\_e/X\_e/X}$ extends to a formal section by Hensel's lemma.


