How to define jet bundles algebraically?

Question

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.

Answer 1

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, 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.

Answer 2

According to Mustață, Jet schemes of locally complete intersection canonical singularities, the $m$-th jet scheme of (the $\mathbb{C}$-scheme) $X$ is the scheme $J_m X$ over $X$ representing the functor $\mathrm{Sch} \to \mathrm{Set}$ given by

$$ S \mapsto \mathrm{Hom}(\Delta^m\times S,X)$$

where $\Delta^m:=\mathrm{Spec}\frac{\mathbb{C}[t]}{\langle t^{m+1} \rangle}\;.$

So the closed points of $J_m X$ are the morhisms of schemes $\Delta^m\to X$. And the fiber of $J_m X\to X$ over $x\in X$ has as set of closed points the set $\mathrm{Hom}(\mathcal{O}_{X,x}\;,\mathbb{C}[t]/\langle t^{m+1} \rangle)$.

Another way of seeing it is: let $\mathbf{F}:\mathrm{Sch}\to\mathrm{Sch}$ be the functor $\mathbf{F}(S)=\Delta^m\times S$, than it has a right adjoint given by $X\mapsto J_m X$.

Remark. According to the above construction we have $J_1 X=TX$, the tangent bundle (or total space of tangent sheaf if $X$ isn't smooth).