Jet Functors in Noncommutative Geometry
Keegan J. Flood, Mauro Mantegazza, Henrik Winther
TL;DR
This work develops a comprehensive noncommutative jet calculus by constructing three families of endofunctors $J^{(n)}_d$, $J^{[n]}_d$, and $J^n_d$ on left $A$-modules with exterior calculus $\Omega^{\bullet}_d$. It extends classical jet theory to the noncommutative setting, defining prolongations, jet projections, and differential operators relative to $\Omega^{\bullet}_d$, and it establishes exact sequences (nonholonomic, semiholonomic, holonomic) under precise flatness and Spencer δ-cohomology conditions. The paper also develops the tensor algebra framework $T^n_d$, the symmetric-forms functor $S^{\bullet}_d$, and a noncommutative analogue of Spencer theory, enabling a robust theory of differential operators and connections in noncommutative geometry. By recovering classical jets in the commutative limit and providing path to noncommutative PDE and D-geometry perspectives, the results offer a foundational toolkit for intrinsic noncommutative differential geometry and its physical applications. See the explicit constructions for $J^1_d$, $J^2_d$, and the associated holonomic/semiholonomic sequences, along with examples on classical manifolds and pathological calculi to illustrate the necessity of flatness hypotheses for exactness.
Abstract
In this article we construct three infinite families of endofunctors $J_d^{(n)}$, $J_d^{[n]}$, and $J_d^n$ on the category of left $A$-modules, where $A$ is a unital associative algebra over a commutative ring $\mathbb{k}$, equipped with an exterior algebra $Ω^\bullet_d$. We prove that these functors generalize the corresponding classical notions of nonholonomic, semiholonomic, and holonomic jet functors, respectively. Our functors come equipped with natural transformations from the identity functor to the corresponding jet functors, which play the rôles of the classical prolongation maps. This allows us to define the notion of linear differential operators with respect to $Ω^{\bullet}_d$. We show that if $Ω^1_d$ is flat as a right $A$-module, the semiholonomic jet functor satisfies the semiholonomic jet exact sequence $0 \rightarrow \bigotimes^n_A Ω^1_d \rightarrow J^{[n]}_d\rightarrow J^{[n-1]}_d \rightarrow 0$. Moreover, we construct a functor of symmetric (in a suitable noncommutative sense) forms $S^n_d$ associated to $Ω^\bullet_d$, and proceed to introduce the corresponding noncommutative analogue of the Spencer $δ$-complex. We give necessary and sufficient conditions under which the holonomic jet functor $J_d^n$ satisfies the (holonomic) jet exact sequence, $0\rightarrow S^n_d \rightarrow J_d^n \rightarrow J_d^{n-1} \rightarrow 0$. In particular, for $n=1$ the sequence is always exact, for $n=2$ it is exact for $Ω^1_d$ flat as a right $A$-module, and for $n\ge 3$, it is sufficient to have $Ω^1_d$, $Ω^2_d$, and $Ω^3_d$ flat as right $A$-modules and the vanishing of the Spencer $δ$-cohomology $H^{\bullet,2}_{δ_d}$.