# Seminar talk, 1 March 2023

We construct an infinite family of endofunctors ${\displaystyle J_{d}^{n}}$ on the category of left ${\displaystyle A}$-modules, where ${\displaystyle A}$ is a unital associative algebra over a commutative ring ${\displaystyle k}$, equipped with an exterior algebra ${\displaystyle \Omega _{d}^{\bullet }}$. We prove that these functors generalize the corresponding classical notion of jet functors. The functor ${\displaystyle J_{d}^{n}}$ comes equipped with a natural transformation from the identity functor to itself, which plays the rôle of the classical prolongation map. This allows us to define the notion of linear differential operator with respect to ${\displaystyle \Omega _{d}^{\bullet }}$. These retain most classical properties of differential operators, and operators such as partial derivatives and connections belong to this class. Moreover, we construct a functor of quantum symmetric forms ${\displaystyle S_{d}^{n}}$ associated to ${\displaystyle \Omega _{d}^{\bullet }}$, and proceed to introduce the corresponding noncommutative analogue of the Spencer ${\displaystyle \delta }$-complex. We give necessary and sufficient conditions under which the jet functor ${\displaystyle J_{d}^{n}}$ satisfies the jet exact sequence, ${\displaystyle 0\rightarrow S_{d}^{n}\rightarrow J_{d}^{n}\rightarrow J_{d}^{n-1}\rightarrow 0}$. This involves imposing mild homological conditions on the exterior algebra, in particular on the Spencer cohomology ${\displaystyle H^{\bullet ,2}}$.