I am learning formal PDE theory for my research and I am currently struggling to have a basic understanding of the operations involved in completing a (say, linear) PDE system to an involutive one (Cartan-Kuranishi procedure) in terms of the partial differential operators involved.
First, let us put some context in order to make the question a bit more interesting to a larger audience and to fix notation. In what follows, all manifolds are smooth, Hausdorff, paracompact, connected and oriented, and all maps between any two of them are assumed to be smooth. Given two vector bundles $\pi:E\rightarrow M$, $\pi':E'\rightarrow M$ over the base manifold $M$, a (linear) partial differential operator of type $E\rightarrow E'$ and order $k\geq 0$ is a linear map $$P:\Gamma(\pi)\rightarrow\Gamma(\pi')$$ ($\Gamma(\pi^{(}{}'{}^{)})=\{\phi\in C^\infty(M,E^{(}{}'{}^{)})\ |\ \pi^{(}{}'{}^{)}\circ\phi=\text{id}_M\}$ is the space of sections of $\pi^{(}{}'{}^{)}$) of the form $$P=\Phi_P\circ j^k\ ,$$ where $j^k$ stands for the $k$-th order jet prolongation of sections of vector bundles and $$\Phi_P:J^kE\rightarrow E'$$ is a vector bundle map covering $\text{id}_M$, i.e. $\pi'\circ\Phi_P=\pi\circ\pi^k$, where $\pi^k:J^k E\rightarrow E$ is the projection map of the $k$-th order jet bundle $J^k E$ of $\pi$ onto the base $E$ (with the convention $J^0 E=E$, $\pi^0=\text{id}_E$, $j^0=\text{id}$). The (homogeneous linear) PDE system associated to $P$ is given by $$\mathcal{R}_k=\ker\Phi_P\ .$$ The $q$-th order prolongation of $P$ ($q\geq 0$) is the partial differential operator $\rho_q P$ of type $E\rightarrow J^q F$ and order $k+q$ given by $$\rho_q P\doteq j^q\circ P=\rho_q(\Phi_P)\circ j^{k+q}\ ,$$ so that $\rho_0 P=P$. The vector bundle map $\rho_q(\Phi_P):J^{k+q}E\rightarrow J^q E'$ covering $\text{id}_M$ is uniquely determined by the second equality above (particularly, $\rho_0(\Phi_P)=\Phi_P$). The PDE system associated to $\rho_q P$, called the $q$-th order prolongation of the PDE system $\mathcal{R}_k$, is then denoted by $$\mathcal{R}_{k+q}=\ker\rho_q(\Phi_P)\ .$$ Therefore, the operation of prolongation of PDE systems has a clear, global interpretation in terms of prolongation of the associated partial differential operators. In what follows, we assume in addition that $P$ is regular, that is, $\rho_q(\Phi_P)$ has constant rank for all $q\geq 0$ - this is demonstrably equivalent to assuming that $\mathcal{R}_{q+k}$ is a vector sub-bundle of $J^{k+q}E$ for all $q\geq 0$. Conversely, if $\mathcal{R}_k$ is a vector sub-bundle of $J^kE$, it is locally the kernel of $\Phi_P$ for some partial differential operator $P$ or order $k$.
Finally, let $$\pi^{r+s}_r:J^{r+s}E\rightarrow J^r E\ ,\quad r,s\geq 0$$ be the natural jet projection maps, so that $$\pi^r_0=\pi^r\ ,\quad\pi^{r+s}_r\circ\pi^{r+s+t}_{r+s}=\pi^{r+s+t}_r$$ for all $r,s,t\geq 0$. It can be shown that $\pi^{k+q}_{k+r}(\mathcal{R}_{k+q})\subset\mathcal{R}_{k+r}$ for all $0\leq r\leq q$. If $P$ is regular and $\pi^{k+q}_{k+r}:\mathcal{R}_{k+q}\rightarrow\mathcal{R}_{k+r}$ have constant rank for all such $r,q$, we say that $P$ is sufficiently regular. Particularly, if $P$ is regular and the above maps are surjective for all such $r,q$, we say that $P$ is formally integrable.
The other key operation employed in the Cartan-Kuranishi completion algorithm besides prolongation is projection. Both operations together unveil the hidden integrability conditions of $P$. In terms of the PDE system $\mathcal{R}_k$ associated to $P$ and its prolongations $\mathcal{R}_{k+q}$ of order $q>0$, projection is a simple operation to describe (unlike prolongation, which is more straightforwardly described in terms of $P$ as done above). Namely, the projection $\mathcal{R}^{(r)}_{k+q}$ of order $r\geq 0$ of $\mathcal{R}_{k+q+r}$ is given by $$\mathcal{R}^{(r)}_{k+q}=\pi^{k+q+r}_{k+q}(\mathcal{R}_{k+q+r})\subset\mathcal{R}_{k+q}\ .$$ For the purposes of the Cartan-Kuranishi algorithm, it suffices to consider $r$-th order projections of PDE systems with $r=1$.
Now we have reached a point where I can ask my
Question: Suppose that $P$ is sufficiently regular. Is there a simple, global formula for a partial differential operator to which the projected PDE system $\mathcal{R}^{(1)}_{k+q}=\pi^{k+q+1}_{k+q}(\mathcal{R}_{k+q+1})$ is associated? (I understand that such an operator should not be unique)
I simply could not find such a formula in the standard literature on the subject (e.g. the 1964 thesis of Quillen and the books of Pommaret, Bryant et al. and Seiler). My conjecture is that composing $\rho_{q+1}P=\rho_{q+1}(\Phi_P)\circ j^{k+q+1}=j^{q+1}\circ\Phi_P\circ j^k$ with the natural projection $\psi_{q+1}$ onto the cokernel of its principal symbol $\sigma(\rho_{q+1} P)$ would provide such an operator and hence a positive answer to my question. In other words,
Subquestion: Is there a vector bundle map $$\rho^{(1)}_q(\Phi_P):J^{k+q}E\rightarrow\text{coker}(\sigma(\rho_{q+1} P))$$ covering $\text{id}_M$ such that $$\rho^{(1)}_q(\Phi_P)\circ\pi^{k+q+1}_{k+q}=\psi_{q+1}\circ\rho_{q+1}(\Phi_P)\ ?$$ If that is the case, is it true that $\mathcal{R}^{(1)}_{k+q}=\ker(\rho^{(1)}_q(\Phi_P))$?
(Remark: if $P$ is sufficiently regular, then $\sigma(\rho_{q+1} P)$ has constant rank and therefore both $\ker(\sigma(\rho_{q+1} P))$ and $\text{coker}(\sigma(\rho_{q+1} P))$ are vector bundles)
If true, this would provide a nice formula for the partial differential operator $D$ produced by the Cartan-Kuranishi algorithm such that $DP$ is involutive and equivalent to $P$, in the same spirit as the compatibility complex associated to an involutive (or, more generally, a formally integrable) partial differential operator, construced by Quillen in his thesis. Moreover, a variation of this procedure would also possibly work with quasilinear partial differential operators.
