I am working through the book The Geometry of Jet Bundles by D.J. Saunders and came across the formula
$$d_h+d_v = \pi_{k+1,k} ^\star \circ d,$$
which I am unable to prove. Here $d$ denotes the exterior derivative, $d_h$ and $d_v$ the horizontal and vertical differentials respectively, and $\pi_{k+1,k}^\star$ the pullback along $\pi_{k+1,k}$. This statement might be completely trivial but I've been staring at it for the better part of two days now and I just can't get my head around it.
I am unsure of how standard the notation used in the book is, so I'll briefly introduce the relevant constructions.
We are considering a fiber bundle $ \pi \colon E \longrightarrow M$ and its associated jet manifolds $J^k \pi$. The relevant bundle projections are denoted \begin{align*}\label{} \pi _k\colon J^k \pi & \longrightarrow M\\ j^k_p\phi & \longmapsto p,\\\\ \pi _{k,0}\colon J^k \pi & \longrightarrow E\\ j^k_p\phi & \longmapsto \phi(p),\\\\ \pi _{k,l}\colon J^k \pi & \longrightarrow J^l \pi \\ j^k_p\phi & \longmapsto j^l_p \phi. \end{align*} and we denote the tangent and cotangent bundles by \begin{align*}\label{} \tau_{J^k\pi}\colon TJ^k \pi & \longrightarrow J^k\pi,\\ \tau^*_{J^k\pi}\colon T^*J^k \pi & \longrightarrow J^k\pi. \end{align*}
The relevant construction starts with the fact that the pullback bundle $ (\pi ^\star_{k+1,k}(TJ^k \pi) , \pi ^\star _{k+1,k}(\tau _{J^k \pi }),J^k \pi ) $ may be written as the direct sum of vertical and holonomic elements: $$ (\pi ^\star_{k+1,k}(V \pi _k )\oplus H \pi_{k+1,k}, \pi ^\star _{k+1,k}(\tau _{J^k \pi }), J^k \pi ), $$ and the corresponding decomposition of $ (\pi ^\star _{k+1,k}(T^*J^k \pi ),\pi ^\star _{k+1,k}(\tau ^*_{J^k \pi }),J^{k+1} \pi )$ into horizontal and contact elements: $$ (\pi ^\star_{k+1,k}(\pi ^\star _k(T^*M))\oplus C^* \pi _{k+1,k}, \pi ^\star _{k+1,k}(\tau ^*_{J^k \pi }), J^{k+1} \pi ). $$ Using these decompositions we can define the vector bundle endomorphisms $h$ and $v$:
Def: The vector bundle endomorphisms $ (h, \operatorname{id} _{J^k \pi }) $ and $ (v, \operatorname{id} _{J^k \pi }) $ of $ \pi ^\star _{k+1,k}(\tau _{J^k \pi }) $ are defined as \begin{align*}\label{} h(\xi ^h + \xi ^v) &= \xi ^h\\ v(\xi ^h + \xi ^v) &= \xi ^v, \end{align*} where $ \xi^h \in H \pi _{k+1,k} $ and $ \xi ^v \in \pi ^\star _{k+1,k}(V \pi _k) $.
Def: The vector bundle endomorphisms $ (h, \operatorname{id} _{J^k \pi }) $ and $ (v, \operatorname{id} _{J^k \pi }) $ of $ \pi ^\star _{k+1,k}(\tau^* _{J^k \pi }) $ are defined as \begin{align*}\label{} h(\eta ^h + \eta ^v) &= \eta ^h\\ v(\eta ^h + \eta ^v) &= \eta ^v, \end{align*} where $ \eta ^h \in \pi ^\star _{k+1,k}(\pi ^\star_k(T^*M)) $ and $ \eta^v \in C^* \pi _{k+1,k} $.
The endomorphisms $h$ and $v$ now allow us to construct the following vector valued 1-forms also called $h$ and $v$:
Def: The vector valued 1-forms $ h $ and $ v $ are the sections of the bundle $ \pi ^\star _{k+1,k}(\tau ^*_{J^k \pi })\otimes \pi ^\star _{k+1,k}(\tau _{J^k \pi }) $ defined by \begin{align*}\label{} h_{j^{k+1}_p \phi }( \xi , \eta ) &= \eta (h(\xi ))\\ v_{j^{k+1}_p \phi }( \xi , \eta ) &= \eta (v(\xi )), \end{align*} for $ \xi \in \pi ^\star _{k+1,k}(TJ^k \pi )_{j^{k+1}_p \phi } $ and $ \eta \in \pi ^\star _{k+1,k}(T^*J^k \pi )_{j^{k+1}_p \phi } $.
The map \begin{align} \pi ^\star_{k+1,k} (T^*J^k\pi) &\longrightarrow T^*J^{k+1}\pi\\ (\eta , j^{k+1}_p \phi ) &\longmapsto (\pi_{k+1,k} ^\star \,\eta)_{j^{k+1}_p \phi} \end{align} allows us to consider $ \pi ^\star _{k+1,k}(\tau ^*_{J^k \pi })\otimes \pi ^\star _{k+1,k}(\tau _{J^k \pi }) $ as a subbundle of $ \tau ^*_{J^{k+1} \pi }\otimes \pi ^\star _{k+1,k}(\tau _{J^k \pi }) $, and so we shall regard $h$ and $v$ as vector-valued 1-forms along $\pi_{k+1,k}$.
Def: If $\xi$ is a vector-valued 1-form along $\pi_{k+1,k}$, we define the interior multiplication by $\xi$ along $\pi_{k+1,k}$, denoted $\imath_\xi$, by $$ \imath _ \xi f =0 \text{ for all } f \in \mathcal{C} ^\infty(J^k\pi), $$ and $$ (\imath_ \xi \vartheta) (X_1, \ldots, X_{s}):= \sum _\sigma (-1)^\sigma \vartheta \Big( \xi (X_{\sigma (1)}), \pi _{k+1,k\star}\, X_{\sigma (2)}, \ldots, \pi _{k+1,k\star} \,X_{\sigma (s)}\Big), $$ for $ \vartheta \in \bigwedge ^s J^k\pi $ ($s \geq 1 $) and $ X_1, \ldots, X_{s} \in \mathfrak{X} (J^{k+1}\pi) $, where we only sum over those permutations $\sigma$ that satsify $\sigma(2)<\ldots<\sigma(s)$.
Having defined the interior multiplications $\imath_h$ and $\imath_v$ we can now define the horizontal and vertical differentials: \begin{align} d_h &:= \imath_h \circ d - d \circ \imath_h,\\ d_v &:= \imath_v \circ d - d \circ \imath_v.\\ \end{align}
The relevant statement in the book
One consequence of the relationship $h+v= \pi _{k+1,k}$ is that $d_h +d_v = \pi _{k+1,k\star} \circ d$; this yields ...
can be found on page 216. I believe that the pushforward in this statement should in fact be the pullback but this is probably just a typo.
I think I figured it out while typing up the question. The solution is indeed almost trivial and, as the author states, the only specific property of $h$ and $v$ necessary to prove the statement is $$ h+v=\pi_{k+1,k\star}.$$ Since $d_h+d_v$ is a derivation along $\pi_{k+1,k}$ (this is easy to show) and derivations are uniquely specified by how they act on functions and 1-forms it suffices to check how $d_h+d_v$ acts on functions $f \in \mathcal{C}^\infty(J^k\pi)$ and 1-forms $\vartheta \in \bigwedge^1 J^k\pi$.
First notice that for 1-forms $$ (\imath_h+\imath_v) \vartheta = \pi_{k+1,k}^\star \vartheta,$$ which gives \begin{align} (d_h+d_v)f&= (\imath_h+\imath_v)df\\ &= \pi_{k+1,k}^\star df, \end{align} where we used that $df$ is a 1-form and $\imath_hf=\imath_vf=0$. Now let $ X_0,X_1 \in \mathfrak{X}(J^{k+1}\pi) $, the definition of $ \imath_h $ simply gives $$ (\imath_h d \vartheta )(X_0,X_1) = d \vartheta (hX_0, \pi _{k+1,k\star} X_1) - d \vartheta (hX_1, \pi _{k+1,k\star} X_0). $$ Together with the analogous result for $ \imath_v $ this shows \begin{align*}\label{} ((\imath_h+\imath_v)d \vartheta )(X_0,X_1) &= d \vartheta (\pi_{k+1,k\star} X_0, \pi _{k+1,k\star} X_1) - d \vartheta (\pi_{k+1,k\star} X_1, \pi _{k+1,k\star} X_0)\\ &= 2 (\pi _{k+1,k}^\star \, d \vartheta) (X_0,X_1), \end{align*} which already gives $$ (d_h+d_v) \vartheta = \pi _{k+1,k}^\star\, d \vartheta. $$ So we have shown that $d_h+d_v$ and $\pi_{k+1,k}^\star\circ d$ agree on functions as well as on 1-forms, which, since both are derivations, implies that they are equal.
Is this correct?