Example

Take $X = K^n$, and denote by $u_1, \ldots, u_n$ the coordinate functions on $K^n$; then the fibre $P^k_0(X)$ of $P^k(X)$ at 0 has as basis the family of jets of order $k$ of the monomials $u_1^{m_1} \ldots u_n^{m_n}$, with $m_i \geqslant 0, \sum_{i=0}^n m_i \leqslant k$.

12.6.3. Let $d$ be an integer $\geqslant 0$, let $E_1, \ldots, E_d, F$ be vector bundles of class $C^r$ with base $X$, and let $u : E_1 \times_X \cdots \times_X E_d \to F$ be a multilinear morphism (7.3.1) of class $C^r$. There then exists one and only one multilinear morphism

such that

for every open set U of X and every sequence of sections $s_i \in \mathscr{S}_{E_i}'(U)$ for $1 \leq i \leq d$.

If A is a bundle of algebras (7.3.2) with base X, the morphism from $P^k(A) \times_X P^k(A)$ into $P^k(A)$ induced by the multiplication $A \times_X A \to A$ makes $P^k(A)$ a bundle of algebras; if A is a bundle of associative algebras, $P^k(A)$ is a bundle of associative algebras; if, in addition, M is a bundle of A-modules (7.3.3), $P^k(M)$ is a bundle of $P^k(A)$-modules. In particular, $P^k(X)$ is an associative, commutative and unital bundle of algebras; if E is a vector bundle, $P^k(E)$ is a bundle of $P^k(X)$-modules. If $u : E \to F$ is a morphism of vector bundles, then $P^k(u) : P^k(E) \to P^k(F)$ is a $P^k(X)$-homomorphism.

12.6.4. If $E \xrightarrow{u} F \xrightarrow{v} G$ is a locally split exact sequence of vector bundles with base X, then the same is true of the sequence

where the homomorphisms in question are $P^k(u)$ and $P^k(v)$.

12.6.5. Let $k'$ and $k''$ be two positive integers whose sum is $k$, and let E be a vector bundle of class $C^r$ with base X. The morphism

induces a morphism of vector bundles

which is of class $C^{r-k}$. If U is an open set of X and $f \in \mathscr{S}_E'(U)$, we have

When K is of characteristic zero, $\beta$ is an isomorphism of $P^k(E)$ onto a vector subbundle of $P^{k''}(P^{k'}(E))$.

12.6.6. Let $k'$ be an integer such that $0 \leq k' \leq k$, and let E be a vector bundle of class $C^r$ with base X. The mapping

is a locally direct surjective morphism of class $C^{r-k}$. Its kernel $N^{k, k'}(E)$ consists of section jets of E having contact of order $\geq k'$ with the zero section.

12.6.7 (The vectorial functor $P_m$). With the notation of 7.6, 7.7, 7.8, put $I_+ = \{0\}, I_- = \{1\}$ and let m be an integer $\geq 0$. If $\mathscr{V} = (V_0, V_1)$ is a pair of Banach spaces, denote by $\tau_m(\mathscr{V})$ the Banach space $P_m(V_1; V_0)$ of homogeneous continuous polynomials of degree m on $V_1$ with values in $V_0$ (A.2). Similarly, if $f = (f_0, f_1)$. where $f_0 : V_0 \to V'_0$ and $f_1 : V'_1 \to V_1$ are morphisms of Banach spaces, denote by $\tau_m(f)$ the morphism $p \mapsto f_0 \circ p \circ f_1$ from $P_m(V_1; V_0)$ into $P_m(V'_1; V'_0)$. We thus obtain a vectorial functor $\tau_m$ of class $C^\omega$. If $E_0$ and $E_1$ are two vector bundles with base $X$, denote by $P_m(E_1; E_0)$ the vector bundle deduced from $(E_0, E_1)$ by means of $\tau_m$ (7.6.2).

12.6.8. Let us resume the notation and hypotheses of 12.6.1. There exists one and only one morphism of vector bundles

such that, whatever the vectorial chart $c_0 = (U, \varphi, F_0)$ of $E$ and the chart $c_1 = (U, \psi, F_1)$ of $X$, the following diagram is commutative:

where: 1) $\iota|U$ is the restriction of $\iota$ to $P_k(T(X), E)|U$; 2) $\theta$ is the bijection defined in 12.6.1; 3) $i$ is the mapping $(u, p) \mapsto (u, 0, \ldots, 0, p)$; 4) $\eta$ is deduced, by means of the vectorial functor $\tau_k$, from the vectorial chart $c'_1$ of $T(X)$ (cf. 8.1.1) and the vectorial chart $c_0$ of $E$.

The morphism $\iota$ is an isomorphism of $P_k(T(X); E)$ onto the vector subbundle $N^{k, k-1}(E)$ of $P^k(E)$ (cf. 12.6.6); the sequence

is a locally direct exact sequence of vector bundles.

More generally, put $N_0 = P^k(E)$, $N_m = N^{k, m-1}(E)$ for $m \geq 1$, so that the $N_m$ form a decreasing sequence of vector subbundles of $P^k(E)$:

For $0 \leq m \leq k$, the projection $r^{k, m} : P^k(E) \to P^m(E)$ defines an isomorphism of $N_m/N_{m+1}$ onto $N^{m, m-1}(E)$; viewed in the light of the foregoing, we thus obtain isomorphisms

In particular, $N_0/N_1 \simeq E$ and $N_1/N_2 \simeq \mathscr{L}(T(X); E)$. For $k = 1$, this gives an exact sequence: