Let $p_{A_M}:A_M\longrightarrow M$ be a Lie algebroid and $I:=[0, 1]$. Then $$p_{A_M}\times \textrm{id}:A_M\times I\longrightarrow M\times I,$$ is a vector bundle. There is a $C^\infty(M\times I)$-module isomorphism $$\Gamma(A_M\times I)\simeq C^\infty(M\times I)\otimes_{C^\infty(M)} \Gamma(A_M),$$ where $\Gamma(A_M\times I)$ stands for the space of sections of $A_M\times I$.
Technical remarks:
(i) $C^\infty(M\times I)$ is a $C^\infty(M\times I)$-module using the product of the algebra $C^\infty(M\times I)$;
(ii) The projection $\textrm{pr}_1:M\times I\longrightarrow M$ induces a morphism of $\mathbb R$-algebras $$\textrm{pr}_1^*:C^\infty(M)\longrightarrow C^\infty(M\times I)$$ so that $C^\infty(M\times I)$ becomes a right $C^\infty(M)$-module and the tensor product $$C^\infty(M\times I)\otimes_{C^\infty(M)}\Gamma(A_M)$$ makes sense.
(iii) $C^\infty(M\times I)\otimes_{C^\infty(M)}\Gamma(A_M)$ is a left $C^\infty(M\times I)$-module with $$f\cdot (g\otimes \alpha):=(fg)\otimes \alpha,$$ where $fg$ stands for the product of $f$ and $g$ in the algebra $C^\infty(M\times I)$.
I'd like to define a lie algebroid structure on $A_M\times I$. Using the above isomorphism I guess the lie bracket on $\Gamma(A_M\times I)$ should be given by $$[1_{\mathbb R}\otimes \alpha, 1_{\mathbb R}\otimes \beta]=1_{\mathbb R}\otimes [\alpha, \beta]_{A_M},$$ where $\alpha, \beta\in \Gamma(A_M)$ and $[\cdot, \cdot]_{A_M}$ is the lie bracket on $\Gamma(A_M)$.
Obs. I'm defining on elements of the form $1\otimes \alpha$ because using the $C^\infty(M\times I)$-module structure given on $(iii)$ we have $f\otimes \alpha=f\cdot (1_{\mathbb R}\otimes \alpha)$ so that it suffices to define the bracket on $1_{\mathbb R}\otimes \alpha$ and we extend by the Leibniz rule.
My questions are:
(a) How is the anchor defined?
The anchor must be given by a $C^\infty(M\times I)$-linear map $$\sharp:C^\infty(M\times I)\otimes_{C^\infty(M)}\Gamma(A_M)\longrightarrow \mathfrak{X}(M\times I)$$ where $\mathfrak{X}(M\times I)$ is the space of vector fields on $M\times I$.
(b) How are the differentials $d_{A_M}$ and $d_{A_M\times I}$ related?
Recall, that for a Lie algebroid $A_M\longrightarrow M$ we have a chain complex of vector spaces associated $(\Omega^\bullet(A_M), d_A)$ where $$\Omega^k(A_M):=\Gamma(\Lambda^k A_M^*)\simeq \textrm{Hom}_{C^\infty(M)}(\Lambda^k \Gamma(A), C^\infty))\simeq \textrm{Alt}^k_{C^\infty(M)}(\Gamma(A), C^\infty(M)),$$ and $$(d_A \varepsilon)(\alpha_0, \ldots, \alpha_k)=\sum_{j=0}^k (-1)^j \mathcal{L}_{\sharp^A \alpha_j}(\varepsilon(\alpha_0, \ldots, \widehat{\alpha_j}, \ldots, \alpha_k))+\sum_{i<j} (-1)^{i+j} \varepsilon([\alpha_i, \alpha_j]_A, \alpha_0, \ldots, \widehat{\alpha_i}, \ldots, \widehat{\alpha_j}, \ldots, \alpha_k).$$
Thanks.
