$\wedge^\bullet B^*$; ${\mathrm{d}_{{B}}}$ is the usual Lie algebroid cohomology differential; and ${\mathrm{\widetilde{d}}_{{B}}}f={\mathrm{d}_{{B}}}\log f=\tfrac{{\mathrm{d}_{{B}}}f}{f}$, for all $f\in{C^{\infty}(U,{\mathbb{C}}^{\times})}$, where $U$ is an arbitrary open subset of $M$. We denote its hypercohomology by ${\mathrm{\widetilde{H}}^{\bullet}}(B,{\mathbb{C}})$. Note that in Eq. , if we replace ${\mathcal{\widetilde{S}}^{0} }$ by ${\mathcal{S}^{0} }$, the sheaf of smooth complex valued functions on $M$, and $\tilde{d}_B$ by the usual Lie algebroid differential $d_B$, the hypercohomology of the resulting complex of sheaves $$\label{Athens} {\mathcal{S}^{0} }
\stackrel{d_B}{\longrightarrow} {\mathcal{S}^{1} }
\stackrel{d_B}{\longrightarrow} {\mathcal{S}^{2} } \cdots
\stackrel{d_B}{\longrightarrow} {\mathcal{S}^{r} },$$ is isomorphic to the usual Lie algebroid cohomology ${\mathrm{H}^{\bullet}}(B,{\mathbb{C}})$ of the complex Lie algebroid $B$ with trivial coefficients ${\mathbb{C}}$ since each ${\mathcal{S}^{\bullet} }$ is a soft sheaf. The exponential sequence $$0\to{\mathbb{Z}}\to{\mathcal{S}^{} }\to{\mathcal{\widetilde{S}}^{} }\to 0 ,$$ where ${\mathcal{S}^{} }$ (resp. ${\mathcal{\widetilde{S}}^{} }$) stands for the the complex of sheaves (resp. ) and the locally constant sheaf ${\mathbb{Z}}$ is regarded as a complex of sheaves concentrated in degree $0$, induces the long exact sequence $$\cdots\to{\mathrm{H}^{i}}(M,{\mathbb{Z}})\to{\mathrm{H}^{i}}(B,{\mathbb{C}}) 
\to{\mathrm{\widetilde{H}}^{i}}(B,{\mathbb{C}})\to{\mathrm{H}^{i+1}}(M,{\mathbb{Z}})\to\cdots$$

Note that ${\mathrm{\widetilde{H}}^{\bullet}}(B,{\mathbb{C}})$ can be computed as the total cohomology of the Čech double complex $$\label{Diagram:CechLsNonzero} 
\xymatrix{ 
\cdots & \cdots & \cdots & \\ 
{\check{C}^{2}({\mathcal{U}};{{\mathcal{\widetilde{S}}^{0} }})} \ar[r]^{{\mathrm{\widetilde{d}}_{{B}}}} \ar[u]^{{\delta}} &
{\check{C}^{2}({\mathcal{U}};{{\mathcal{S}^{1} }})} \ar[r]^{{\mathrm{d}_{{B}}}} \ar[u]^{{\delta}} &
{\check{C}^{2}({\mathcal{U}};{{\mathcal{S}^{2} }})} \ar[r]^{{\mathrm{d}_{{B}}}} \ar[u]^{{\delta}} & \cdots \\ 
{\check{C}^{1}({\mathcal{U}};{{\mathcal{\widetilde{S}}^{0} }})} 