  We study the blow-down map in cohomology in the context of real projective
blowups of Lie algebroids. Using the blow-down map in cohomology we compute the
Lie algebroid cohomology of the blowup of transversals of arbitrary
codimension, generalising the Mazzeo-Melrose theorem on b-cohomology. To prove
the result we develop a Gysin sequence for Lie algebroids. As another example
we use the developed tools to compute the Lie algebroid cohomology of the
action Lie algebroid $\mathfrak{so}(3)\ltimes \mathbb{R}^3$, a result known in
Poisson geometry literature. Moreover, we use similar techniques to compute the
de Rham cohomology of real projective blowups.
