  This article is the last one about the isomorphism between Lubin-Tate and
Drinfeld towers. We prove the existence of an isomorphism between the compactly
supported etale cohomology of the Lubin-Tate and Drinfeld towers, and more
generally their equivariant cohomology complex. We also prove the existence of
a geometric local Jacquet-Langlands correspondence between some equivariant
rigid etale sheaves on Gross-Hopkins period space $\mathbb{P}^{n-1}$ and
Drinfeld one $\Omega$.
