The main tool in Ngô's proof of the Langlands-Shelstad fundamental lemma, is a theorem on the support of the relative cohomology of the elliptic part of the Hitchin fibration. For $\mathrm{GL}(n)$ and a divisor of degree $> 2g-2$, the theorem says that the relative cohomology is completely determined by its restriction to any dense open subset of the base of the Hitchin fibration. In this talk, we will explain our extension of that theorem to the whole Hitchin fibration, including the global nilpotent cone (for $\mathrm{GL}(n)$ and a divisor of degree $> 2g-2$). (based on a joint work with G. Laumon).