  Let $G$ be a connected reductive algebraic group over an algebraically closed
field $\Bbbk$ of characteristic $p \ge 0$, and let $\mathcal{N}$ be its
nilpotent cone. Under mild hypotheses, we construct for each nilpotent
$G$-orbit $C$ and each indecomposable tilting vector bundle $T$ on $C$ a
certain complex $S(C,T)$ of $G \times \mathbb{G}_m$-equivariant coherent
sheaves on $\mathcal{N}$. We prove that these objects are (up to shift)
precisely the indecomposable objects in the coheart of a certain
co-$t$-structure.
  We then show that if $p$ is larger than the Coxeter number, then the
hypercohomology $H^\bullet(S(C,T))$ is identified with the cohomology of a
tilting module for $G$. This confirms a conjecture of Humphreys on the support
of the cohomology of tilting modules.
