  We obtain sharp interior Schauder estimates for solutions to nonlocal Poisson
problems driven by fractional powers of nondivergence form elliptic operators
$(-a^{ij}(x) \partial_{ij})^s$, for $0<s<1$, in bounded domains under minimal
regularity assumptions on the coefficients $a^{ij}(x)$. Solutions to the
fractional problem are characterized by a local degenerate/singular extension
problem. We introduce a novel notion of viscosity solutions for the extension
problem and implement Caffarelli's perturbation methodology in the
corresponding degenerate/singular Monge--Amp\`ere geometry to prove Schauder
estimates in the extension. This in turn implies interior Schauder estimates
for solutions to the fractional nonlocal equation. Furthermore, we prove a new
Hopf lemma, the interior Harnack inequality and H\"older regularity in the
Monge--Amp\`ere geometry for viscosity solutions to the extension problem.
