We prove the nondegeneracy of the positive solutions for the following fractional Choquard equation: \((-\Delta)^{s}u=(\mid{x}\mid^{-\alpha}*\mid{u}\mid^{2_{\alpha,s}^*})\mid{u}\mid^{2_{\alpha,s}^*-1},\;x\in\mathbb{R}^N,\) where s ∈ (0, 1), N > 2s, 0 < α < N, * stands for the convolution, \({2_{\alpha,s}^*}=\frac{2N-\alpha}{N-2s}\) is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality, and 2*α,s ⩾ 2. The proof is based on the stereographic projection and the Funk-Hecke formula of the spherical harmonic functions which are essential to deal with the doubly nonlocal terms. As applications, we first establish a remainder term result for a nonlocal Sobolev-type inequality. Besides, we also obtain an existence result of a perturbed equation under some proper assumptions.
