We report $^{195}$Pt-NMR and $^{75}$As-NQR measurements for the locally non-centrosymmetric superconductor SrPtAs where the As-Pt layer breaks inversion symmetry while globally the compound is centrosymmetric. The nuclear spin lattice relaxation rate $1/T_1$ shows a well-defined coherence peak below $T_c$ and decreases exponentially at low temperatures. The spin susceptibility measured by the Knight shift also decreases below $T_c$ down to $T<T_c/6$. These data together with the penetration depth obtained from the NMR spectra can be consistently explained by assuming a spin-singlet superconducting state with a full gap. Our results suggest that the spin-orbit coupling due to the local inversion-breaking is not large enough to bring about an exotic superconducting state, or the inter-layer hopping interaction is larger than the spin-orbit coupling.
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