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Let \( R \) be a complete discrete valuation ring with fraction field of characteristic 0 and algebraically closed residue field of characteristic \( p > 0 \). Let \( X_{R} \to {\text{Spec}}(R) \) be a smooth projective morphism of relative dimension 1. We prove that, given a line bundle \( {\mathcal{L}}_{{\mathcal{R}}} \) the moduli space of Gieseker stable torsion-free sheaves of rank \( r \ge 2 \) over \( X_{R} \), with determinant \( {\mathcal{L}}_{{\mathcal{R}}} \), is smooth over \( R \).

Smoothness of Moduli Space of Stable Torsion-free Sheaves with Fixed Determinant in Mixed Characteristic