Deformation cohomology for cyclic groups acting on polynomial rings

We examine the Hochschild cohomology governing graded deformations for finite cyclic groups acting on polynomial rings. We classify the infinitesimal graded deformations of the skew group algebra $S⋊G$ for a cyclic group $G$ acting on a polynomial ring $S$. This gives all graded deformations of the first order. We are particularly interested in the case when the characteristic of the underlying field divides the order of the acting group, which complicates the determination of cohomology.