  We prove that a gradient shrinking Ricci soliton with fourth order
divergence-free Riemannian tensor is rigid. For the $4$-dimensional case, we
show that any gradient shrinking Ricci soliton with fourth order
divergence-free Riemannian tensor is either Einstein, or a finite quotient of
the Gaussian shrinking soliton $\mathbb{R}^4$, $\mathbb{R}^2\times\mathbb{S}^2$
or the round cylinder $\mathbb{R}\times\mathbb{S}^3$. Under the condition of
fourth order divergence-free Weyl tensor, we have the same results.
