  The main purpose of this paper is to investigate the curvature behavior of
four dimensional shrinking gradient Ricci solitons. For such soliton $M$ with
bounded scalar curvature $S$, it is shown that the curvature operator
$\mathrm{Rm}$ of $M$ satisfies the estimate $|\mathrm{Rm}|\le c\,S$ for some
constant $c$. Moreover, the curvature operator $\mathrm{Rm}$ is asymptotically
nonnegative at infinity and admits a lower bound $\mathrm{Rm}\geq -c\,\left(\ln
r\right)^{-1/4},$ where $r$ is the distance function to a fixed point in $M$.
As application, we prove that if the scalar curvature converges to zero at
infinity, then the manifold must be asymptotically conical.
  As a separate issue, a diameter upper bound for compact shrinking gradient
Ricci solitons of arbitrary dimension is derived in terms of the injectivity
radius.
