The Schwarzschild solution is meant to be a solution of the vacuum Einstein equations. That is
So, the Ricci tensor must be null for $r>0$.
Now, if the scalar curvature is nothing but the Ricci tensor contracted, and the Ricci tensor is null, the cuvature should be zero.
Nonetheless, I have been told that the curvature of the Schwarzschild solution (in the usual coordinates) is
which is obviously non zero.
What am I making wrong?