The Schwarzschild solution is meant to be a solution of the vacuum Einstein equations. That is

$R_{\mu\nu}=0$

So, the Ricci tensor must be null.

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

$\frac{12r_s^2}{r^6}$

which is obviously non zero.

What am I making wrong?

The Schwarzschild solution is meant to be a solution of the vacuum Einstein equations. That is

$$R_{\mu\nu}=0.$$

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

$$\frac{12r_s^2}{r^6},$$

which is obviously non zero.

What am I making wrong?