ag.algebraic geometry - In what degrees does Ext(S/(f),S) vanish?

Let $S=k[x_0,...,x_n]$ be the polynomial ring over a field $k$ and $fin S$ non-zero and homogeneous. Is it true that $Ext^m(S/(f),S)$ is zero?

This would help me to show that $Ext^m(S/fI,S)cong Ext^m(S/I,S)(deg f)$ for $mgeq 2$ and a homogeneous ideal $I$ of codim $geq 2$. I tried the following approach:
Applying the long exact sequence of $Ext$ to the exact sequence of graded $S$-modules
$$0to S/Ixrightarrow{cdot f} S/fI(deg f)xrightarrow{tau}S/(f)(deg f)to 0,$$
where $tau$ is the canonical morphism $s+fImapsto s+(f)$,
brings
$$ldotsto Ext^m(S/(f)(deg f),S)to Ext^m(S/fI(deg f),S)to Ext^m(S/I,S)to$$
$$Ext^{m+1}(S/(f)(deg f),S)to Ext^{m+1}(S/fI(deg f),S)to Ext^{m+1}(S/I,S)toldots$$
and two zeros on the left would suffice.