Let $R=k[x_1,\dots,x_n]$ be a polynomial ring over some field $k$, let $M$ be an $R$-module and $M^{\text{sat}}$ the saturation of $M$. Denote by $m$ the maximal ideal of $R$. Is it true that: $$ H^i_{m}(M) \simeq H^i_{m}(M^{\text{sat}}) \text{ for }i\geq 1\ ?$$
Here every $R$-module is graded, and by $M^{\text{sat}}$ I mean the global sections of the associated sheaf in the projective space $\text{Proj}(S)$.
With your comment, this is in general false. As an example, take $R=k[x_1,\ldots,x_4]$ and $S=k[u^4,u^3v,uv^3,v^4]$ which is a quotient of $R$ by sending $x_1\mapsto u^4, x_2\mapsto u^3v$ etc. Then, the saturation of $S$ is $T=k[u^4,u^3v,u^2v^2, uv^3, v^4]$. $S$ has depth one and $T$ has depth 2. Thus, the first local cohomology of $S$ is non-zero, but that of $T$ is zero.