Let $S=k[x\_1,...,x\_n]$ be a polynomial ring over field $k$ with maximal ideal $m=(x\_1,...,x\_n)$. I wanna make a $3$-dimensional $S$-module $M$ such that $H^0\_m(M)=H^1\_m(M)=0$ and $H^2\_m(M)\neq 0$ be finitely generated (or in general case: $H^i\_m(M)$ be finite for all $i=0,1,2$ ). Is there a simple way to create similar examples(for any dimension)?


**Background:**


$H^i\_m(M)$ means $i$'th local cohomology module of $M$.



example of Local cohomology
Take $M$ to be the second syzygy of $k$ over $S=k[x\_1,x\_2,x\_3]$. Then a graded version of local duality tells us that $H^2\_m(M)$ is dual to $Ext^1(M,R)= Ext^3(k,R)$, the last one is $k$ either by direct computation or duality again. 


One can easily generalize this, the $j$ syzygy of $k$ in $n$ variables will have local cohomologies vanish up to degree $j-1$ and finitely generated up to $n-1$. 


