TITLE: example of Local cohomology
QUESTION [8 upvotes]: 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 $M$.

REPLY [11 votes]: 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 $n-1$.