Hema Srinivasan

R is a polynomial ring over a field $k$ in $d$ variables and $m$ is the irrelevant maximal ideal. For any proper homogenous ideal $I$ of $R$, we will discuss the growth of the length of local cohomology modules $H^0_m(R/I^n)$ for large $n$ and show that the ratio ${\lambda(H^0_m(R/I^n))}\over {n^d}$ has a limit as $n$ tends to infinity and that this limit is not always rational. Thus, $\lambda(H^0_m(R/I^n))$ will not in general be a polynomial in $n$.