Length of local cohomology of powers of ideals

Abstract:

Let $R$ be a polynomial ring over a field $k$ with irrelevant ideal $\mathfrak {m}$ and dimension $d$. Let $I$ be a homogeneous ideal in $R$. We study the asymptotic behavior of the length of the modules $\operatorname {H}^{i}_{\mathfrak {m}}(R/I^n)$ for $n\gg 0$. We show that for a fixed number $\alpha \in \mathbb {Z}$, $\limsup _{n\rightarrow \infty }\frac {\lambda (\operatorname {H}^{i}_{\mathfrak {m}}(R/I^n)_{\geqslant -\alpha n}) }{n^d}<\infty .$ It follows that $\limsup _{n\rightarrow \infty }\frac {\lambda (\operatorname {H}^{i}_{\mathfrak {m}}(R/I^n)) }{n^d}<\infty$ when $X = \operatorname {Proj} R/I$ is locally a complete intersection (lci) and $i\leqslant \dim X$. We also establish that the actual limit exists and is rational for certain classes of monomial ideals $I$ such that the lengths of local cohomology of $I^n$ are eventually finite. Our proofs use Gröbner deformation and Presburger arithmetic. Finally, we utilize more traditional commutative algebra techniques to show that $\liminf _{n\rightarrow \infty }\frac {\lambda (\operatorname {H}^{i}_{\mathfrak {m}}(R/I^n)}{n^d}>0$ under certain conditions when $R/I$ is either $F$-pure or lci.