Length of local cohomology of powers of ideals

TL;DR

This paper investigates the asymptotic behavior of the length of local cohomology modules of powers of ideals in polynomial rings, establishing finiteness, rationality of limits, and positivity under certain conditions.

Contribution

It proves the finiteness and rationality of the asymptotic limits of local cohomology lengths for powers of ideals, using Gr"obner deformation and Presburger arithmetic.

Findings

Limit superior of normalized local cohomology lengths is finite in many cases.

The actual limit exists and is rational for certain monomial ideals.

The limit inferior is positive when the quotient ring has 'nice' singularities.

Abstract

Let $R$ be a polynomial ring over a field $k$ with irrelevant ideal $\frak m$ and dimension $d$. Let $I$ be a homogeneous ideal in $R$. We study the asymptotic behavior of the length of the modules $H^{i}_{\frak m}(R/I^n)$ for $n\gg 0$. We show that for a fixed number $\alpha \in \mathbb Z$, $\limsup_{n\rightarrow \infty}\frac{\lambda(H^{i}_{\frak m}(R/I^n)_{\geq -\alpha n})}{n^d}<\infty.$ Combining this with recent strong vanishing results gives that $\limsup_{n\rightarrow \infty}\frac{\lambda(H^{i}_{\frak m}{R/I^n})}{n^d}<\infty$ in many situations. 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 Gr\"obner deformation and Presburger arithmetic. Finally, we utilize more traditional commutative algebra techniques to show that $\liminf_{n\rightarrow \infty}\frac{\lambda(H^{i}_{\frak m}(R/I^n))}{n^d}>0$ when $R/I$ has "nice" singularities in both zero and positive characteristics.