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 $H_{\mathfrak{m}}^i(R/I^n)$ for $n\gg 0$. We show that for a fixed number $\alpha \in \mathbb{Z}$, ${\mathrm{limsup}}_{n\to \infty }\frac{\lambda (H_{\mathfrak{m}}^i(R/I^n{)}_{\ge -\alpha n})}{n^d}<\infty .$ Combining this with recent strong vanishing results gives that ${\mathrm{limsup}}_{n\to \infty }\frac{\lambda (H_{\mathfrak{m}}^{i}R/I^n)}{n^d}<\infty$ 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 Gröbner deformation and Presburger arithmetic. Finally, we utilize more traditional commutative algebra techniques to show that ${\mathrm{liminf}}_{n\to \infty }\frac{\lambda (H_{\mathfrak{m}}^i(R/I^n))}{n^d}>0$ when $R/I$ has "nice" singularities in both zero and positive characteristics.

We were able to extend one previous result to all l.c.i varieties, thanks to a vanishing result explained to us by Robert Lazarsfeld. See Corollary 5.8. To appear in Transactions of AMS

References in corpus (4)

• Irrational Hilbert-Kunz multiplicities
• On generalized Hilbert-Kunz function and multiplicity
• Generalized Hilbert-Kunz function in graded dimension two
• Local Okounkov bodies and limits in prime characteristic

Cited by in corpus (1)

• Local Cohomology and Degree Complexes of Monomial Ideals