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Length of local cohomology of powers of ideals. (English) Zbl 1409.13036
The main aim of the paper is to investigate the asymptotic behavior of the length of local cohomology modules of powers of an ideal \(I\). More precisely, if \(I\) is an ideal in a \(d\)-dimensional local noetherian ring \((R, \mathfrak{m}, k)\) or a polynomial ring over a field \(k\) and maximal homogeneous ideal \(\mathfrak{m}\), the authors study situations when the length \(\lambda(H_{\mathfrak{m}}^i (R/I^n))\) is finite for \(n \gg 0\) and consider the existence of the limit \[ \lim_{n \to \infty}\frac{\lambda(H_{\mathfrak{m}}^i (R/I^n))}{n^d}\tag{*} \] for \(i > 0\). The paper is motivated by results of S. D. Cutkosky [Adv. Math. 264, 55–113 (2014; Zbl 1350.13032)] who proved that this limit exists if \(i=0\) and \(R\) is analytically unramified. The most concrete results of the paper are obtained for monomial ideals in a polynomial ring. In this case, it is proved that \(\lambda(H_{\mathfrak{m}}^i (R/I^n))< \infty \) for \(n \gg 0\) and the sequence \(\{\lambda(H_{\mathfrak{m}}^i (R/I^n))\}_{n \geq 0}\) agrees with a quasi-polynomial of degree \(d\) for \(n \gg 0\). Moreover, the generating function of this sequence is rational. Even though the limit (*) might not exist, a better behavior is obtained by replacing the filtration of powers of \(I\) with the filtration of integral closures \(\{\overline{I^n}\}_{n \geq 1}\), in which case \(\lim_{n \to \infty}\frac{\lambda(H_{\mathfrak{m}}^i (R/\overline{I^n}))}{n^d}\) exists and is a rational number. In the more general case when \(I\) is a homogeneous ideal, the authors prove that \[ \limsup_{n \to \infty}\frac{\lambda(H_{\mathfrak{m}}^i (R/I^n)_{\geq -\alpha n})}{n^d} < \infty \] for every \(\alpha \in \mathbb{Z}\) and leave open the question of whether or not \(\limsup_{n \to \infty}\frac{\lambda(H_{\mathfrak{m}}^i (R/I^n))}{n^d}\) is always finite. The authors also raise questions about situations when the limit (*) is rational and about finding interpretations of it in terms of volumes of geometric bodies associated with the ideal \(I\). The paper also discusses several classes of ideals for which \(\liminf_{n \to \infty}\frac{\lambda(H_{\mathfrak{m}}^i (R/I^n))}{n^d}\) is positive.

13D45 Local cohomology and commutative rings
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
14B05 Singularities in algebraic geometry
05E40 Combinatorial aspects of commutative algebra
Macaulay2; Normaliz
Full Text: DOI arXiv
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