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Asymptotic behavior of the length of local cohomology. (English) Zbl 1095.13015

Let \(R=k[X_1,\dots,X_d]\) be the polynomial ring over a field \(K\) of characteristic 0, \(m\) the graded maximal ideal and \(I\) a proper homogeneous ideal. The authors investigate the asymptotic growth of \(\lambda(\text{ Ext}^d_R(R/I^n),R)\) as a function of \(n\), where \(\lambda(M)\) denotes the length of an \(R\)-module \(M\). When \(R\) is a local Gorenstein ring and \(I\) is an \(m\)-primary ideal, then this is easily seen to be equal to \(\lambda(R/I^n)\) and hence a polynomial in \(n\). It is also previously known that this extends to \(m\)-primary ideals in local Cohen-Macaulay rings \(R\). The authors consider homogeneous ideals in polynomial rings which are not \(m\)-primary and show that this limit exists asymptotically. The authors show that by local duality
\[ \lambda(\text{ Ext}^d_R(R/I^n,R(-d)))=\lambda(H_m^0(R/I^n)) \]
and the limit \[ \lim_{n\rightarrow\infty}\frac{\lambda(H_m^0(R/I^n))}{n^d}=\lim_{n\rightarrow\infty}\frac{\lambda(\text{ Ext}^d_R(R/I^n,R(-d)))}{n^d}. \]
The authors also give an example of a regular ring \(S\) of dimension 4 (which is essentially of finite type over the field of complex numbers) and an ideal \(J\) of \(S\) such that the limit \[ \lim_{n\rightarrow \infty}\frac{\lambda(\text{ Ext}^d_S(S/J^n,S))}{n^d} \] is an irrational number. In particular \(\lambda(\text{ Ext}^d_S(S/J^n,S))\) is not a polynomial nor a quasi-polynomial for large \(n\). The authors’ work uses ideas and techniques from [D. Kirby, Math. Proc. Cambridge Philos. Soc. (3)105, 441–446 (1989; Zbl 0689.13004), V. Kodiyalam, Proc. Am. Math. Soc. (3) 118, 757–764 (1993; Zbl 0780.13007), E. Theodorescu, Math. Proc. Camb. Philos. Soc. 132, 75–88 (2002; Zbl 1054.13007)] on polynomial growth of the length of extension functors.

MSC:

13D45 Local cohomology and commutative rings
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
14B15 Local cohomology and algebraic geometry
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