Powers of ideals and the cohomology of stalks and fibers of morphisms. (English) Zbl 1270.13008

Chardin, Marc

Algebra Number Theory 7, No. 1, 1-18 (2013).

Let \(R\) be a Noetherian commutative ring with identity and \(A\) be a finitely generated \(R\)-algebra which is positively \(\mathbb{Z}\)-graded. Let \(I\) be a graded ideal of \(A\) and \(M\) a nonzero finitely generated graded \(A\)-module. Then the local cohomology modules \(H_{A_+}^i(M)\) are graded \(A\)-modules. So, we can have the following definitions: \[ a^i(M):=\max \{l|H_{A_+}^i(M)_{l}\neq 0\} \] and \[ \mathrm{reg}(M):=\max \{a^i(M)+i|i\in \mathbb{Z}\}. \] In the case \(A\) is an standard polynomial ring over a field, Kodiyalam and Cutkosky, Herzog and Trung have proved that \(\mathrm{reg}(I^t)\) is eventually a linear function in \(t\); (see, V. Kodiyalam [Proc. Am. Math. Soc. 128, No. 2, 407–411 (2000; Zbl 0929.13004)] and S. D. Cutkosky, J. Herzog and N. V. Trung [Compos. Math. 118, No. 3, 243–261 (1999; Zbl 0974.13015)]).
Later, Trung and Wang generalized this result to standard graded \(R\)-algebras; (see, N. V. Trung and H.-J. Wang [J. Pure Appl. Algebra 201, No. 1–3, 42–48 (2005; Zbl 1100.13024)]).
Here, the author proves the following improvement of this result:
Let \(d:=\min\{l|\text{there exists} \;p, \;(I_{\leq l})I^pM=I^{p+1}M\}.\) Then \[ \lim_{t\rightarrow \infty} (a^i(I^tM)+i-td)\in \mathbb{Z}\cup \{-\infty\}. \] exists for any \(i\), and is at least equal to the initial degree of \(M\) for some \(i\). As applications of this result, the author gives a positive answer to a conjecture posed by Hà and also he extends a result of Eisenbud and Harris.

Reviewer: Kamran Divaani-Aazar (Tehran)

MSC:

13D02	Syzygies, resolutions, complexes and commutative rings
13A30	Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13D45	Local cohomology and commutative rings
14A15	Schemes and morphisms

Keywords:

cohomology; stalks; Rees algebras; fibers of morphisms; power of ideals; Castelnuovo-Mumford regularity

Citations:

Zbl 0929.13004; Zbl 0974.13015; Zbl 1100.13024