Bounds for regularity and coregularity of graded modules. (English) Zbl 1134.13014

For a finite module \(M\) over a standard graded ring \(R\) with local base ring \(R_0\), to the local cohomology modules of \(M\) with support in the irrelevant ideal of \(R\) there are associated numerical invariants \(\mathrm{reg}^k(M)\) for natural numbers \(k\) (for \(k=0\) this gives the Castelnuovo-Mumford regularity of \(M\)). Brodmann, Matteotti and Minh established upper bounds for \(\mathrm{reg}^k(M)\) in case \(R_0\) is Artinian. In this paper related bounds are proven for the case \(\dim(R_0)=1\). Similar remarks apply to lower bounds of \(\mathrm{coreg}^k(M)\). Finally, sharper bounds are proven in situations where \(R_0\) has nice properties (e. g. if it is regular).


13D45 Local cohomology and commutative rings
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
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