## 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).

### MSC:

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

### Keywords:

regularity; coregularity; local cohomology
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### References:

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