## Local cohomology over homogeneous rings with one-dimensional local base ring.(English)Zbl 1041.13012

Let $$R$$, the direct product of rings $$R_n$$, $$n>0$$ or $$n=0$$, be a homogeneous Noetherian ring with local base ring $$R_0$$, $$M$$ a finitely generated graded $$R$$-module. The main result in this paper shows that certain quotients and submodules of the local cohomology module are Artinian, provided the base ring $$R_0$$ is of dimension $$<1$$ or $$=1$$. Using the results above, the authors give some properties of a local cohomology modules. In addition, some examples related to previous results are presented.

### MSC:

 13D45 Local cohomology and commutative rings 13E10 Commutative Artinian rings and modules, finite-dimensional algebras 13A02 Graded rings
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