For polynomial rings, every nonzero local cohomology or related Lyubeznik module has nonzero pieces in all degrees at or below -m, and the top nonvanishing module has infinite-dimensional pieces.
Schenzel, On the use of local cohomology in algebra and geometry , in, Six lectures on commutative algebra ( B ellaterra, 1996) , Progr
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Graded components of local cohomology modules over polynomial rings
For polynomial rings, every nonzero local cohomology or related Lyubeznik module has nonzero pieces in all degrees at or below -m, and the top nonvanishing module has infinite-dimensional pieces.