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.
Lyubeznik, Finiteness Properties of Local Cohomology Modules (an Application of D-modules to Commutative Algebra), Inv
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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.