For associated graded rings of powers of the maximal ideal, local cohomology vanishes in high degrees exactly when the quotient ring has dimension at least two and its projective scheme is connected.
Hartshorne, Cohomological dimension of algebraic varieties, Ann
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Cohomology Vanishing theorems over some rings containing nilpotents
For associated graded rings of powers of the maximal ideal, local cohomology vanishes in high degrees exactly when the quotient ring has dimension at least two and its projective scheme is connected.